Asymptotic robustness of standard errors in multilevel structural equation models
|
|
- Daniella Barber
- 6 years ago
- Views:
Transcription
1 ournal of Multivariate Analysis 97 (2006) Asymptotic robustness of standard errors in multilevel structural equation models Ke-Hai Yuan a,, Peter M. Bentler b a University of Notre Dame, IN, USA b University of California, Los Angeles, CA, USA Received 20 May 2004 Available online 28 uly 2005 Abstract Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical methodology was developed to analyze such data. Most of the procedures for analyzing multilevel data are derived from maximum likelihood based on the normal distribution assumption. Standard errors for parameter estimates in these procedures are obtained from the corresponding information matrix. Because practical data typically contain heterogeneous marginal skewnesses and kurtoses, this paper studies how nonnormally distributed data affect the standard errors of parameter estimates in a two-level structural equation model. Specifically, we study how skewness and kurtosis in one level affect standard errors of parameter estimates within its level and outside its level. We also show that, parallel to asymptotic robustness theory in conventional factor analysis, conditions exist for asymptotic robustness of standard errors in a multilevel factor analysis model Elsevier Inc. All rights reserved. AMS 1991 subect classification: primary 62H05; secondary 62H25; 62P25 Keywords: Nonnormal data; Skewness; Kurtosis; Standard errors; Asymptotic robustness This research was supported by NSF Grant DMS and Grants DA01070 and DA00017 from the National Institute on Drug Abuse. Corresponding author. Fax: address: kyuan@nd.edu (K.-H. Yuan) X/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi: /.mva
2 1122 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Introduction Data in social and behavioral sciences often exhibit a hierarchical structure. For example, households are nested within neighborhoods, neighborhoods are nested within cities, and cities are further nested within countries; students are nested within classes, classes are nested within schools, and schools are further nested within school districts. Because observations within a cluster are generally correlated, statistical methods for such data have been developed to explicitly account for these correlations in order to get accurate results. When predictors are error free, using the hierarchical linear model (HLM) leads to correct analysis of the hierarchical data [8,10,11,13,17,27,31]. When predictors contain measurement errors, HLM is not appropriate because it does not account for these errors. The multilevel structural equation model (SEM) has to be used to obtain consistent parameter estimates [2,6,14 16,20,23,24]. Actually, when errors in predictors are modeled explicitly, a HLM model automatically becomes a multilevel SEM model [9,18,21,25]. All the above literature deals with model inference through methods that require the multivariate normality assumption for hierarchical data. Real data typically have larger skewness and kurtosis than those of a normal distribution [29]. For example, Micceri [22] reported that among 440 large sample achievement and psychometric measures taken from ournal articles, research proects, and tests, all were significantly nonnormally distributed. In reality, the normality assumption used in modeling should be considered as only a working assumption. When extra kurtoses exist in the data, the actual standard errors (SEs) of parameter estimates will be larger than those based on the normality assumption; the likelihood ratio statistics for testing variance components or the overall model structure also tend to be more significant than they actually are. Consequently, inference based on the normality assumption may be no longer valid when modeling practical data. A few studies on the multilevel model with violations of the normality assumption also exist [5,26,34,35]. Within the context of HLM, Cheong et al. [5] studied the effect of nonnormal data on SEs of regression parameter estimates using simulation. They found that the SEs based on a sandwich-type covariance matrix are quite robust even when the model is misspecified. For two-level HLM and SEM, [34] discussed the strength of several possible sandwich-type covariance matrices. Neither of these papers analytically studied the effect of skewness and kurtosis on SEs. Yuan and Bentler [34 37] studied the behavior of the normal theory based likelihood ratio statistic with distribution violations; they also proposed several alternative statistics for overall model evaluation. However, the effect of nonnormal data on SEs of the parameter estimates in the context of multilevel models has not been well studied. For example, there are level-1 and level-2 random components in a two-level SEM model. It is not clear how the kurtosis in level-1 affects the SEs of level-1 and level-2 parameter estimates. How the skewnesses of level-1 and level-2 components affect the SEs of parameter estimates at different levels is not clear either. When data are elliptically distributed (see [7,12]), SEs of factor loading estimates within the conventional SEM context can be obtained by adusting the corresponding normal distribution based SEs [4,28,30] using the relative kurtosis. It is of interest to see whether this result can be extended to a multilevel confirmatory factor model. In the context of conventional factor analysis, there also exist results of asymptotic robustness on SEs. For
3 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) example, Anderson andamemiya [1] found that the SEs of factor loading estimates using the normal distribution assumption can be asymptotically valid for nonnormal data. A further result in this direction was obtained in [32] by considering skewed data whose fourth-order moments are identical to those of an elliptical distribution. It is of interest to see whether parallel results hold in a multilevel factor analysis model. Our purpose is to systematically study the effect of nonnormal data on SEs of parameter estimates in multilevel analysis. Although aiming to study SEs for multilevel SEM, we only explicitly consider the two-level model; generalization from a two-level model to a higher level model can be performed in a parallel way. Our study for SEs will be based on a quite general model formulated in [16]. Within this model, we will study the effect of skewness and kurtosis on SEs for estimates of parameters that are shared by the mean, the betweenand within-level covariance structures. We also study the effect of skewness and kurtosis on SEs for estimates of parameters that are unique to the mean, the between-level covariance matrix, and the within-level covariance matrix. In Section 2 we characterize the effect of skewness (third-order moments) and kurtosis (fourth-order moments) on SEs of parameter estimates at either level. By focusing on a class of nonnormal distributions in Section 3, we study the asymptotic robustness of SEs for parameter estimates in a two-level factor analysis model. We will state and discuss the results in Sections 2 and 3. Proofs of these results will be provided in the appendix. 2. The effect of skewness and Kurtosis on standard errors Any multilevel data must have a hierarchical structure. In addition, real data may contain additional variables that are only observed at the highest level. For example, while scores of student achievement are nested within schools, variables of school resources such as the number of computer labs or the number of elective courses are only measured at the school level. The two-level SEM model studied in [34] does not contain variables that are only observed at level-2. Liang and Bentler [16] formulated a more general model that contains both level-1 and level-2 observed variables. This model can be expressed as ( ) ( ) ( ) z z 0 = +, i = 1, 2,...,n y i v u ; = 1, 2,...,, (1) i where the observable variables are on the left side and the hypothesized generating variables are on the right side. There are two types of observable variables: The q-dimensional vector z varies at level-2 (between) only, and the p-dimensional vector y i varies at both level- 1 (within) and level-2. The vector u i represents the level-1 components of y i, while v represents the level-2 components; both are p-dimensional. With model (1) it is typically assumed that b = (z, v ) and u i are statistically independent. Following Liang and Bentler s [16] setup, we assume E(u i ) = 0. Denote ( ) ( ) ( ) ( ) z µz z Σzz Σ µ = E =, Σ v µ b = Cov = zv, Σ v v Σ vz Σ w = Cov(u i ). vv
4 1124 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Let ȳ. = n i=1 y i /n. Then the covariance matrix of t = (z, ȳ. ) is ( Σzz Σ zv Σ = Cov(t ) = Σ vz Σ vv + n 1 Σ w ). With interesting structures µ(θ), Σ b (θ) and Σ w (θ), 2 times the normal distribution based log likelihood function of (1) is (see [16,35]) where N = n 1 + +n, with S y = 1 N l(θ) = (N ){log Σ w (θ) +tr[σ 1 w (θ)s y]} + {log Σ (θ) +tr[σ 1 (θ)r (θ)]}, (2) n S =1 =1 S = 1 n n (y i ȳ. )(y i ȳ. ) i=1 and R (θ) =[t µ(θ)][t µ(θ)]. Let ˆθ be the parameter value that minimizes (2). Liang and Bentler [16] gave an EMalgorithm for obtaining ˆθ. Yuan and Bentler [35] provided alternative statistics for overall model evaluation. However, with nonnormal data, formulae for obtaining a consistent covariance matrix of parameter estimates for this model have not been provided in the literature. In this section, we will first present the asymptotic distribution of ˆθ and a consistent estimator of its covariance matrix before relating the covariance matrix to skewness and kurtosis. We will implicitly assume the standard regularity conditions as in [38], which ensures the consistency and asymptotically normality of ˆθ. The consistency and asymptotic normality hold in general only when. Special results can be obtained when the average level-1 sample size n = N/ also approaches infinity, which will be explicitly stated. For a p p symmetric matrix A, let vec(a) be the vectors of stacking the columns of A and vech(a) be the subvector of vec(a) that only contains the elements on and below the diagonal of A. Then there is a duplication matrix D p such that vec(a) = D p vech(a) [19]. Notice that Σ is a matrix of dimension p + q. Denote s y = vech(s y ), s = vech(s ), r (θ) = vech[r (θ)], σ w = vech(σ w ), σ b = vech(σ b ), σ = vech(σ ), W w = 2 1 D p (Σ 1 w Σ 1 w )D p, W = 2 1 D (p+q) (Σ 1 Σ 1 )D (p+q).
