New families of estimators and test statistics in log-linear models

Transcription

1 Journal of Multivariate Analysis ew families of estimators and test statistics in log-linear models irian Martín a,, Leandro Pardo b a Department of Statistics and O. R. III, Scool of Statistics, Complutense University of Madrid, 8040 Madrid, Spain b Department of Statistics and O. R. I, Faculty of Matematics, Complutense University of Madrid, 8040 Madrid, Spain Received 18 Marc 006 Available online 9 January 008 Abstract In tis paper we consider categorical data tat are distributed according to a multinomial, productmultinomial or Poisson distribution wose expected values follow a log-linear model and we study te inference problem of ypotesis testing in a log-linear model setting. Te family of test statistics considered is based on te family of φ-divergence measures. Te unknown parameters in te log-linear model under consideration are also estimated using φ-divergence measures: Minimum φ-divergence estimators. A simulation study is included to find test statistics tat offer an attractive alternative to te Pearson ci-square and likeliood-ratio test statistics. c 008 Elsevier Inc. All rigts reserved. AMS 1991 subject classification: 6H17 Keywords: Asymptotic distributions; ested ypoteses; Poisson sampling; Multinomial sampling; Product-multinomial sampling; Minimum φ-divergence estimator; φ-divergence test statistics 1. Introduction Consider a table wit k cells, containing counts, n n 1,..., n k T. In wat is to follow, we assume tat n is te result of eiter Poisson, multinomial, or product-multinomial sampling. Define mθ m 1 θ,..., m k θ T E[n]. If we denote by log mθ te vector log m 1 θ,..., log m k θ T, te log-linear model is defined by log mθ = Xθ, 1 Tis work was partially supported by Grants MTM and UCM Corresponding autor. addresses: nirian@estad.ucm.es. Martín, lpardo@mat.ucm.es L. Pardo X/$ - see front matter c 008 Elsevier Inc. All rigts reserved. doi: /j.jmva

2 . Martín, L. Pardo / Journal of Multivariate Analysis were X is a known k t full rank matrix suc tat t < k and θ R t. Te range of models is summarized by CX {log mθ : log mθ = Xθ; θ R t }. We can observe tat CX represents te span of te columns of te matrix X. For Poisson sampling, te total k n i is random; in tis case, we define = E[ k n i ], te expected total. If one ten conditions on te total, te vector n = n 1,..., n k T becomes multinomial. In tis case, we define = k n i and n is multinomial wit parameters ; π 1 θ,..., π k θ, were π i θ = m i θ/ k =1 m θ; i = 1,..., k. If te observations come from a product-multinomial sampling sceme, certain of te margins are fixed. Assume tat tere are r independent multinomial distributions, j ; π j1 θ,..., π jk j θ; j = 1,..., r, were = r j=1 j and r j=1 k j = k. Witout loss of generality, we can write mθ = 1 π 11 θ,..., 1 π 1k1 θ,..., r π r1 θ,..., r π rkr θ T. If r = 1, te sampling sceme reduces to te simple multinomial. otice tat {1,..., k} is partitioned into r sets, M j = { j 1 k + 1,..., =0 } j k ; j = 1,..., r, =0 were k 0 = 0. Let te p 1 vector of 1 s be defined by, J p 1,..., 1 T, and te p 1 vector of 0 s be defined by, O p 0,..., 0 T. Ten we write te k 1 vector x j as follows: x j O T k 1,..., J T k j,..., O T k r T ; j = 1,..., r. By te product-multinomial sampling sceme, we ave n T x j = j, 3 were j is te fixed sample size for te j-t multinomial. Ten = r j=1 j = k n i, wic is also fixed, and mθ T x j = k E [n i ] x i j = j = n T x j ; j = 1,..., r. 4 For product-multinomial sampling, it will be convenient to combine te vectors x j into a k r matrix, say X 0 = x 1,..., x r. For multinomial sampling, it is clear tat X 0 = J k, and for Poisson sampling, X 0 can be taken as te vector O k. Ten for log-linear models tere are two restrictions on te parameters, namely, and log mθ CX, n T X 0 = mθ T X 0. ote tat in te case of te Poisson sampling, 6 is not a restriction itself. We are going to consider an important assumption for te purpose of normalization, if te sampling is 5 6

3 159. Martín, L. Pardo / Journal of Multivariate Analysis multinomial or product-multinomial CX 0 CX, 7 and if te sampling is Poisson J k CX, 8 wic is verified in multinomial and product-multinomial sampling as well, as a consequence of 7. We consider for multinomial sampling, witout loss of generality, tat te first column vector of X is J k. It can be seen in Cristensen [3] tat te maximum likeliood estimator θ in te log-linear model 1, for Poisson sampling, multinomial sampling, and product-multinomial sampling, is te same and can be obtained maximizing l θ, mθ k n i log m i θ For more details, see formula 5 in Cristensen [3] page 399. ow we introduce by D φ n, mθ = k k m i θ. 9 ni m i θφ, φ Φ, 10 m i θ te φ-divergence measure between te expected vectors n and m θ, were Φ is te class of all convex and differentiable functions φ : [0, R { }, suc tat at x = 1, φ 1 = φ 1 = 0, φ 1 > 0. In 10 we sall assume te conventions 0φ 0/0 0 and 0φ p/0 p lim u φ u /u. For more details about φ-divergence measures see Pardo [15]. Te function φ x = x log x x +1 Φ gives te Kullback Leibler divergence, D Kull n, mθ. It is immediate tat D Kull n, mθ = k n i log n i k n i lθ, mθ. On te basis of tis expression maximizing lθ, mθ is equivalent to minimizing D Kull n, mθ and terefore we can define te maximum likeliood estimator, θ, by were θ = arg min θ Θ D Kulln, mθ, 11 Θ = {θ R t : n T X 0 = mθ T X 0 }. 1 Wen multinomial sampling is understood troug te Poisson sampling conditioned on te sample size, it is important to empasize tat te dimension of te parameter space Θ R t is one unit greater compared wit te traditional parameter space case were no constraint is taken into account Θ = R t 1. Tis extra component of parameter θ is associated wit 8, and is directly obtained from te constraint once te rest of te t 1 components are determined. Te same idea remains for te r extra components of te parameter in product-multinomial sampling. For more details, see Lang [1] and references terein.