5 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) We will use a dot on top of a function to denote the derivative. For example, µ(θ) = dµ(θ)/dθ. The argument of a function will be omitted when it is evaluated at the population value. The notation L implies convergence in distribution. Let B (1) A (1) = 1 = ( n 1) σ w W w σ w σ W σ + 1 =1 σ w W wvar(n s )W w σ w + 1 =1 =1 µ, (3) σ W Var(r )W σ + 1 =1 σ w W wcov(n s, r )W σ + 1 =1 =1 =1 σ w W wcov(n s, t )Σ 1 µ + 1 σ W Cov(r, t )Σ 1 µ + 1 =1 =1 σ W Cov(r,n s )W w σ w =1 =1 Cov(t,n s )W w σ w µ Cov(t, r )W σ (4) and Ω (1) = A (1) 1 (1) B A(1) 1. (5) The following result characterizes the asymptotic distribution of ˆθ. Theorem 1. When Ω (1) = lim Ω (1) exists, we have (ˆθ θ 0 ) L N(0, Ω (1) ). (6) A consistent estimator of Ω (1) is given by ˆΩ (1) =  (1) 1 are obtained when replacing the unknown parameters in A (1) and B (1) Var(r ),Cov(n s, r ),Cov(n s, t ) and Cov(r, t ) in B (1) Var(n s ) =[n s (n 1)ˆσ w ][n s (n 1)ˆσ w ], Var(r ) =[r ( ˆµ) ˆσ ][r ( ˆµ) ˆσ ], Ĉov(n s, r ) =[n s (n 1)ˆσ w ][r ( ˆµ) ˆσ ], Ĉov(n s, t ) =[n s (n 1)ˆσ w ](t ˆµ), Ĉov(r, t ) =[r ( ˆµ) ˆσ ](t ˆµ). ˆB (1)  (1) 1, where  (1) and ˆB (1) by ˆθ and Var(n s ), by, respectively, Note that the consistency of ˆΩ (1) is with respect to while the n s are uniformly bounded. The distribution in Theorem 1 becomes degenerate when n s are unbounded
6 1126 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) such that n. We will discuss this particular case in the following subsections. More elaborate estimators of Ω (1) for a less general two-level SEM were discussed in [34] For parameters that are shared by µ(.), Σ b (.) and Σ w (.) The model parameterization in (2) does not distinguish parameters in different parts of the model. In practice, parameters in µ(.), σ b (.) and σ w (.) may not totally overlap, θ contains parameters that are shared by µ(.), σ b (.) and σ w (.) and those that are not shared by them. Our characterization of SEs in this subsection is only for estimates of parameters that are shared by µ(.), σ b (.), σ w (.), and we will refer them as the common parameters. Note that ( ) ( ) 0q 1 0q p = u u i I i = Cu i. p Let b 0 = b µ 0 and denote Δ b = E[b 0 vech (b 0 b 0 )], Γ b = Var[vech(b 0 b 0 )], Δ u = E[u i vech (u i u i )], Γ u = Var[vech(u i u i )], Δ cu = E{(Cu i )vech (u i u i )}=CΔ u, Γ cu = Cov{vech[(Cu i )(Cu i ) ], vech(u i u i )}=D+ p+q (C C)D pγ u, Δ cuc = E{(Cu i )vech [(Cu i )(Cu i ) ]} = CΔ u D p (C C )D + p+q, Γ cuc = Var{vech[(Cu i )(Cu i ) ]} = D + p+q [(C C)D pγ u D p (C C )]D + p+q, V b = 2D + p+q (Σ b Σ b )D + p+q, V w = 2D + p (Σ w Σ w )D + p, V = 2D + p+q (Σ Σ )D + p+q, V cw = 2D + p+q (CΣ w) (CΣ w )D + p, V cwc = 2D + p+q (CΣ wc ) (CΣ w C )D + p+q. It can be easily verified that W w = Vw 1, W = V 1, and for normal data Γ b, Γ u, Γ cu and Γ cuc reduce to V b, V w, V cw and V cwc, respectively. Relating the covariance matrix in (5) to the third- and fourth-order moment matrix, we have the following result. Theorem 2. The covariance matrix Ω (1) in (5) can be decomposed as Ω (1) = A (1) 1 (1) 1 (1) + A E A(1) 1, (7) where A (1) given in (3) corresponds to the information matrix based on the normal distribution assumption and E (1) = n σ w W w(γ u V w )W w σ w n =1
7 A (1) K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) [ ] σ W (Γ b V b ) + 1 n 3 (Γ cuc V cwc ) W σ =1 (n 1) { σ w W w(γ cu V cw )W σ + σ W (Γ cu V cw )W w σ w } =1 n ) (1 1n { Δ cu W w σ w + σ w W wδ cu Σ 1 µ} =1 { ( ) + 1 Δ b + 1 n 2 Δ cuc W σ =1 ( ) } + σ W Δ b + 1 n 2 Δ cuc Σ 1 µ. (8) Note that when data are normally distributed, E (1) = 0, Ω (1) = lim A (1) 1. The matrix does not depend on the underlying distributions of z and y i. Nonnormal data affect Ω (1) by E (1) through skewnesses Δ b, Δ u and kurtoses Γ b, Γ u. Specifically, the kurtosis Γ u of the within-level components u i affects Ω (1) mainly through the first term in (8). This influence is also accelerated by the average level-1 sample size n. The kurtosis Γ u of u i also appears in the second and third terms in (8), but its effect is downplayed by the level-1 sample sizes n. The kurtosis Γ b of the between-level components b affects Ω (1) by the second term in (8). The skewnesses Δ u and Δ b of the within- and between-level affect Ω (1) through the fourth and fifth terms in (8). It follows from Theorem 2 that the additional variance covariance matrix of the common parameter estimate ˆθ is proportional to the level-1 and level-2 kurtoses as well as level-1 and level-2 skewnesses. The above conclusion is based on and a small n. When n is large, we can use n as an approximation to obtain a more elegant conclusion. Denote by O(a n ) a matrix whose elements are of order O(a n ) (see [3, Chapter 14]). It follows from (3), (7) and (8) that A (1) = O( n), Ω (1) = O( n 1 ). We need to restandardize (6) so that the limiting distribution is not degenerate when n. Let Ω (2) = nω (1), then the asymptotic covariance matrix of N(ˆθ θ 0 ) is Ω (2) = lim Ω (2) and we have the following result. Corollary 1. There exist Ω (2) = (A (1) / n) 1 + (A (1) / n) 1 σ w W w(γ u V w )W w σ w (A (1) / n) 1 + O( n 1 ) (9a) and Ω (2) = ( σ w W w σ w ) 1 + ( σ w W w σ w ) 1 σ w W w(γ u V w )W w σ w ( σ w W w σ w ) 1. (9b)
8 1128 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) The matrix (A (1) / n) 1 is the covariance matrix of N(ˆθ θ 0 ) corresponding to the normal distribution assumption. The second term in (9a) or (9b) is due to nonnormal data. So for large n the larger SEs of the common parameter estimates ˆθ are mainly caused by Γ u, the within-level kurtosis For parameters that are not shared by µ(.), Σ b (.) and Σ w (.) To facilitate the study of SEs of estimates for parameters that are not shared by µ(.), Σ b (.) and Σ w (.), we assume their parameters are totally separated. That is θ = (θ m, θ b, θ w ) and the structures are µ(θ m ), Σ b (θ b ) and Σ w (θ w ). Denote W c = 2 1 D p (C Σ 1 ) (C Σ 1 )D p+q and W c c = 2 1 D p (C Σ 1 C) (C Σ 1 C)D p. Let ˆθ = (ˆθ m, ˆθ b, ˆθ w ) minimizes (2); where where A (3) = A mm = 1 A bb = 1 A mm A bb A bw 0 A wb A ww =1 =1, (10a) µ, (10b) σ b W σ b, A bw = 1 n 1 σ b W c σ w, (10c) =1 A wb = A bw, A ww = ( n 1) σ w W w σ w + 1 E (3) = E mb = 1 0 E mb E mw E bm E bb E bw, E wm E wb E ww =1 E mw = 1 =1 n 2 σ w W c c σ w, (10d) ( ) Δ b + 1 n 2 Δ cuc W σ b, (11a) ) (1 1n =1 + 1 =1 1 n Δ cu W w σ w ( ) Δ b + 1 n 2 Δ cuc W c σ w, (11b)
9 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) [ ] E bm = E mb, E bb = 1 σ b W (Γ b V b ) + 1 n 3 (Γ cuc V cwc ) W σ b, =1 (11c) E bw = 1 (n 1) σ b W (Γ cu V cw )W w σ w =1 n =1 [ ] 1 σ b n W (Γ b V b ) + 1 n 3 (Γ cuc V cwc ) W c σ w, (11d) E wm = E mw, E wb = E bw, E ww = n σ w W w(γ u V w )W w σ w n = n 2 =1 [ ] σ w W c (Γ b V b ) + 1 n 3 (Γ cuc V cwc ) W c σ w + 1 (n 1) [ σ w W w(γ cu V cw )W c σ w =1 n 3 + σ w W c (Γ cu V cw )W w σ w ] (11e) and = A (3) 1 (3) + A Ω (3) 1 (3) E A(3) The following theorem characterizes the distribution of ˆθ. 1. (12) Theorem 3. When Ω (3) = lim Ω (3) exists, we have (ˆθ θ 0 ) L N(0, Ω (3) ). (13) Note that the A (3) in (10a) corresponds to normal distribution based information matrix. It follows from Theorem 3 and (12) that the larger SEs of ˆθ are caused by E (3). When data are normally distributed, E (3) = 0, SEs based on ˆΩ (3) = Â (3) 1 are asymptotically correct. Many terms in (10) and (11) involve 1 =1 n 1. To better understand the extra SEs
10 1130 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) caused by E (3), we need to properly quantify the magnitude of 1 =1 n 1. For such a purpose, we assume that the n s are evenly distributed on [n a + 1,n a + ]. Wehavethe following result. Lemma 1. Let 1 = Then A (3) A mm A bb A bw. 0 A wb A ww A mm = A 1 mm = O(1), A bb = A 1 bb + O[ n 1 2 ln 2 (1 + /n a )], A 1 bb = ( σ b W b σ b ) 1 + O[ 1 ln(1 + /n a )], A ww = A 1 ww + O[ n 2 2 ln 2 (1 + /n a )], A 1 ww = ( n 1) 1 ( σ w W w σ w ) 1 + O( n 2 1 ), A bw = O[ n 1 1 ln(1 + /n a )]. (14a) (14b) (14c) (14d) (14e) (14f) Lemma 1 implies that, for normally distributed data, ˆθ b and ˆθ w are approximately independent when either n or is large. It also implies that, as n, the SEs of (ˆθ w θ w0 ) approach zero for normally distributed data. As we shall see, this is also true in general for nonnormally distributed data. In order to better study the effect of skewness and kurtosis on the SEs of the within-level parameter estimates ˆθ w, we further let ˆγ = (ˆθ m, ˆθ b,( n 1)1/2 ˆθ w ) and G = diag(i m, I b,( n 1) 1/2 I w ), where I m, I b, I w are identity matrices of proper dimensions. Let then = GΩ (3) G = GA(3) 1 G + H, Ω (4) GA (3) 1 G = H = GA (3) A mm A bb ( n 1) 1/2 A bw, 0 ( n 1) 1/2 A wb ( n 1)A ww 1 G = 1 (3) E A(3) H mm H mb H mw H bm H bb H bw. H wm H wb H ww The asymptotic distribution of ˆγ is characterized by the following result. (15a) (15b) (15c) Theorem 4. Let Ω (4) = lim Ω (4), we have (ˆγ γ0 ) L N(0, Ω (4) ), (16)
11 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) and there exist H mm = 0, H mb = A 1 mm 1 =1 Δ b W σ b A bb + O(1/ ), (17a) H mw = ( n 1) 1/2 A 1 mm 1 =1 Δ cu W w σ w A ww + O[ n 1/2 1 ln(1 + /n a )], (17b) H bb = A 1 bb 1 σ b W (Γ b V b )W σ b A 1 bb =1 + O[ n 1 2 ln 2 (1 + /n a )], (17c) H bw = O[ n 1/2 1 ln(1 + /n a )], (17d) H ww = ( n 1)( n 2)A 1 ww [ σ w W w(γ u V w )W w σ w ]A 1 ww + O[ n 1 1 ln(1 + /n a )]. (17e) Because SEs of (ˆγ γ 0 ) using ˆΩ (4) = GÂ (3) 1 G are the normal distribution based procedure, H mm = 0 in (17a) implies that the skewness or kurtosis of between- or withinlevel random components does not affect SEs of ˆθ m. It also follows from (17a) that Δ b is mainly responsible for the covariances between ˆθ m and ˆθ b. Eq. (17b) implies that, when n is small and is large, Δ u is responsible for the covariances between ˆθ m and ˆθ w. It follows from (14d) and (14e) that A ww = O( n 1 ), (17b) implies that ˆθ m and ( n 1)ˆθ w are not correlated when n approaches infinity even when Δ u = 0. It follows from (17c) that, when n or approaches infinity, the within-level kurtosis has no effect on the SEs of the between-level parameter estimates. Eq. (17d) implies that ˆθ b and ( n 1)ˆθ w are asymptotically independent when either or n approaches infinity. It follows from (17e) that Γ u is mainly responsible for the larger SEs of the within-level parameter estimates ˆθ w. When n or approaches infinity, the between-level kurtosis has no effect on the SEs of the within-level parameter estimates. 3. Asymptotic robustness of standard errors In order to characterize the asymptotic robustness of SEs we first introduce a class of nonnormal distributions given by Yuan and Bentler [33]. Our study of the asymptotic SEs is based on this class of distributions. Properties of SEs when the between- or within-level random components follow the class of elliptical distributions are a special case of the more general results presented below.
12 1132 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Let ξ 1,...,ξ m be independent random variables with E(ξ ) = 0, E(ξ 2 ) = 1, E(ξ3 ) = ζ, E(ξ 4 ) = κ, and ξ = (ξ 1,...,ξ m ). Let r be a random variable which is independent of ξ, E(r 2 ) = 1, E(r 3 ) = γ, and E(r 4 ) = τ. Also, let m d and L = (l i ) = (l 1,...,l m ) be a d m matrix of rank d such that LL = Σ, where l = (l 1,...,l d ). Then the random vector x = rlξ (18) generally follows a nonnormal distribution. Different distributions are obtained by choosing a different set of ξ s, L and r. It is easy to see that the population covariance matrix of x is given by Σ. Yuan and Bentler [33] obtained the fourth-order moment matrix Γ = Cov[vech(xx )] as Γ = 2τD + p (Σ Σ)D+ p + (τ 1)σσ + τ m (κ 3)vech(l l )vech (l l ). (19) When all the κ s equal 3, then the Γ in (19) reduces to that corresponding to an elliptical distribution (see [4]).Yuan and Bentler [33] called the corresponding distribution of x in (18) a pseudo-elliptical distribution, since it is no longer symmetric. When τ = 1 in addition to κ = 3, the corresponding distribution of x in (18) was called a pseudo-normal distribution. It was noted by [33] that for a given matrix L, different marginal skewnesses will not affect the Γ matrix in (19). We assume that the between-level random vector b 0 follows (18) with d = p + q and has a fourth-order moment matrix Γ b = τ b V b + (τ b 1)σ b σ b + τ b m b (κ (b) =1 =1 3)vech(l (b) l (b) )vech (l (b) l (b) ). (20) We can only generalize the asymptotic robustness property of SEs from conventional SEM to the multilevel SEM context when all the parameters are separated as in Section 2.2. Let Ω mm Ω mb Ω mw Ω (4) = Ω bm Ω bb Ω bw. Ω wm Ω wb Ω ww It follows from (15) that Ω bb = A bb + H bb. Combining (14b), (14c) and (17c) yields Ω bb = ( σ b W b σ b ) 1 ( σ b W bγ b W b σ b )( σ b W b σ b ) 1 + O[ 1 ln(1 + /n a )]. (21) Suppose the between-level covariance structure Σ b (θ b ) has q b parameters with θ b = (θ b1, θ b2,...,θ bqb ). Let R( σ b ) be the space spanned by the column vectors of σ b.we need the following condition for asymptotic robustness: Condition B. For each of the = 1,...,m b, vech(l (b) l (b) ) R( σ b ).
13 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Note that Condition B implies that there exist vectors c () b = (c () b1,c() b2,...,c() bq b ) such that vech(l (b) l (b) ) = σ b c () b. Because Σ b = m b l (b) l (b), Condition B also implies σ b =1 R( σ b ). Thus there exists a vector c b = (c b1,c b2,...,c bqb ) such that σ b = σ b c b. Consequently, we can rewrite (21) as Ω bb = τ b ( σ b W b σ b ) 1 + (τ b 1)c b c b m b + τ b (κ (b) 3)c () b c() b + O[ 1 ln(1 + /n a )]. (22) =1 When κ (b) = 3, the Γ b in (20) is identical to that when b follows an elliptical symmetric distribution. So, Ω bb = lim Ω bb corresponding to a pseudo-elliptical distribution is exactly the same as that corresponding to an elliptical distribution. When τ b = 1 and κ (b) = 3, Ω bb = ( σ b W b σ b ) 1. This indicates that, when is large, SEs for ˆθ b based on the normal distribution assumption can be used for a skewed data set sampled from a pseudo-normal distribution. For some models, c () b1 = = c() br = 0, = 1,...,m b and c b1 = = c br = 0 hold from the model hypothesis. This simplifies the upper left r r submatrix of Ω bb to Ω (r) bb = τ b[( σ b W b σ b ) 1 ] (r). Consider the confirmatory factor model for the between-level components and where b = µ + Λ b f b + ε b Σ b (θ b ) = Λ b Φ b Λ b + Ψ b, Λ b = λ b λ bkb and all the λ b s are vectors so that each between-level random component only measures one factor, Φ b = Cov(f b ), and Ψ b = Cov(ε b ) = diag(ψ b11,...,ψ b(p+q)(p+q) ). Let the model be identified by fixing one of the factor loadings at 1.0 for each factor. When data are generated by (18) with L = L b = (Λ b Φ 1/2 b, Ψ 1/2 b ), then m b = (p + q) + k b in (20). For such a setup, Yuan and Bentler [32] showed that Condition B is satisfied and the asymptotic covariance matrix for estimates of the r = p + q k b free factor loadings is τ b [( σ b W b σ b ) 1 ] (r). When τ b = 1, data generated by (18) can have arbitrary skewness and kurtosis. But SEs of the between-level factor loading estimates provided by the normal distribution assumption are still valid when. This
14 1134 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) extends the result of Anderson and Amemiya [1], Shapiro and Browne [30], Satorra and Bentler [28] andyuan and Bentler [32] from the context of conventional confirmatory factor analysis to the between-level factor model within a two-level SEM framework. Note that the within-level structure Σ w (θ w ) can be any identifiable structure which does not interfere with the asymptotic robustness of the between-level SEs. Similarly, the normal distribution based SEs for the within-level parameter estimates ˆθ w can also be asymptotically robust. It follows from (15), (14d), (14e) and (17e) that Ω ww = ( σ w W w σ w ) 1 ( n 2) + ( n 1) ( σ w W w σ w ) 1 [ σ w W w(γ u V w )W w σ w ]( σ w W w σ w ) 1 + O[ n 1 1 ln(1 + /n a )]. (23) Suppose the within-level vector u i is generated by (18) and has a fourth-order moment matrix Γ w = τ w V w + (τ w 1)σ w σ w m w +τ w (κ (w) 3)vech(l (w) =1 l (w) )vech (l (w) l (w) ). (24) Parallel to Condition B, we need the following condition for the asymptotic robustness at the within-level. Condition W. For each of the = 1,...,m w, vech(l (w) l (w) ) R( σ w ) When Condition W is satisfied, there exist c () w Ω ww = ( σ w W w σ w ) 1 + m w + τ w (κ (w) =1 and c w such that ( n 2) ( n 1) (τ w 1)[( σ w W w σ w ) 1 + c w c w ] 3)c () w c() w + O[ n 1 1 ln(1 + /n a )]. (25) When κ (w) = 3, the Ω ww = lim Ω ww is identical to that when u i follows an elliptical symmetric distribution. So, when n or, the Ω ww matrix corresponding to u i following a pseudo-elliptical distribution is exactly the same as that corresponding to u i following an elliptical distribution. When τ w = 1 and κ (w) = 3, Ω ww = ( σ w W w σ w ) 1. This indicates that, when or n is large, SEs for ˆθ w based on the normality assumption can be used for skewed data sampled from a pseudo normal distribution. When Σ w = Λ w Φ w Λ w + Ψ w and u i can be represented by (18) with L = L w = (Λ w Φw 1/2, Ψ1/2 w ), the SEs for the factor loading estimates based on the normality assumption are asymptotically valid for many nonnormal distributions with heterogeneous skewness and kurtoses. So the results of Anderson and Amemiya [1], Shapiro and Browne [30], Satorra and Bentler [28] and Yuan and Bentler [32] are still true for within-level factor analysis models in a two-level SEM regardless of the between-level structure.