4 . Martín, L. Pardo / Journal of Multivariate Analysis In Cressie and Pardo [4] and Cressie et al. [6] tere were introduced and studied te minimum φ-divergence estimator in log-linear models wit multinomial sampling as well as a family of φ-divergence test statistics for testing a nested sequence of log-linear models wit multinomial sampling. Te minimum φ-divergence estimator was defined in [4] as a natural extension of te maximum likeliood estimator and its asymptotic properties were obtained. Tis estimator was ten used in a φ-divergence goodness-of-fit statistic, wic was te basis of two new families of statistics for solving te problem of testing a nested sequence of log-linear models. In [6], under te assumption of multinomial sampling, tere were collected togeter te operating caracteristics of te ypotesis test based on bot asymptotic using large-sample teory and finite-sample using a designed simulation study results. On te oter and, Cressie and Pardo [5] considered φ-divergence test statistics for Poisson sampling, multinomial sampling, and product-multinomial sampling, using a maximum likeliood estimator. If we pay attention to papers [4,5] and [6] an open problem appears: Is it possible to consider φ-divergence test statistics and minimum φ-divergence estimators, at te same time, for Poisson, multinomial and product-multinomial sampling? Tis provides te main purpose of tis paper: extending te results obtained in [4,5] and [6] in te sense of considering, at te same time, te minimum φ-divergence estimator and φ-divergence test statistics for log-linear models wen te sampling sceme is Poisson, multinomial and product-multinomial. In Section we introduce te minimum φ-divergence estimator, θ φ, for te tree different kinds of sampling scemes, as a natural extension of te maximum likeliood estimator, and we study its asymptotic properties as well as te asymptotic beavior of te estimated log-linear model m θ φ. In Section 3 we introduce te problem of goodness-of-fit. We consider a family of φ-divergence test statistics wic contains as special cases te classical likeliood-ratio test and Pearson ci-square test. Its asymptotic distribution is obtained under bot te null ypotesis and contiguous alternative ypoteses. On te basis of tis family of φ-divergence test statistics a family of residuals is given and its asymptotic distribution is obtained. Tis family of residuals is an extension of te family of residuals considered in Gupta et al. [10] for log-linear models wit multinomial sampling. Te family of residuals introduced in tis paper is valid for te tree different kinds of sampling scemes, not only for multinomial sampling. Te inference problem tat we consider in Section 4 is tat of ypotesis testing in a log-linear model setting. Te null ypotesis is a composite ypotesis nested witin te alternative. Two new families of φ-divergence test statistics are introduced and studied. Teir asymptotic distributions under te null as well as under contiguous alternative ypoteses are obtained. Section 5 contains a simulation study in wic members of te class of power divergence statistics are compared and it is found tat some test statistics offer an attractive alternative to te Pearson ci-square test and likeliood-ratio test statistics in terms of bot exact size and power.. Minimum φ-divergence estimator On te basis of 10 we can define a new family of estimators for Poisson, multinomial and product-multinomial sampling. Definition 1. For a log-linear model 1 wit parameter space 1, te minimum φ-divergence estimator is given by θ φ = arg min θ Θ D φn, mθ, 13 wit D φ n, mθ defined by 10.

5 1594. Martín, L. Pardo / Journal of Multivariate Analysis Te formulation 13 is equivalent to te usual definition using φ-divergences between probabilities instead of nonnegative vectors, and is also valid for te Poisson sampling. Tis family of estimators represents a natural extension of te maximum likeliood estimator for tree types of sampling because tis can be obtained by coosing φ x = x log x x + 1. In establising asymptotic properties, we let tend to infinity. For product-multinomial sampling, we assume tat, as, te probabilities in eac cell remain fixed and j /, j = 1,..., r, remain also fixed. For Poisson sampling, recall tat = E[ k n i ], and we assume tat te ratio of any pair of expected values remains fixed. Define te normalized vector m m 1,..., m k T, as follows: * Poisson sampling: m i = m i /; i = 1,..., k. * Multinomial sampling: m i = π i = m i /; i = 1,..., k. * Product-multinomial sampling: m = m 1 T,..., m r T T = m/, were m j π j1,..., π jk j T j / ; j = 1,..., r. Tat is, m is proportional to m, but m as been normalized to be a probability vector. ow, since log m = log m + log J k and we ave assumed tat J k CX, te restriction log m CX in 5 is equivalent to log m CX. In te following teorem we present a decomposition for te minimum φ-divergence estimator independently of te type of sampling. Teorem. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose a function φ Φ, were Φ is defined in Section 1; we ave φ 1 θ = θ 0 + X T D m θ 0 X Λ X T n n m θ 0 + o m θ 0, were θ 0 Θ is te true and unknown value of te parameter, wit D m θ 0 diag{m 1 θ 0,..., m k θ 0}, O t O T t, Poisson, Λ = e 1 e T 1, multinomial, 1 1 X T X X T X 0 D 1 XT 0 X X X T, product-multinomial, { 1 D diag,..., } r = X0 T D m θ 0 X 0 and e i 0,..., 1 i,..., 0 T. Proof. We omit te proof because its steps are similar to ones given in Martín and Pardo [13]. In te next teorem we obtain te asymptotic distribution of θ φ as well as of m θ φ. Teorem 3. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose a function φ Φ, were Φ was defined in Section 1, we ave