15 4. Conclusions K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Motivated by typical nonnormal data in social and behavioral sciences and the increasingly popular normal distribution based multilevel methodology, we study how skewness and kurtosis in one level affect SEs of parameter estimates within its level and outside its level. To facilitate the study with different level-1 sample sizes we have assumed that the level-1 sample sizes n s are evenly distributed. For parameters that are shared by the mean, the between- and the within-level covariance structures, the effect of skewnesses and kurtoses on SEs of their estimates depends on the average level-1 sample size n = N/. When n is small, both the between- and the within-level skewnesses and kurtoses affect the SEs linearly. When n is large, however, it is mainly the within-level kurtosis that affects the SEs of the common parameter estimates. For parameters that are unique to the mean, the between-level covariance structure and the within-level covariance structure, effects of skewness and kurtosis are different on different parameter estimates. First, skewness or kurtosis at either level does not affect the asymptotic SEs of the mean parameter estimates. The estimates of the mean parameters and parameters of the between-level covariance structure are asymptotically correlated in general; this correlation is determined by the between-level skewness. The estimates of the mean parameters and those of parameters of the within-level covariance structure are asymptotically independent when n but not when. When either or n, the within-level kurtosis has no effect on the SEs of the between-level parameter estimates. Similarly, when or n, the between-level kurtosis has no effect on the SEs of the within-level parameter estimates. When either or n, the between-level and within-level covariance parameter estimates are asymptotically independent. We also showed that, parallel to the asymptotic robustness of SEs in the conventional SEM model, asymptotic robustness may exist for SEs in a multilevel factor analysis model. For example, under proper conditions, SEs of factor loading estimates at both the betweenand within-level are asymptotically robust. Unfortunately, the same limitation as with conventional SEM model holds: The results of asymptotic robustness may only be observed when n or are large enough, and the asymptotic robustness conditions B and W are not verifiable. When data cannot be represented by (18), asymptotic robustness may not hold. In practice, then, it seems more appropriate to compute SEs using (6) than to accept normal theory SEs in the hope that asymptotic robustness would resolve issues related to nonnormality. In the study we have assumed that the within-level random component vector u i s, i = 1,...,n, follow the same distribution for all the clusters = 1,...,. It is possible that u i s are distributed differently for different. Additional study in this direction is needed and will provide more detailed results for SEs. Acknowledgements We are thankful to a referee and an editor for their comments that have led the paper to a significant improvement over the previous version.
16 1136 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Appendix A. This appendix will provide the proofs for Theorems 1 4, Corollary 1 and Lemma 1. Proof of Theorem 1. On differentiating the l(θ) in (2) we obtain dl(θ) = (N )tr{σ 1 w (θ)[s y Σ w (θ)]σ 1 w (θ)[dσ w(θ)]} tr{σ 1 (θ)[r (θ) Σ (θ)]σ 1 [dσ (θ)]} 2 =1 [dµ (θ)]σ 1 (θ)[t µ(θ)]. (A.1) =1 It follows from (A.1) that the normal estimating function corresponding to l(θ) is g(θ) = ( n 1) σ w (θ)w w(θ)[s y σ w (θ)] + 1 σ (θ)w (θ)[r (θ) σ (θ)] + 1 =1 =1 µ (θ)σ 1 (θ)[t µ(θ)]. Applying the Taylor expansion on g(ˆθ) = 0 at θ 0 leads to (ˆθ θ 0 ) = ġ 1 (θ 0 ) g(θ 0 ) + o p (1). Theorem 1 directly follows from (A.2) by noticing that A (1) = E[ġ(θ 0 )] and B (1) Var[g(θ 0 )]. (A.2) = Proof of Theorem 2. Following the outline given in the appendix ofyuan and Bentler [36] we have ) Var(n s ) = (n 2 + 1n Γ u + 2(n 1) D + p n (Σ w Σ w )D + p ) = (n 1)V w + (n 2 + 1n (Γ u V w ), (A.3) Var(r ) = Γ b + 1 n 3 Γ cuc + 2 n D + p+q [(CΣ wc ) Σ b + Σ b (CΣ w C )]D + p+q + 2(n 1) n 3 D + p+q [(CΣ wc ) (CΣ w C )]D + p+q = V + (Γ b V b ) + 1 n 3 (Γ cuc V cwc ), (A.4) n Cov(r, s ) = (n 1) n 2 D + p+q (C C)[D pγ u D p 2(Σ w Σ w )]D + p
17 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) n Cov(t, s ) = = (n 1) n 2 (Γ cu V cw ), (A.5) ) (1 1n Δ cu, (A.6) Cov(t, r ) = Δ b + 1 n 2 Δ cuc. (A.7) Combining (4) with (A.3) (A.7) leads to B (1) = A (1) + E (1). Theorem 2 follows from (5) and (A.8). (A.8) Proof of Corollary 1. When n, it follows from (8) that E (1) = n σ w W w(γ u V w )W w σ w + O(1). (A.9) Because A (1) 1 = O( n 1 ), the corollary follows from (7) and (A.9). Proof of Theorem 3. The estimating function corresponding to the parameterization θ = (θ m, θ b, θ w ) is h 1 (θ) h(θ) = h 2 (θ), h 3 (θ) where h 1 (θ) = 1 h 2 (θ) = 1 =1 µ (θ m )Σ 1 (θ)[t µ(θ m )], σ b (θ b)w (θ)[r (θ m ) σ (θ)], =1 h 3 (θ) = σ w (θ w)w w (θ w )[n s (n 1)σ w (θ w )] =1 =1 1 n σ w (θ w)w c (θ)[r (θ m ) σ (θ)]. Because ˆθ satisfies h(ˆθ) = 0, its asymptotic distribution is characterized by (see [38]) (ˆθ θ 0 ) L N(0, Ω (3) ),
18 1138 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) where Ω (3) = lim ḣ 1 (θ 0 ){ Var[h(θ 0 )]}ḣ 1 (θ 0 ). It follows from (A.3) to (A.7) that Var[h(θ 0 )]=A (3) + E (3). The theorem follows by noticing that A (3) = E[ḣ(θ 0 )]. Proof of Lemma 1. Because n s are evenly distributed on [n a + 1,n a + ], there exists O 1 n 1 = O 1 (n a + ) 1 =1 =1 = O ( 1 = O 1 dx ) n a + x ] [ 1 ln(1 + /n a ). (A.10) It follows from (10) that A mm = O(1), A bb = O(1), A bw = O[ 1 ln(1 + /n a )] and A ww = O( n). Applying the rule of matrix inversion for partitioned matrices to (10a), we have A mm = A 1 mm = O(1), A ww = (A ww A wb A 1 bb A bw) 1 ={A ww O[ 2 ln 2 (1 + /n a )]} 1 = A 1 ww + O[ n 2 2 ln 2 (1 + /n a )], A bb = A 1 bb + A 1 bb A bwa ww A wb A 1 bb = A 1 bb + O[ n 1 2 ln 2 (1 + /n a )], A bw = A 1 bb A bwa ww = A 1 bb A bwa 1 ww + O[ n 2 3 ln 3 (1 + /n a )] = O[ n 1 1 ln(1 + /n a )]. Because Σ = Σ b + n 1 CΣ w C = Σ b + O(n 1 ), ( 1 W = W b + O n Eq. (10c) implies that ). A bb = σ b W b σ b + O[ 1 ln(1 + /n a )], and thus (14c) follows. Eq. (10d) implies that A ww = ( n 1) σ w W w σ w + O( 1 ), and thus (14e) follows.