6 a. Martín, L. Pardo / Journal of Multivariate Analysis θ φ θ 0 L 1 O t, X T D m θ 0 X Λ, 14 were Λ is defined in Teorem, and b 1 m θ φ L mθ 0 O k, Wθ 0 W 0, 15 were Wθ 0 D m θ 0 X X T D m θ 0 X 1 X T D m θ 0 and O k O T k, W 0 m θ 0 m θ 0 T, D m θ 0 X 0 D 1 XT 0 D m θ 0, Poisson, multinomial, product-multinomial. Proof. Result a follows by Teorem and taking into account see Haberman [11] tat were 1 L n mθ 0 O k, D m θ 0 W 0, 16 D m θ 0 W 0 = r j=1 j D π j θ 0 π j θ 0 π j θ 0 T wit π j θ 0 = π j1 θ 0,..., π jk j θ 0 T, wen sampling is product-multinomial. By r j=1 A j we are denoting te direct sum of te matrices A 1,..., A r, i.e., r j=1 A j = diag{a 1,..., A r }. Part b follows by a and applying te delta metod. Sections and in Agresti [] present a detailed study of te delta metod. 3. Residuals and goodness-of-fit test statistics Te classical Pearson ci-square test statistic for goodness-of-fit in te log-linear model is given by X θ = k n i m i θ, 17 m i θ were θ is te maximum likeliood estimator of θ, and te likeliood-ratio test statistic by G θ = k n i n i log m i θ n i m i θ. 18 Te two classical test statistics can be extended using te minimum φ-divergence estimator instead of te maximum likeliood estimator. Tese test statistics are particular cases of te

7 1596. Martín, L. Pardo / Journal of Multivariate Analysis family of φ-divergence test statistics, T φ 1,φ = φ 1 1 D φ 1 n, m θ φ 19 because for φ 1 x = 1 x 1 we get X θ φ and for φ 1 x = x log x x +1 we get G θ φ. If we replace te maximum likeliood estimator by te minimum φ-divergence estimator θ φ, in oter words φ x = x log x x + 1, we obtain te classical Pearson ci-square, X θ, and te likeliood-ratio test statistic, G θ. In te following teorem we establis tat te asymptotic distribution of te family of φ- divergence test statistics, T φ 1,φ, is a ci-square wit k t degrees of freedom χk t. Terefore, we do not accept 5 if T φ 1,φ > c, were c is specified so tat te size of te test is α, Pr Q > c H ull = α, i.e. Pr χk t c = 1 α were c χk t 1 α is te 1 α- t quantile of a χk t distribution. Teorem 4. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose te function φ 1, φ Φ, were Φ is defined in Section 1. Ten, for testing H ull : log m CX against H Alt : log m CX, 0 te asymptotic null distribution of te test statistic T φ 1,φ, given in 19, is ci-squared wit k t degrees of freedom. Proof. We consider te function f x, y = xφ 1 y/x. A second-order Taylor s expansion of f n i, m i θ φ about mi θ 0, mi θ 0 gives ni f, m i θ φ = φ 1 1 ni m i θ φ mi θ + o P 1 ; i = 1,..., k. 0 Taking into account were T φ 1,φ = φ 1 1 k f ni, m i θ φ = Z i = n i m i θ φ ; i = 1,..., k, mi θ 0 we obtain te following vectorial expression: wit T φ 1,φ = Z T Z + o P 1, Z = D 1 mθ 0 n m θ φ. k Zi + o P 1, Te random vector Z is asymptotically normally distributed wit mean vector zero and variance covariance matrix T = I k k Aθ 0, wit Aθ 0 = D 1 m θ 0 XXT D m θ 0 X 1 X T D 1 m θ 0. 1

8 . Martín, L. Pardo / Journal of Multivariate Analysis Ten, te asymptotic distribution of te φ-divergence test statistic T φ 1,φ will be a ci-square iff te matrix T is idempotent and symmetric. It is clear tat T is symmetric, and to establis tat T = I k k Aθ 0 it will be enoug to see tat Aθ 0 is an idempotent matrix. Te degrees of freedom of te ci-squared distributed statistic T φ 1,φ coincide wit te trace of te matrix T, i.e. k t. For ypotesis test 0 referred to previously it would be interesting to know te asymptotic distribution of 19 for te case were H ull is not true. Consider te vector of means m mθ 0 + d, log mθ 0 C X and θ 0 unknown, were d d 1,..., d k T is a fixed k 1 vector suc tat k d i = 0 for Poisson and multinomial sampling, and M i=m 1 d i = 0, = 1,..., r, for product-multinomial sampling. It can be seen tat = k m i θ 0 = k m i for te tree kinds of sampling and = M i=m 1 m i θ 0 = M i=m 1 m i, = 1,..., r, for product-multinomial sampling. As, te sequence of normalized vectors of means { m } converges to a normalized vector of means m θ 0 in H ull at a rate O 1/. We call H Alt, : m mθ 0 + d, log mθ 0 C X and θ 0 unknown, 3 a sequence of contiguous alternative ypoteses, ere contiguous to te null ypotesis. ow consider testing H ull against H Alt, ; te power of te test statistic T φ 1,φ is π Pr T φ 1,φ > c HAlt,. In wat is to follow, we sow tat under te contiguous alternative ypotesis H Alt,, and as, T φ 1,φ converges in distribution to a noncentral ci-squared random variable wit noncentrality parameter µ, were µ is given in Teorem 5, and k t degrees of freedom χk t,µ, i.e. π Pr χk t,µ > c. 4 Teorem 5. Suppose tat data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Under te alternative ypotesis, te statistic T φ 1,φ is ci-square wit k t degrees of freedom and noncentrality parameter µ = d T D 1 m θ 0 XXT D m θ 0 X 1 X T d. Proof. A second-order Taylor s expansion gives were T φ 1,φ = Z T Z + o P 1, Z = T D 1 mθ 0 n mθ 0 L µ, Σ, 5

9 1598. Martín, L. Pardo / Journal of Multivariate Analysis wit µ = T D 1 m θ 0 d and Σ = T. Te matrix Σ is idempotent and symmetric and its trace is k t. Te following lemma: Suppose tat Z is µ, Σ. If Σ is idempotent and Σµ=µ, te distribution of Z T Z is noncentral ci-square wit degrees of freedom equal to rank of te matrix Σ and noncentrality parameter µ = µ T µ, appears in Ferguson [9] page 63. Te result Σµ=µ follows because T is an idempotent matrix, Σµ = T T D 1 m θ 0 d = T D 1 m θ 0 d = µ. ow we are going to get te noncentrality parameter, µ T µ = d D T 1 m θ 0 I kd 1 m θ 0 1 D m θ 0 Aθ 0D 1 m θ 0 d = µ. ow te result follows. Remark 6. Teorem 5 can be used to obtain an approximation to te power function of T φ 1,φ for testing 0, as follows. Write, m θ φ = n + 1 m θ φ n and define m m θ φ + d, were d = 1 m θ φ n. Ten substitute d in place of d in te definition of µ, and finally µ into te rigt side of 4. Goodness-of-fit statistics provide only broad summaries of ow models fit data. Te pattern of lack of fit is revealed in cell-by-cell comparisons of observed and fitted cell counts. One can study te quality of fit in a more precise way. Te residuals can sow wy a model fits poorly, peraps suggesting an alternative model, or igligt cells tat display lack of fit in a model wose fit is generally good. Te standardized and deviance residuals based on X θ φ and G θ φ are given respectively by and r S i θ φ = n i m i θ φ m i θ φ 1 r D i θ φ = signn i m i θ φ n i 6 1/ n i log n i m i θ φ. 7 m i θ φ Te expressions for te residuals given in 6 and 7 are an extension of classical residuals tat we can obtain by replacing te maximum likeliood estimator by te minimum φ-divergence estimator θ φ. Ten it is intuitively clear tat we can define a family of residuals based on te expression of T φ 1,φ in te following way. Definition 7. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose te function φ 1, φ Φ, were Φ is defined in Section 1. We sall define φ 1 -residuals based on te minimum φ -divergence estimator as r φ 1 i θ φ = signn i m i θ φ φ 1 1m i θ φ φ 1 n i m i θ φ 1/ ; i = 1,..., k.

10 . Martín, L. Pardo / Journal of Multivariate Analysis ote tat te sum of te squares of φ 1 -residuals based on te minimum φ -divergence estimator is precisely T φ 1,φ. For φ 1 x = 1 x 1 and φ 1 x = x log x x + 1 we get ri S θ φ and ri D θ φ respectively as particular cases of r φ 1 i θ φ, and if we replace te maximum likeliood estimator by te minimum φ-divergence estimator θ φ, in oter words if φ x = x log x x + 1, ten we obtain te classical standardized and deviance residuals. In te following teorem we sall obtain te asymptotic distribution of r φ 1 i θ φ. Teorem 8. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose te function φ 1, φ Φ, were Φ is defined in Section 1. Te asymptotic distribution of φ 1 -residuals based on te minimum φ -divergence estimator, r φ 1 i θ φ, i = 1,..., k, is normal wit mean zero and variance 1 a ii θ 0, were a ii θ 0 is te i-t diagonal element of te square matrix Aθ 0 defined in 1. Proof. Since r φ 1 i θ φ = Z i + o P 1; i = 1,..., k, by Slutsky s Teorem, r φ 1 i θ φ beaves asymptotically like Z i. Te standardized residuals 6 are also known as adjusted residuals. Te term adjusted was introduced to distinguis tese from te Pearson residuals ri S θ φ /1 a ii θ φ 1 because Pearson s residuals are often referred as standardized residuals. On te basis of Teorem 8 we can define te standardized φ 1 -residuals based on te minimum φ -divergence estimator in te following way: Definition 9. Suppose tat te data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Coose te function φ 1, φ Φ, were Φ is defined in Section 1. We sall define standardized φ 1 -residuals based on te minimum φ -divergence estimator as r φ 1 i θ φ = r φ 1 i θ φ ; i = 1,..., k, 1 a ii θ φ were te r φ 1 i θ φ were given in Definition 7. Remark 10. In te particular case were we consider multinomial sampling we get a ii θ φ = e T i D 1 m θ φ XXT D m θ φ X 1 X T D 1 m θ φ e i; i = 1,..., k, i.e., we get te standardized φ 1 -residuals based on te minimum φ -divergence estimator obtained in Gupta et al. [10]. Terefore Definition 9 is an extension of te definition given in Gupta et al. [10] in te sense tat it is valid not only for multinomial sampling but also for te tree different kinds of sampling scemes simultaneously. 4. φ-divergence test statistics and nested ypoteses In tis section we are going to consider nested sequences of ierarcical log-linear models CX l CX l 1 CX 1, 8