19 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) Proof of Theorem 4. Direct matrix multiplication in (15c) leads to H mm = 0, H mb = A 1 mm (E mba bb + E mw A wb ), H mw = n 1/2 A 1 mm (E mba bw + E mw A ww ), H bb = A bb (E bb A bb + E bw A wb ) + A bw (E wb A bb + E ww A wb ), (A.11a) (A.11b) H ww = ( n 1)A wb (E bb A bw + E bw A ww ) + ( n 1)A ww (E wb A bw + E ww A ww ), (A.11c) H bw = ( n 1) 1/2 A bb (E bb A bw + E bw A ww ) + ( n 1) 1/2 A bw (E wb A bw + E ww A ww ), (A.11d) H bm = H mb, H wm = H mw, H wb = H bw. Combining (A.10) and (11a) (11e) yields E mb = 1 E mw = 1 =1 =1 Δ b W σ b + O(1/ ), (A.12a) Δ cu W w σ w + O[ 1 ln(1 + /n a )], (A.12b) E bb = 1 σ b W (Γ b V b )W σ b + O(1/ ), =1 E bw = O[ 1 ln(1 + /n a )] (A.12c) (A.12d) and E ww = ( n 2) σ w W w(γ u V w )W w σ w + O[ 1 ln(1 + /n a )]. (A.12e) It follows from Lemma 1, (A.11a) and (A.12a) that H mb = A 1 mm 1 Δ b W σ b A bb + O(1/ ). =1 Combining Lemma 1, (A.11a) and (A.12b) yields H mw = ( n 1) 1/2 A 1 1 mm =1 + O[ n 1/2 1 ln(1 + /n a )]. Δ cu W w σ w A ww
20 1140 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) It follows from Lemma 1, (A.11b) and (A.12) that H bb = A 1 bb 1 σ b W (Γ b V b )W σ b A 1 bb + O[ n 1 2 ln 2 (1 + /n a )]. =1 It follows from Lemma 1, (A.11c) and (A.12) that H ww = ( n 1)A ww E ww A ww + O[ n 1 2 ln 2 (1 + /n a )] = ( n 1)( n 2)A 1 ww [ σ w W w(γ u V w )W w σ w ]A 1 ww + O[ n 1 1 ln(1 + /n a )]. Finally, Lemma 1, (A.11d) and (A.12) imply H bw = O[ n 1/2 1 ln(1 + /n a )]. References [1] T.W. Anderson,Y. Amemiya, The asymptotic normal distribution of estimators in factor analysis under general conditions, Ann. Statist. 16 (1988) [2] P.M. Bentler,. Liang, Two-level mean and covariance structures: maximum likelihood via an EM algorithm, in: S. Reise, N. Duan (Eds.), Multilevel Modeling: Methodological Advances, Issues, and Applications, Erlbaum, Mahwah, N, 2003, pp [3] Y.M.M. Bishop, S.E. Fienberg, P.W. Holland, Discrete MultivariateAnalysis: Theory and Practice, MIT Press, Cambridge, [4] M.W. Browne, Asymptotic distribution-free methods for the analysis of covariance structures, British. Math. Statist. Psychol. 37 (1984) [5] Y.F. Cheong, R.P. Fotiu, S.W. Raudenbush, Efficiency and robustness of alternative estimators for two- and three-level models: the case of NAEP,. Educational Behav. Statist. 26 (2001) [6] S. du Toit, M. du Toit, Multilevel structural equation modeling, in:. de Leeuw (Ed.), The Analysis of Multilevel Models, Springer, New York, pp , in press. [7] K.-T. Fang, K.W. Kotz, Symmetric Multivariate and Related Distributions, Chapman & Hall, London, [8] H. Goldstein, Multilevel Statistical Models, second ed., Arnold, London, [9] H. Goldstein, R.P. McDonald, A general model for the analysis of multilevel data, Psychometrika 53 (1988) [10] R.H. Heck, S.L. Thomas, An Introduction of Multilevel Modeling Techniques, Erlbaum, Mahwah, N, [11].. Hox, Multilevel Analysis: Techniques and Applications, Erlbaum, Mahwah, N, [12] Y. Kano, Consistency property of elliptical probability density functions,. Multivariate Anal. 51 (1994) [13] I. Kreft,. de Leeuw, Introducing Multilevel Modeling, Sage, London, [14] S.-Y. Lee, Multilevel analysis of structural equation models, Biometrika 77 (1990) [15] S.-Y. Lee, W.-Y. Poon, Analysis of two-level structural equation models via EM type algorithms, Statist. Sinica 8 (1998) [16]. Liang, P.M. Bentler, An EM algorithm for fitting two-level structural equation models, Psychometrika 69 (2004) [17] N.T. Longford, A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects, Biometrika 74 (1987) [18] N.T. Longford, Regression analysis of multilevel data with measurement error, British. Math. Statist. Psychol. 46 (1993) [19].R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, revised edition, Wiley, New York, 1999.
21 K.-H. Yuan, P. M. Bentler / ournal of Multivariate Analysis 97 (2006) [20].. McArdle, F. Hamagami, Multilevel models from a multiple group structural equation perspective, in: G.A. Marcoulides, R.E. Schumacker (Eds.), Advanced Structural Equation Modeling Techniques, Erlbaum, Mahwah, N, 1996, pp [21] R.P. McDonald, H. Goldstein, Balanced versus unbalanced designs for linear structural relations in two-level data, British. Math. Statist. Psychol. 42 (1989) [22] T. Micceri, The unicorn, the normal curve, and other improbable creatures, Psychol. Bull. 105 (1989) [23] B. Muthén, Multilevel covariance structure analysis, Sociol. Methods Res. 22 (1994) [24] B. Muthén, Latent variable modeling of longitudinal and multilevel data, in: A. Raftery (Ed.), Sociological Methodology, Blackwell Publishers, Boston, 1997, pp [25] B. Muthén, A. Satorra, Complex sample data in structural equation modeling, in: P.V. Marsden (Ed.), Sociological Methodology 1995, Blackwell Publishers, Cambridge, MA, 1995, pp [26] W.-Y. Poon, S.-Y. Lee, A distribution free approach for analysis of two-level structural equation model, Comput. Statist. Data Anal. 17 (1994) [27] S.W. Raudenbush, A.S. Bryk, Hierarchical Linear Models, second ed., Sage, Newbury Park, [28] A. Satorra, P.M. Bentler, Corrections to test statistics and standard errors in covariance structure analysis, in: A. von Eye, C.C. Clogg (Eds.), Latent Variables Analysis: Applications for Developmental Research, Sage, Thousand Oaks, CA, 1994, pp [29].L. Schafer,.W. Graham, Missing data: our view of the state of the art, Psychol. Methods 7 (2002) [30] A. Shapiro, M. Browne, Analysis of covariance structures under elliptical distributions,. Amer. Statist. Assoc. 82 (1987) [31] T. Sniders, R. Bosker, Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling, Sage, Thousand Oaks, CA, [32] K.-H. Yuan, P.M. Bentler, On asymptotic distributions of normal theory MLE in covariance structure analysis under some nonnormal distributions, Statist. Probab. Lett. 42 (1999) [33] K.-H. Yuan, P.M. Bentler, On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions, Statist. Sinica 9 (1999) [34] K.-H. Yuan, P.M. Bentler, On normal theory based inference for multilevel models with distributional violations, Psychometrika 67 (2002) [35] K.-H. Yuan, P.M. Bentler, Eight test statistics for multilevel structural equation models, Comput. Statist. Data Anal. 44 (2003) [36] K.-H. Yuan, P.M. Bentler, On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions, Psychometrika 69 (2004) [37] K.-H. Yuan, P.M. Bentler, Asymptotic robustness of the normal theory likelihood ratio statistic for two-level covariance structure models,. Multivariate Anal. 94 (2005) [38] K.-H.Yuan, R.I. ennrich, Asymptotics of estimating equations under natural conditions,. Multivariate Anal. 65 (1998)
Testing Structural Equation Models: The Effect of Kurtosis
Testing Structural Equation Models: The Effect of Kurtosis Tron Foss, Karl G Jöreskog & Ulf H Olsson Norwegian School of Management October 18, 2006 Abstract Various chi-square statistics are used for
More informationSpecifying Latent Curve and Other Growth Models Using Mplus. (Revised )
Ronald H. Heck 1 University of Hawai i at Mānoa Handout #20 Specifying Latent Curve and Other Growth Models Using Mplus (Revised 12-1-2014) The SEM approach offers a contrasting framework for use in analyzing
More informationComputationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models
Computationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models Tihomir Asparouhov 1, Bengt Muthen 2 Muthen & Muthen 1 UCLA 2 Abstract Multilevel analysis often leads to modeling
More informationNesting and Equivalence Testing
Nesting and Equivalence Testing Tihomir Asparouhov and Bengt Muthén August 13, 2018 Abstract In this note, we discuss the nesting and equivalence testing (NET) methodology developed in Bentler and Satorra
More informationHypothesis Testing for Var-Cov Components
Hypothesis Testing for Var-Cov Components When the specification of coefficients as fixed, random or non-randomly varying is considered, a null hypothesis of the form is considered, where Additional output
More informationANALYSIS OF TWO-LEVEL STRUCTURAL EQUATION MODELS VIA EM TYPE ALGORITHMS
Statistica Sinica 8(1998), 749-766 ANALYSIS OF TWO-LEVEL STRUCTURAL EQUATION MODELS VIA EM TYPE ALGORITHMS Sik-Yum Lee and Wai-Yin Poon Chinese University of Hong Kong Abstract: In this paper, the maximum
More informationStrati cation in Multivariate Modeling
Strati cation in Multivariate Modeling Tihomir Asparouhov Muthen & Muthen Mplus Web Notes: No. 9 Version 2, December 16, 2004 1 The author is thankful to Bengt Muthen for his guidance, to Linda Muthen
More informationModel fit evaluation in multilevel structural equation models
Model fit evaluation in multilevel structural equation models Ehri Ryu Journal Name: Frontiers in Psychology ISSN: 1664-1078 Article type: Review Article Received on: 0 Sep 013 Accepted on: 1 Jan 014 Provisional
More informationScaled and adjusted restricted tests in. multi-sample analysis of moment structures. Albert Satorra. Universitat Pompeu Fabra.