11 1600. Martín, L. Pardo / Journal of Multivariate Analysis for eac sceme of sampling, Poisson, multinomial, and product-multinomial. Ten as in 5 8 we ave an assumption for multinomial or product-multinomial sampling CX 0 CX l, and anoter one for Poisson sampling J k CX l. Defining t = rank X, notice tat t +1 < t, = 1,..., l 1, rank X 0 = r < t l if te sampling is multinomial or product-multinomial, and t l > 1 if te sampling is Poisson. Taking a sequence of nested ypoteses H : log mθ 0 CX ; = 1,..., l, 9 our goal is to present φ-divergence test statistics to test successively H ull : H +1 against H Alt : H H +1 ; = 1,..., l 1, 30 were we continue to test as long as te null ypotesis is accepted and we infer an integer m, suc tat 1 m l, to be te first value for wic H +1 is rejected as null ypotesis. For testing 30 te classical likeliood-ratio test is given by G θ +1 θ = k m i θ log m i θ m i θ +1 m i θ + m i θ +1 = D Kull n, m θ +1 D Kull n, m θ, 31 were θ +1 and θ are te maximum likeliood estimators of θ under H ull and H Alt respectively. It is easy to prove see page 3 in Cristensen [3] tat G θ +1 θ = D Kull m θ, m θ On te basis of 31 and 3 we ave introduced in tis paper two families of φ-divergence test statistics for testing 30. Tese two families are given by and S φ 1,φ = φ 1 1 D φ1 n, m θ φ +1 D φ 1 n, m θ φ T φ 1,φ = φ 1 1 D φ 1 m θ φ, m θ φ +1, 34 were θ φ and θ φ +1 are defined by 13. For eac test statistic Q {T φ 1,φ, S φ 1,φ }, we reject H ull in 30 if Q > c, were c is specified so tat te size of te test is α, Pr Q > c H +1 = α. Te special case of φ 1 x = 1 x 1 and φ x = x log x x + 1 yields in S φ 1,φ a statistic based on te difference of Pearson ci-square statistics wit maximum likeliood estimation used to obtain te expected frequencies e.g., Agresti [1] page 197, namely k n i m i θ m i θ k n i m i θ +1. m i θ +1 33

12 . Martín, L. Pardo / Journal of Multivariate Analysis However, te nonnegativity of S φ 1,φ does not old wen φ 1 φ. Tus, for te case above, considered by Agresti [1], te difference of te Pearson ci-square statistics is not necessarily nonnegative. In te following we sall consider φ 1 = φ = φ and we denote te family of test statistics S φ 1,φ by S φ. ow it is clear tat for all, Sφ 0, because D φ n, m θ φ +1 D φn, m θ φ. otice tat wen φ 1 x = φ x = x log x x + 1, we obtain te usual likelioodratio test from bot S φ 1,φ and T φ 1,φ test statistics, and tat wen φ 1 x = 1 x 1 and φ x = x log x x + 1 we obtain te Pearson test statistic from T φ 1,φ. Terefore te family contains as a particular case te classical likeliood-ratio test as well as te Pearson test T φ 1,φ statistic wile te S φ only contains te classical likeliood-ratio test. Tis interesting fact as motivated to us to develop te simulation study using te family T φ 1,φ. In te next teorem we establis tat under H ull : H +1, te test statistics T φ 1,φ and S φ 1,φ converge in distribution to a ci-square wit t t +1 degrees of freedom χ t t +1 ; = 1,..., l 1. Tus, c could be cosen as c χ t t +1 1 α. Teorem 11. Suppose tat data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. Consider te nested sequence of ypoteses given in 8. Coose functions φ 1 and φ Φ. Ten for testing 30, te asymptotic null distribution of te φ- divergence test statistics T φ 1,φ and S φ 1,φ, given in 34 and 33, is a ci-square wit t t +1 degrees of freedom. Proof. A Taylor s expansion similar to te one given in Teorem 4 yields were T φ 1,φ = Z T Z +o P 1, 35 Z = D 1 mθ 0 m θ φ m θ φ +1 is distributed asymptotically as a normal distribution wit mean vector zero and variance covariance matrix T = A θ 0 A +1 θ 0 wit A θ 0 = D 1 m θ 0 X X T D m θ 0 X 1 X T D 1 m θ Ten, te asymptotic distribution of te φ-divergence test statistic T φ 1,φ will be a ci-square because te matrix T is idempotent and symmetric. Te degrees of freedom of te ci-squared distributed statistic T φ 1,φ coincide wit te trace of te matrix T, i.e. t t +1. Similar arguments give te asymptotic distribution of S φ 1,φ. To test te nested sequence of ypoteses {H : = 1,..., l} referred to previously, we need an asymptotic independence result for te sequence of test statistics T φ 1,φ 1, T φ 1,φ,..., T φ 1,φ m or S φ 1,φ 1, S φ 1,φ,..., S φ 1,φ m, were m is te integer 1 m l for wic H m is true but H m+1 is not true. Tis result is given in te teorem below.