Scaled and adjusted restricted tests in multi-sample analysis of moment structures Albert Satorra Universitat Pompeu Fabra July 15, 1999 The author is grateful to Peter Bentler and Bengt Muthen for their
More informationHigh-dimensional asymptotic expansions for the distributions of canonical correlations
Journal of Multivariate Analysis 100 2009) 231 242 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva High-dimensional asymptotic
More informationFisher information for generalised linear mixed models
Journal of Multivariate Analysis 98 2007 1412 1416 www.elsevier.com/locate/jmva Fisher information for generalised linear mixed models M.P. Wand Department of Statistics, School of Mathematics and Statistics,
More informationEdgeworth Expansions of Functions of the Sample Covariance Matrix with an Unknown Population
Edgeworth Expansions of Functions of the Sample Covariance Matrix with an Unknown Population (Last Modified: April 24, 2008) Hirokazu Yanagihara 1 and Ke-Hai Yuan 2 1 Department of Mathematics, Graduate
More informationMisspecification in Nonrecursive SEMs 1. Nonrecursive Latent Variable Models under Misspecification
Misspecification in Nonrecursive SEMs 1 Nonrecursive Latent Variable Models under Misspecification Misspecification in Nonrecursive SEMs 2 Abstract A problem central to structural equation modeling is
More informationLINEAR MULTILEVEL MODELS. Data are often hierarchical. By this we mean that data contain information
LINEAR MULTILEVEL MODELS JAN DE LEEUW ABSTRACT. This is an entry for The Encyclopedia of Statistics in Behavioral Science, to be published by Wiley in 2005. 1. HIERARCHICAL DATA Data are often hierarchical.
More informationSTRUCTURAL EQUATION MODELS WITH LATENT VARIABLES
STRUCTURAL EQUATION MODELS WITH LATENT VARIABLES Albert Satorra Departament d Economia i Empresa Universitat Pompeu Fabra Structural Equation Modeling (SEM) is widely used in behavioural, social and economic
More informationTitle. Description. Remarks and examples. stata.com. stata.com. Variable notation. methods and formulas for sem Methods and formulas for sem
Title stata.com methods and formulas for sem Methods and formulas for sem Description Remarks and examples References Also see Description The methods and formulas for the sem commands are presented below.
More informationMODEL IMPLIED INSTRUMENTAL VARIABLE ESTIMATION FOR MULTILEVEL CONFIRMATORY FACTOR ANALYSIS. Michael L. Giordano
MODEL IMPLIED INSTRUMENTAL VARIABLE ESTIMATION FOR MULTILEVEL CONFIRMATORY FACTOR ANALYSIS Michael L. Giordano A thesis submitted to the faculty at the University of North Carolina at Chapel Hill in partial
More informationUCLA Department of Statistics Papers
UCLA Department of Statistics Papers Title A Unified Approach to Multi-group Structural Equation Modeling with Nonstandard Samples Permalink https://escholarship.org/uc/item/0q8492c0 Authors Yuan, Ke-Hai
More informationON NORMAL THEORY AND ASSOCIATED TEST STATISTICS IN COVARIANCE STRUCTURE ANALYSIS UNDER TWO CLASSES OF NONNORMAL DISTRIBUTIONS
Statistica Sinica 9(1999), 831-853 ON NORMAL THEORY AND ASSOCIATED TEST STATISTICS IN COVARIANCE STRUCTURE ANALYSIS UNDER TWO CLASSES OF NONNORMAL DISTRIBUTIONS Ke-Hai Yuan and Peter M. Bentler University
More informationDetermining Sample Sizes for Surveys with Data Analyzed by Hierarchical Linear Models
Journal of Of cial Statistics, Vol. 14, No. 3, 1998, pp. 267±275 Determining Sample Sizes for Surveys with Data Analyzed by Hierarchical Linear Models Michael P. ohen 1 Behavioral and social data commonly
More informationStreamlining Missing Data Analysis by Aggregating Multiple Imputations at the Data Level
Streamlining Missing Data Analysis by Aggregating Multiple Imputations at the Data Level A Monte Carlo Simulation to Test the Tenability of the SuperMatrix Approach Kyle M Lang Quantitative Psychology
More informationRANDOM INTERCEPT ITEM FACTOR ANALYSIS. IE Working Paper MK8-102-I 02 / 04 / Alberto Maydeu Olivares
RANDOM INTERCEPT ITEM FACTOR ANALYSIS IE Working Paper MK8-102-I 02 / 04 / 2003 Alberto Maydeu Olivares Instituto de Empresa Marketing Dept. C / María de Molina 11-15, 28006 Madrid España Alberto.Maydeu@ie.edu
More informationMultilevel Analysis of Grouped and Longitudinal Data
Multilevel Analysis of Grouped and Longitudinal Data Joop J. Hox Utrecht University Second draft, to appear in: T.D. Little, K.U. Schnabel, & J. Baumert (Eds.). Modeling longitudinal and multiple-group
More informationHierarchical Linear Models. Jeff Gill. University of Florida
Hierarchical Linear Models Jeff Gill University of Florida I. ESSENTIAL DESCRIPTION OF HIERARCHICAL LINEAR MODELS II. SPECIAL CASES OF THE HLM III. THE GENERAL STRUCTURE OF THE HLM IV. ESTIMATION OF THE
More informationMultilevel Analysis, with Extensions
May 26, 2010 We start by reviewing the research on multilevel analysis that has been done in psychometrics and educational statistics, roughly since 1985. The canonical reference (at least I hope so) is
More informationAsymptotic inference for a nonstationary double ar(1) model
Asymptotic inference for a nonstationary double ar() model By SHIQING LING and DONG LI Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong maling@ust.hk malidong@ust.hk
More informationROBUSTNESS OF MULTILEVEL PARAMETER ESTIMATES AGAINST SMALL SAMPLE SIZES
ROBUSTNESS OF MULTILEVEL PARAMETER ESTIMATES AGAINST SMALL SAMPLE SIZES Cora J.M. Maas 1 Utrecht University, The Netherlands Joop J. Hox Utrecht University, The Netherlands In social sciences, research
More informationTesting structural equation models: the effect of kurtosis. Tron Foss BI Norwegian Business School. Karl G. Jøreskog BI Norwegian Business School
This file was downloaded from the institutional repository BI Brage - http://brage.bibsys.no/bi (Open Access) Testing structural equation models: the effect of kurtosis Tron Foss BI Norwegian Business
More informationForecasting 1 to h steps ahead using partial least squares
Forecasting 1 to h steps ahead using partial least squares Philip Hans Franses Econometric Institute, Erasmus University Rotterdam November 10, 2006 Econometric Institute Report 2006-47 I thank Dick van
More informationThe properties of L p -GMM estimators
The properties of L p -GMM estimators Robert de Jong and Chirok Han Michigan State University February 2000 Abstract This paper considers Generalized Method of Moment-type estimators for which a criterion
More informationPIRLS 2016 Achievement Scaling Methodology 1
CHAPTER 11 PIRLS 2016 Achievement Scaling Methodology 1 The PIRLS approach to scaling the achievement data, based on item response theory (IRT) scaling with marginal estimation, was developed originally
More informationInferences on a Normal Covariance Matrix and Generalized Variance with Monotone Missing Data
Journal of Multivariate Analysis 78, 6282 (2001) doi:10.1006jmva.2000.1939, available online at http:www.idealibrary.com on Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone
More informationStatistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach
Statistical Methods for Handling Incomplete Data Chapter 2: Likelihood-based approach Jae-Kwang Kim Department of Statistics, Iowa State University Outline 1 Introduction 2 Observed likelihood 3 Mean Score
More informationESTIMATION OF NONLINEAR BERKSON-TYPE MEASUREMENT ERROR MODELS
Statistica Sinica 13(2003), 1201-1210 ESTIMATION OF NONLINEAR BERKSON-TYPE MEASUREMENT ERROR MODELS Liqun Wang University of Manitoba Abstract: This paper studies a minimum distance moment estimator for
More informationInference with Heywood cases
Inference with Joint work with Kenneth Bollen (UNC) and Victoria Savalei (UBC) NSF support from SES-0617193 with funds from SSA September 18, 2009 What is a case? (1931) considered characterization of
More informationMISCELLANEOUS TOPICS RELATED TO LIKELIHOOD. Copyright c 2012 (Iowa State University) Statistics / 30
MISCELLANEOUS TOPICS RELATED TO LIKELIHOOD Copyright c 2012 (Iowa State University) Statistics 511 1 / 30 INFORMATION CRITERIA Akaike s Information criterion is given by AIC = 2l(ˆθ) + 2k, where l(ˆθ)
More informationModel Estimation Example
Ronald H. Heck 1 EDEP 606: Multivariate Methods (S2013) April 7, 2013 Model Estimation Example As we have moved through the course this semester, we have encountered the concept of model estimation. Discussions
More informationAn Introduction to Multivariate Statistical Analysis
An Introduction to Multivariate Statistical Analysis Third Edition T. W. ANDERSON Stanford University Department of Statistics Stanford, CA WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Contents
More informationA Threshold-Free Approach to the Study of the Structure of Binary Data
International Journal of Statistics and Probability; Vol. 2, No. 2; 2013 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education A Threshold-Free Approach to the Study of
More informationMultilevel Modeling: When and Why 1. 1 Why multilevel data need multilevel models
Multilevel Modeling: When and Why 1 J. Hox University of Amsterdam & Utrecht University Amsterdam/Utrecht, the Netherlands Abstract: Multilevel models have become popular for the analysis of a variety
More informationCENTERING IN MULTILEVEL MODELS. Consider the situation in which we have m groups of individuals, where
CENTERING IN MULTILEVEL MODELS JAN DE LEEUW ABSTRACT. This is an entry for The Encyclopedia of Statistics in Behavioral Science, to be published by Wiley in 2005. Consider the situation in which we have
More informationJournal of Multivariate Analysis. Sphericity test in a GMANOVA MANOVA model with normal error
Journal of Multivariate Analysis 00 (009) 305 3 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva Sphericity test in a GMANOVA MANOVA
More informationMultilevel Regression Mixture Analysis
Multilevel Regression Mixture Analysis Bengt Muthén and Tihomir Asparouhov Forthcoming in Journal of the Royal Statistical Society, Series A October 3, 2008 1 Abstract A two-level regression mixture model
More informationAn Alternative to Cronbach s Alpha: A L-Moment Based Measure of Internal-consistency Reliability
Southern Illinois University Carbondale OpenSIUC Book Chapters Educational Psychology and Special Education 013 An Alternative to Cronbach s Alpha: A L-Moment Based Measure of Internal-consistency Reliability
More informationPartitioning variation in multilevel models.
Partitioning variation in multilevel models. by Harvey Goldstein, William Browne and Jon Rasbash Institute of Education, London, UK. Summary. In multilevel modelling, the residual variation in a response
More informationAn Efficient Estimation Method for Longitudinal Surveys with Monotone Missing Data
An Efficient Estimation Method for Longitudinal Surveys with Monotone Missing Data Jae-Kwang Kim 1 Iowa State University June 28, 2012 1 Joint work with Dr. Ming Zhou (when he was a PhD student at ISU)
More informationHigh Dimensional Empirical Likelihood for Generalized Estimating Equations with Dependent Data
High Dimensional Empirical Likelihood for Generalized Estimating Equations with Dependent Data Song Xi CHEN Guanghua School of Management and Center for Statistical Science, Peking University Department
More informationMinimax design criterion for fractional factorial designs
Ann Inst Stat Math 205 67:673 685 DOI 0.007/s0463-04-0470-0 Minimax design criterion for fractional factorial designs Yue Yin Julie Zhou Received: 2 November 203 / Revised: 5 March 204 / Published online:
More informationMultilevel regression mixture analysis
J. R. Statist. Soc. A (2009) 172, Part 3, pp. 639 657 Multilevel regression mixture analysis Bengt Muthén University of California, Los Angeles, USA and Tihomir Asparouhov Muthén & Muthén, Los Angeles,
More informationMARGINAL HOMOGENEITY MODEL FOR ORDERED CATEGORIES WITH OPEN ENDS IN SQUARE CONTINGENCY TABLES
REVSTAT Statistical Journal Volume 13, Number 3, November 2015, 233 243 MARGINAL HOMOGENEITY MODEL FOR ORDERED CATEGORIES WITH OPEN ENDS IN SQUARE CONTINGENCY TABLES Authors: Serpil Aktas Department of
More informationA note on profile likelihood for exponential tilt mixture models
Biometrika (2009), 96, 1,pp. 229 236 C 2009 Biometrika Trust Printed in Great Britain doi: 10.1093/biomet/asn059 Advance Access publication 22 January 2009 A note on profile likelihood for exponential
More informationA Goodness-of-fit Test for Copulas
A Goodness-of-fit Test for Copulas Artem Prokhorov August 2008 Abstract A new goodness-of-fit test for copulas is proposed. It is based on restrictions on certain elements of the information matrix and
More informationLocal Influence and Residual Analysis in Heteroscedastic Symmetrical Linear Models
Local Influence and Residual Analysis in Heteroscedastic Symmetrical Linear Models Francisco José A. Cysneiros 1 1 Departamento de Estatística - CCEN, Universidade Federal de Pernambuco, Recife - PE 5079-50
More informationJournal of Multivariate Analysis. Use of prior information in the consistent estimation of regression coefficients in measurement error models
Journal of Multivariate Analysis 00 (2009) 498 520 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva Use of prior information in
More informationKRUSKAL-WALLIS ONE-WAY ANALYSIS OF VARIANCE BASED ON LINEAR PLACEMENTS
Bull. Korean Math. Soc. 5 (24), No. 3, pp. 7 76 http://dx.doi.org/34/bkms.24.5.3.7 KRUSKAL-WALLIS ONE-WAY ANALYSIS OF VARIANCE BASED ON LINEAR PLACEMENTS Yicheng Hong and Sungchul Lee Abstract. The limiting
More informationCONSTRUCTION OF COVARIANCE MATRICES WITH A SPECIFIED DISCREPANCY FUNCTION MINIMIZER, WITH APPLICATION TO FACTOR ANALYSIS
CONSTRUCTION OF COVARIANCE MATRICES WITH A SPECIFIED DISCREPANCY FUNCTION MINIMIZER, WITH APPLICATION TO FACTOR ANALYSIS SO YEON CHUN AND A. SHAPIRO Abstract. The main goal of this paper is to develop
More informationManabu Sato* and Masaaki Ito**
J. Japan Statist. Soc. Vol. 37 No. 2 2007 175 190 THEORETICAL JUSTIFICATION OF DECISION RULES FOR THE NUMBER OF FACTORS: PRINCIPAL COMPONENT ANALYSIS AS A SUBSTITUTE FOR FACTOR ANALYSIS IN ONE-FACTOR CASES
More informationFACTOR ANALYSIS AS MATRIX DECOMPOSITION 1. INTRODUCTION
FACTOR ANALYSIS AS MATRIX DECOMPOSITION JAN DE LEEUW ABSTRACT. Meet the abstract. This is the abstract. 1. INTRODUCTION Suppose we have n measurements on each of taking m variables. Collect these measurements
More informationEstimation and Testing for Common Cycles
Estimation and esting for Common Cycles Anders Warne February 27, 2008 Abstract: his note discusses estimation and testing for the presence of common cycles in cointegrated vector autoregressions A simple
More informationChapter 4: Factor Analysis
Chapter 4: Factor Analysis In many studies, we may not be able to measure directly the variables of interest. We can merely collect data on other variables which may be related to the variables of interest.
More informationStandard errors in covariance structure models: Asymptotics versus bootstrap
1 British Journal of Mathematical and Statistical Psychology (2006), 00, 1 22 q 2006 The British Psychological Society The British Psychological Society www.bpsjournals.co.uk Standard errors in covariance
More informationEstimation: Problems & Solutions
Estimation: Problems & Solutions Edps/Psych/Stat 587 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2017 Outline 1. Introduction: Estimation of
More informationA Monte Carlo Power Analysis of Traditional Repeated Measures and Hierarchical Multivariate Linear Models in Longitudinal Data Analysis
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Developmental Cognitive Neuroscience Laboratory - Faculty and Staff Publications Developmental Cognitive Neuroscience Laboratory
More informationStatistical Inference On the High-dimensional Gaussian Covarianc
Statistical Inference On the High-dimensional Gaussian Covariance Matrix Department of Mathematical Sciences, Clemson University June 6, 2011 Outline Introduction Problem Setup Statistical Inference High-Dimensional
More informationReview of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley
Review of Classical Least Squares James L. Powell Department of Economics University of California, Berkeley The Classical Linear Model The object of least squares regression methods is to model and estimate
More informationAsymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½
University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 1998 Asymptotic Nonequivalence of Nonparametric Experiments When the Smoothness Index is ½ Lawrence D. Brown University
More informationOnline Appendix. j=1. φ T (ω j ) vec (EI T (ω j ) f θ0 (ω j )). vec (EI T (ω) f θ0 (ω)) = O T β+1/2) = o(1), M 1. M T (s) exp ( isω)
Online Appendix Proof of Lemma A.. he proof uses similar arguments as in Dunsmuir 979), but allowing for weak identification and selecting a subset of frequencies using W ω). It consists of two steps.