13 160. Martín, L. Pardo / Journal of Multivariate Analysis Teorem 1. Suppose tat data n = n 1,..., n k T are Poisson, multinomial, or productmultinomial distributed. We first test H ull : H against H Alt : H 1, and ten H ull : H +1 against H Alt : H. Ten, under te ypotesis H +1, te statistics T φ 1,φ 1 and T φ 1,φ S φ 1,φ 1 and S φ 1,φ are asymptotically independent. Proof. A second-order Taylor s expansion gives were T φ 1,φ s = Z T s Z s + o P 1; s = 1, 37 Z s = T s D 1 mθ 0 n mθ 0. By Teorem 4 in Searle [16] te quadratic forms T φ 1,φ s = 1 n mθ 0 T D 1 m θ 0 T s D 1 are asymptotically independent if m θ 0 1 n mθ 0 ; s = 1, D 1 m θ 0 T D 1 m θ 0 D m θ 0 W 0 D 1 m θ 0 T 1 D 1 m θ 0 = O ko T k. 38 We ave to take into account an important property; if CX v CX w ten A v θ 0 A w θ 0 = A v θ 0. Due to 39, we ave T T 1 = A θ 0 A 1 θ 0 + A θ 0 A 1 θ 0 A +1 θ 0 A θ 0 = O k O T k, and te first term of 38 is zero; ten for Poisson sampling, as W 0 is zero, 38 olds. For product-multinomial sampling we need to prove tat te second term of 38 is zero, i.e. were D 1 m θ 0 T A 0θ 0 T 1 D 1 m θ 0 = O ko T k, 40 A 0 θ 0 = D 1 m θ 0 W 0D 1 m θ 0 = D 1 m θ 0 X 0X T 0 D m θ 0 X 0 1 X T 0 D 1 m θ 0. We know, as was assumed for te model, tat CX 0 CX 1 ; terefore by 39 we get A 0 θ 0 T 1 = A 0θ 0 A 0 θ 0 = O k O T k, and 40 olds for multinomial and product-multinomial sampling. In a similar way one can obtain te result for S φ 1,φ. In general, teoretical results for te test statistic T φ 1,φ or S φ 1,φ under alternative ypoteses are not easy to obtain. An exception to tis is wen tere is a contiguous sequence of alternatives as we sow in te sequel in tis section. Consider te vector of means. ow we take d +1 instead of d. As, te sequence { m } converges to a normalized vector of means m θ 0 in H +1 at te rate of O 1/. We call H +1, : m mθ 0 + d +1, log mθ 0 CX +1 and θ 0 unknown, 41 39

14 . Martín, L. Pardo / Journal of Multivariate Analysis a sequence of contiguous alternative ypoteses, ere contiguous to te null ypotesis H +1. ow consider testing H ull : H +1 against H Alt : H +1,, using te test statistic Q {T φ 1,φ, S φ 1,φ }. In wat is to follow, we sow tat under te alternative H +1,, and as, Q converges in distribution to a noncentral ci-squared random variable wit noncentrality parameter µ, were µ is given in Teorem 13, and t t +1 degrees of freedom χt t +1,µ. Consequently, under H +1, as, π χ Pr t t +1,µ > c. 4 Teorem 13. Suppose tat n = n 1,..., n k T are Poisson, multinomial, or product-multinomial distributed. Te asymptotic distribution of te statistics T φ 1,φ and S φ 1,φ, under te contiguous alternative ypoteses 41, is ci-square wit t t +1 degrees of freedom and noncentrality parameter µ = d T X X T D m θ 0 X 1 X T X +1X+1 T D m θ 0 X +1 1 X+1 T d. Proof. By 37, we ave T φ 1,φ wit = Z T Z + o P 1, were Z = T 1 D mθ 0 n mθ L 0 µ, Σ, µ = T D 1 m θ 0 d and Σ = T. Te result olds following te same steps as in te proof of Teorem 5. In a similar way one can obtain te result for S φ 1,φ. Remark 14. Teorem 13 can be used to obtain an approximation to te power function of T φ 1,φ for testing 30, as follows. Write, m θ φ = m θ φ m θ φ m θ φ +1 and define m m θ φ +1 + d, were d = 1 m θ φ m θ φ +1. Ten substitute d in place of d in te definition of µ, and finally µ into te rigt side of Simulation study In tis section we are going to present a simulation study designed to analyze te beavior of te new family of estimators, θ φ, as well as te new family of test statistics, T φ 1,φ, introduced and studied in tis paper. In tis simulation study we pay special attention to te family of power divergence measures introduced by Cressie and Read [7]. Tis family of divergences is a particular case of 10 if we consider te family of functions x λ+1 x λ x 1, λ 0, 1, φ λ x = λ λ x log x x + 1, λ = 0, log x + x 1, λ = 1. Te power divergence measure associated wit λ = 0 is te Kullback Leibler divergence. If we apply 43 in 13 we get te family of minimum power divergence estimators, θ φλ, and if we

15 1604. Martín, L. Pardo / Journal of Multivariate Analysis apply 43 in 19 we obtain te following power divergence test statistics based on contingency tables wose expression is given by j=1 =1 n λ 1 +1 i j m λ 1 n + λ j=1 =1 i j θ φ λ 1 φ λ n, λ 1 0, 1, λ 1 λ T φλ1 θ φ λ = n i j n i j log j=1 =1 m i j θ φ n + φ λ, λ λ 1 = 0, n i j m i j θ φ λ m i j θ φ, λ λ 1 = 1, wit φ λ = j=1 =1 m i j θ φ λ and n = j=1 =1 n i j. It is interesting to observe tat θ φ0 θ corresponds to te maximum likeliood estimator, θ φ1 to te minimum cisquare estimator, θ φ /3 to te Cressie Read estimator and θ φ 1/ to te Freeman Tukey estimator. In relation to 44, wen λ 1 = 0, φ 0 = n olds, and terefore we get te likeliood-ratio test for λ 1 = λ = 0 and te Pearson s ci-square test statistic for λ 1 = 1 and λ = 0. Te expressions for tese test statistics were given in 17 and 18, respectively. We sall consider Poisson sampling and te following teoretical models: log m i jk θ = u + θ 1i + θ j + θ 3k + θ 1i j, i, j, k = 1,, 45 were θ 1i = j=1 θ j = k=1 θ 3k = 0, j=1 θ 1i j = 0, i = 1,, and θ 1i j = 0, j = 1,. In particular, we ave taken model 45 wit θ 11 = , θ 1 = 0.670, θ 31 = , θ 111 = , and te u parameter is implicitly fixed from te different values of te expected total = 100, 00, 300, 400. In order to compare te different estimators, we use te mean squared error criterion and we simulate R = 50, 000 samples from te teoretical model considered previously. We compute te estimator of θ = u, θ 11, θ 1, θ 31, θ 111 T θ 1, θ, θ 3, θ 4, θ 5 T, l θ φ λ = l θ φ λ 1, l θ φ λ, l θ φ λ 3, l θ φ λ 4, l θ φ λ 5 T, l = 1,,..., R, for λ = 1, 0, 3, 1,, and we compare different samples wit te true parameter θ, = 1,..., 5, using te mean squared error MSE φ λ θ = 1 R R l θ φ λ θ, = 1,..., 5. l=1 Te joint mean squared error associated wit te true parameter θ is given by MSE φ λ θ = =1 MSE λ θ. In Table 1 we list te MSE φ λ θ 44 for λ = 1, 0, 3, 1,. In Table we present te joint mean, 1, and = 100, 00, 300, 400. square errors, MSE φ λ mθ, for λ = 1, 0, 3 We can observe in Table tat te estimators associated wit λ = 3 te Cressie Read estimator and λ = 1 te minimum ci-squared estimator ave better beaviors tan te maximum likeliood estimators for te expected total values under consideration. In Table 1