More informationImproper Solutions in Exploratory Factor Analysis: Causes and Treatments
Improper Solutions in Exploratory Factor Analysis: Causes and Treatments Yutaka Kano Faculty of Human Sciences, Osaka University Suita, Osaka 565, Japan. email: kano@hus.osaka-u.ac.jp Abstract: There are
More informationWeb Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D.
Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Ruppert A. EMPIRICAL ESTIMATE OF THE KERNEL MIXTURE Here we
More informationAn Extended BIC for Model Selection
An Extended BIC for Model Selection at the JSM meeting 2007 - Salt Lake City Surajit Ray Boston University (Dept of Mathematics and Statistics) Joint work with James Berger, Duke University; Susie Bayarri,
More informationCorrespondence Analysis of Longitudinal Data
Correspondence Analysis of Longitudinal Data Mark de Rooij* LEIDEN UNIVERSITY, LEIDEN, NETHERLANDS Peter van der G. M. Heijden UTRECHT UNIVERSITY, UTRECHT, NETHERLANDS *Corresponding author (rooijm@fsw.leidenuniv.nl)
More informationMeasuring the Sensitivity of Parameter Estimates to Estimation Moments
Measuring the Sensitivity of Parameter Estimates to Estimation Moments Isaiah Andrews MIT and NBER Matthew Gentzkow Stanford and NBER Jesse M. Shapiro Brown and NBER May 2017 Online Appendix Contents 1
More informationMore Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order Restriction
Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 4, pp. 700-716 c 2007, Indian Statistical Institute More Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order
More informationEconometric Analysis of Cross Section and Panel Data
Econometric Analysis of Cross Section and Panel Data Jeffrey M. Wooldridge / The MIT Press Cambridge, Massachusetts London, England Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND
More informationAssessing the relation between language comprehension and performance in general chemistry. Appendices
Assessing the relation between language comprehension and performance in general chemistry Daniel T. Pyburn a, Samuel Pazicni* a, Victor A. Benassi b, and Elizabeth E. Tappin c a Department of Chemistry,
More informationConfirmatory Factor Analysis: Model comparison, respecification, and more. Psychology 588: Covariance structure and factor models
Confirmatory Factor Analysis: Model comparison, respecification, and more Psychology 588: Covariance structure and factor models Model comparison 2 Essentially all goodness of fit indices are descriptive,
More information(Received April 2008; accepted June 2009) COMMENT. Jinzhu Jia, Yuval Benjamini, Chinghway Lim, Garvesh Raskutti and Bin Yu.
960 R. DENNIS COOK, BING LI AND FRANCESCA CHIAROMONTE Johnson, R. A. and Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Sixth Edition. Pearson Prentice Hall. Jolliffe, I. T. (2002).
More informationFor more information about how to cite these materials visit
Author(s): Kerby Shedden, Ph.D., 2010 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution Share Alike 3.0 License: http://creativecommons.org/licenses/by-sa/3.0/
More informationAnalysis of the AIC Statistic for Optimal Detection of Small Changes in Dynamic Systems
Analysis of the AIC Statistic for Optimal Detection of Small Changes in Dynamic Systems Jeremy S. Conner and Dale E. Seborg Department of Chemical Engineering University of California, Santa Barbara, CA
More informationEvaluating Small Sample Approaches for Model Test Statistics in Structural Equation Modeling
Multivariate Behavioral Research, 9 (), 49-478 Copyright 004, Lawrence Erlbaum Associates, Inc. Evaluating Small Sample Approaches for Model Test Statistics in Structural Equation Modeling Jonathan Nevitt
More informationHaruhiko Ogasawara. This article gives the first half of an expository supplement to Ogasawara (2015).
Economic Review (Otaru University of Commerce, Vol.66, No. & 3, 9-58. December, 5. Expository supplement I to the paper Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters
More informationUtilizing Hierarchical Linear Modeling in Evaluation: Concepts and Applications
Utilizing Hierarchical Linear Modeling in Evaluation: Concepts and Applications C.J. McKinney, M.A. Antonio Olmos, Ph.D. Kate DeRoche, M.A. Mental Health Center of Denver Evaluation 2007: Evaluation and
More informationMeasuring the Sensitivity of Parameter Estimates to Estimation Moments
Measuring the Sensitivity of Parameter Estimates to Estimation Moments Isaiah Andrews MIT and NBER Matthew Gentzkow Stanford and NBER Jesse M. Shapiro Brown and NBER March 2017 Online Appendix Contents
More informationParametric Techniques Lecture 3
Parametric Techniques Lecture 3 Jason Corso SUNY at Buffalo 22 January 2009 J. Corso (SUNY at Buffalo) Parametric Techniques Lecture 3 22 January 2009 1 / 39 Introduction In Lecture 2, we learned how to
More informationThe outline for Unit 3
The outline for Unit 3 Unit 1. Introduction: The regression model. Unit 2. Estimation principles. Unit 3: Hypothesis testing principles. 3.1 Wald test. 3.2 Lagrange Multiplier. 3.3 Likelihood Ratio Test.
More informationTHE 'IMPROVED' BROWN AND FORSYTHE TEST FOR MEAN EQUALITY: SOME THINGS CAN'T BE FIXED
THE 'IMPROVED' BROWN AND FORSYTHE TEST FOR MEAN EQUALITY: SOME THINGS CAN'T BE FIXED H. J. Keselman Rand R. Wilcox University of Manitoba University of Southern California Winnipeg, Manitoba Los Angeles,
More informationDESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective
DESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective Second Edition Scott E. Maxwell Uniuersity of Notre Dame Harold D. Delaney Uniuersity of New Mexico J,t{,.?; LAWRENCE ERLBAUM ASSOCIATES,
More informationHigh-Dimensional AICs for Selection of Redundancy Models in Discriminant Analysis. Tetsuro Sakurai, Takeshi Nakada and Yasunori Fujikoshi
High-Dimensional AICs for Selection of Redundancy Models in Discriminant Analysis Tetsuro Sakurai, Takeshi Nakada and Yasunori Fujikoshi Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku,
More informationDiscrete Dependent Variable Models
Discrete Dependent Variable Models James J. Heckman University of Chicago This draft, April 10, 2006 Here s the general approach of this lecture: Economic model Decision rule (e.g. utility maximization)
More informationAn almost sure invariance principle for additive functionals of Markov chains
Statistics and Probability Letters 78 2008 854 860 www.elsevier.com/locate/stapro An almost sure invariance principle for additive functionals of Markov chains F. Rassoul-Agha a, T. Seppäläinen b, a Department
More informationOn asymptotic properties of Quasi-ML, GMM and. EL estimators of covariance structure models
On asymptotic properties of Quasi-ML, GMM and EL estimators of covariance structure models Artem Prokhorov November, 006 Abstract The paper considers estimation of covariance structure models by QMLE,
More informationConsistency of Test-based Criterion for Selection of Variables in High-dimensional Two Group-Discriminant Analysis
Consistency of Test-based Criterion for Selection of Variables in High-dimensional Two Group-Discriminant Analysis Yasunori Fujikoshi and Tetsuro Sakurai Department of Mathematics, Graduate School of Science,
More informationTesting a Normal Covariance Matrix for Small Samples with Monotone Missing Data
Applied Mathematical Sciences, Vol 3, 009, no 54, 695-70 Testing a Normal Covariance Matrix for Small Samples with Monotone Missing Data Evelina Veleva Rousse University A Kanchev Department of Numerical
More informationGoodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach
Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach By Shiqing Ling Department of Mathematics Hong Kong University of Science and Technology Let {y t : t = 0, ±1, ±2,
More informationParametric Techniques
Parametric Techniques Jason J. Corso SUNY at Buffalo J. Corso (SUNY at Buffalo) Parametric Techniques 1 / 39 Introduction When covering Bayesian Decision Theory, we assumed the full probabilistic structure
More informationTesting Restrictions and Comparing Models
Econ. 513, Time Series Econometrics Fall 00 Chris Sims Testing Restrictions and Comparing Models 1. THE PROBLEM We consider here the problem of comparing two parametric models for the data X, defined by
More informationNew insights into best linear unbiased estimation and the optimality of least-squares
Journal of Multivariate Analysis 97 (2006) 575 585 www.elsevier.com/locate/jmva New insights into best linear unbiased estimation and the optimality of least-squares Mario Faliva, Maria Grazia Zoia Istituto
More informationNonparametric Identification of a Binary Random Factor in Cross Section Data - Supplemental Appendix
Nonparametric Identification of a Binary Random Factor in Cross Section Data - Supplemental Appendix Yingying Dong and Arthur Lewbel California State University Fullerton and Boston College July 2010 Abstract
More informationChapter 3. Point Estimation. 3.1 Introduction
Chapter 3 Point Estimation Let (Ω, A, P θ ), P θ P = {P θ θ Θ}be probability space, X 1, X 2,..., X n : (Ω, A) (IR k, B k ) random variables (X, B X ) sample space γ : Θ IR k measurable function, i.e.
More informationGARCH Models Estimation and Inference. Eduardo Rossi University of Pavia
GARCH Models Estimation and Inference Eduardo Rossi University of Pavia Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function
More information