16 . Martín, L. Pardo / Journal of Multivariate Analysis Table 1 Joint mean squared error, MSE φ λ θ λ / Table Joint mean squared error, MSE φ λ mθ λ / te beaviors of te maximum likeliood estimator λ = 0 and te minimum power divergence estimators associated wit λ = 3 and λ = 1 are quite similar because tey depend on te expected total values. To study te beavior of te power divergence test statistics we are going to consider te following testing problem: H ull : te model is 45 against H Alt : te model is not Te essence of our simulation study is to obtain te exact probabilities = Pr T φ λ 1 > χ3 1 α H ull α φ λ 1 and β φ λ 1 = Pr T φ λ 1 > χ3 1 α H Alt. In fact, α φ λ 1 and β φ λ 1 are estimated using R = 50,000 simulations. In Table 3 we present te simulated exact sizes α φ λ 1. First of all, we study te closeness of te exact size to te nominal size α = Following Dale [8], we consider te inequality logit1 α φ λ 1 logit1 α ɛ, 47 were logitp logp/1 p. Te two probabilities are considered to be close if tey satisfy 47 wit ɛ = 0.35 and fairly close if tey satisfy 47 wit ɛ = 0.7. ote tat for α = 0.05, ɛ = 0.35 corresponds to α φ λ 1 [0.0357, ] and ɛ = 0.7 corresponds to α φ λ 1 [0.054, , ]. Tose exact sizes wic are taken as

17 1606. Martín, L. Pardo / Journal of Multivariate Analysis Table 3 Study for te size of te test statistics given in 44 λ 1 = 100 = 00 λ = 0 λ = /3 λ = 1 λ = 0 λ = /3 λ = / = 300 = 400 λ = 0 λ = /3 λ = 1 λ = 0 λ = /3 λ = / Table 4 Values of te parameters for different 48 alternative ypoteses Models θ 11 θ 1 θ 31 θ 111 θ 1311 Model A Model B Model A Model B Model A Model B Model A Model B close according to te Dale s criterion ave been marked in bold in Table 3, and te fairly close exact sizes in italics. In relation to te powers, we ave considered te following alternative ypoteses. By adding a sixt θ 13ik 0 parameter wit k=1 θ 13ik = 0, i = 1,, and θ 1ik = 0, k = 1,, and te constraints of 45, we ave log m i jk θ = u + θ 1i + θ j + θ 3k + θ 1i j + θ 13ik, i, j, k = 1,, 48 wic are as close to H ull as we want, as θ 1311 decreases to zero. In Table 4 selected parameters for te 48 model are sown. Models wit te same subindex are expected to ave similar values of α and β for a fixed ; in fact tey ave an equidistant θ 1311 value wit respect to zero. On te oter and, 1,, 3, 4 subindexes are related to te absolute values of θ 1311, i.e. 0.1, 0.3, 0.5, 0.7 respectively, A being te model wose value of θ 1311 is negative and B te one wose value is positive.

18 . Martín, L. Pardo / Journal of Multivariate Analysis Fig. 1. β φ 1,A, β φ 1,A 1, α φ 1, β φ 1,B, β φ 1 1,B case wit = 00, λ = 0. In Fig. 1, on te ordinate axis we present, wit te left points, te exact powers of Models A, A 1, wit te middle point, te exact size of model 45 and, wit te rigt points, te exact powers of Models B 1, B for = 00 expected total value and te maximum likeliood estimator λ = 0. On te abscissa axis we ave te value of θ 1311 for eac model and in te case of te null ypotesis, θ 1311 = 0. Te different types of lines are related to te different statistics:, λ 1 = 0.5;, λ 1 = 0; +, λ 1 = /3;, λ 1 = 1;, λ 1 =. Observe tat te beaviors of te powers for power divergence test statistics based on λ 1 = 3 and 1 are quite similar. If we consider λ = 3 or λ = 1 we get similar results and altoug we do not sow te figures for oter values, as expected te lines become closer for = 300, 400 and more separated for = 100. An overall measure of performance may elp us to obtain a better interpretation of grapical results presented in te previous figures. For tis purpose te size corrected average gradient is defined as g λ 1,λ = 1 α φ λ k {A i,b i } 4 β φ λ 1 k α φ λ 1 θ 1311 k wic measures te normalized mean rate of power gain, wit respect to te null ypotesis H 0, along te considered alternatives for more details, see Molina and Morales [14] and references terein. Obviously, te proposed criterion is to select te test statistics T φ λ 1 wit g λ 1,λ = arg max λ1,λ g λ 1,λ. Te results for λ 1 = 0.5, 0, /3, 1 and and λ = 0, /3 and 1 as well as for = 100, 00, 300 and 400 are listed in Table 5. If we consider λ = 0, i.e. te maximum likeliood estimator, we can observe tat, except for = 400, te test statistic associated wit λ 1 = 3 as te best beavior. For λ = 3, except for = 100, te test statistic associated wit λ 1 = 1 as te best beavior. Finally if we consider λ = 1, i.e. te minimum ci-square estimator, we get tat te best test statistic is obtained for λ 1 = 1. If we do not ave te estimation procedure fixed we can observe, except for = 100, tat λ 1 = 1, λ = 1 is te best coice for testing and estimation, i.e. considering te Pearson test statistic in wic te known parameters are estimated using te minimum ci-square estimator.,

19 1608. Martín, L. Pardo / Journal of Multivariate Analysis Table 5 Size corrected average gradient values λ λ / / / / / Te second coice is λ 1 = 1, λ = 3, i.e. considering te Pearson test statistic in wic te parameters are estimated using te minimum power divergence estimator wit λ = Conclusions Statistical inferences for log-linear models involve te estimation of parameters in te model, goodness-of-fit testing and coosing te most convenient model from a nested sequence of loglinear models. From a classical point of view te unknown parameters are generally estimated by te metod of maximum likeliood. Pearson and likeliood-ratio ci-square tests provide overall measures of te compatibility of te model and te data as well as a procedure for coosing a model from a nested sequence of log-linear models. Cressie and Pardo [4] presented for te first time te minimum φ-divergence estimator for loglinear models and multinomial sampling wic is seen to be a generalization of te maximum likeliood estimator. Tis estimator is ten used in a φ-divergence goodness-of-fit statistic, wic is te basis of a new statistic for solving te problem of testing a nested sequence of log-linear models. Te adequacy of tis new statistic is establised in Cressie et al. [6]. In Cressie and Pardo [5], an extension to Poisson and product-multinomial sampling is presented using te maximum likeliood estimator. Gupta et al. [10] defined a family of residuals based on φ- divergence measures for multinomial log-linear models. Te main purpose of tis paper is to extend te previous results for Poisson, multinomial and product-multinomial sampling simultaneously using te minimum φ-divergence estimator. We can observe tat tis as not been considered until now. Te previous papers pay special attention to te minimum φ-divergence estimator and multinomial sampling or te maximum likeliood estimator and Poisson, multinomial or product-multinomial sampling. In order to carry out te study we ave ad to consider φ-divergence measures between mean vectors instead of between probability vectors. Assuming Poisson, multinomial or product-multinomial sampling in a log-linear model, we present in tis paper te most general asymptotic result possible for minimum φ-divergence estimation, divergence-based goodness-of-fit testing including φ- divergence residuals based on minimum φ-divergence estimators and coosing a model from a nested sequence of ypoteses. Any estimator or statistic tat is divergence based is covered

20 . Martín, L. Pardo / Journal of Multivariate Analysis by our results. A simulation study is carried out in order to compare members of te class of power divergence statistics wit te minimum power divergence estimation. It is found tat some members of te family of test statistics based on power divergence measures offer an attractive alternative to te Pearson-based and te likeliood test statistics, in terms of bot simulated exact size and simulated exact power. Acknowledgments Te autors tank te referees for comments and suggestions wic ave improved te paper. References [1] A. Agresti, An Introduction to Categorical Data Analysis, Jon Wiley & Sons, ew York, [] A. Agresti, Categorical Data Analysis, Jon Wiley & Sons, ew York, 00. [3] R. Cristensen, Log-Linear Model and Logistic Regression, Springer-Verlag, ew York, [4]. Cressie, L. Pardo, Minimum φ-divergence estimator and ierarcical testing in loglinear models, Statistica Sinica [5]. Cressie, L. Pardo, Model cecking in loglinear models using φ-divergences and MLEs, Journal of Statistical Planning and Inference [6]. Cressie, L. Pardo, M.C. Pardo, Size and power considerations for testing loglinear models using φ-divergence test statistics, Statistica Sinica [7]. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, Journal of te Royal Statistical Society Series B [8] J.R. Dale, Asymptotic normality of goodness-of-fit statistics for sparse product multinomials, Journal of te Royal Statistical Society Series B [9] T.S. Ferguson, A Course in Large Sample Teory, Texts in Statistical Science, Capman & Hall, London, [10] A.K. Gupta, T. guyen, L. Pardo, Residual analysis and outliers in loglinear models based on pi-divergence statistics, Journal of Statistical Planning and Inference [11] S.J. Haberman, Te Analysis of Frequency Data, University of Cicago Press, Cicago, [1] J.B. Lang, On te comparison of multinomial and Poisson log-linear models, Journal of te Royal Statistical Society Series B [13]. Martín, L. Pardo, Minimum Pi-divergence estimators for loglinear models wit linear constraints and multinomial sampling, Statistical Papers [14] I. Molina, D. Morales, Rényi statistics for testing ypoteses in mixed linear regression models, Journal of Statistical Planning and Inference [15] L. Pardo, Statistical Inference based on Divergence Measures, in: Textbooks and Monograps, Capman & Hall/CRC, ew York, 006. [16] S.R. Searle, Linear Models, Jon Wiley & Sons, ew York, 1971.