On Perturbed Symmetric Distributions Associated with the Truncated Bivariate Elliptical Models

Transcription

1 Communications of the Korean Statistical Society Vol. 15, No. 4, 8, pp On Perturbed Symmetric Distributions Associated with the Truncated Bivariate Elliptical Models Hea-Jung Kim 1) Abstract This paper proposes a class of perturbed symmetric distributions associated with the bivariate elliptically symmetricor simply bivariate elliptical) distributions. The class is obtained from the nontruncated marginals of the truncated bivariate elliptical distributions. This family of distributions strictly includes some univariate symmetric distributions, but with extra parameters to regulate the perturbation of the symmetry. The moment generating function of a random variable with the distribution is obtained and some properties of the distribution are also studied. These developments are followed by practical examples. Keywords: Perturbed symmetric distribution; truncated bivariate elliptical distribution; Skew-elliptical distribution. 1. Introduction A distribution of k 1 random vector X, written X EC k θ, Σ, g k) ), is said to have k-variate elliptically symmetricor simply elliptical) distribution with location vector θ R k and a k k positive definite) dispersion matrix Σ and the density generating function g k). The density of X distribution is given by fx θ, Σ) = Σ 1 g k) [ x θ) Σ 1 x θ) ], 1.1) for some density generator function g k) u), u, such that ) k Γ u k 1 g k) u)du =. 1.) By varying the function g k), distributions with longer or shorter tails than the k-variate normal can be obtained. The density 1.1) has appeared at various places in the literature, sometimes in a somewhat casual manner. A systematic treatment of this distribution has been given by Fang et al. 199) and Fang and Zhang 199). Various univariate/multivariate extensions of the distribution is considered by Azzalini 1985), This work was partially supported by the Korea Research Foundation Grant funded by the Korean Government MOEHRD) KRF C16). 1) Professor, Department of Statistics, Dongguk University, 3 Pil-Dong, Chung-Gu, Seoul 1-715, Korea. kim3hj@dongguk.edu π k

2 484 Hea-Jung Kim Branco and Dey 1), Azzalini and Capitanio 3), Ma and Genton 4), Arellano- Valle et al. 6) and Kim 7, 8) among others. The major goal of this paper is to study a class of perturbed symmetric distributions as an extension of univariate elliptically symmetric distribution. The class is obtained from deriving the distribution of Z [X a < Y < b], 1.3) where X, Y ) is a bivariate elliptical variable with the location parameter θ 1, θ ) and scale matrix Σ. Although cases of perturbed distributionsobtained from a conditioning mechanism) are well addressed in the literature, so far as we know, there are few results concerning the class of perturbed distributions associated with a doubly truncated bivariate elliptical distribution. This motivates the investigation in this article. The interest in studying the distribution 1.3) comes from both theoretical and applied directions. On the theoretical side, it provides a class of distributions that enjoys a number of formal properties which resemble those of the elliptical distributions as given in Fang and Zhang 199) and produces a general class of distributions that strictly includes many symmetric distributions as well as the univariate skewed distributions considered by Arellano-Valle et al. 6). From the applied viewpoint, the distribution is a unimodal empirical distribution with presence of skewness and possibly heavyor light) tail see, Figure 3.1). This implies that 1.3) is useful to modeling random phenomena which have heavieror lighter) tails than the normal as well as having some skewness. Moreover, the class of distributions obtained from 1.3) provides yet other models that enable us to analyze a screened data in terms of the sum of truncated and untruncated observations.. Preliminaries Prior to suggesting the class of perturbed symmetric distributions, we provide lemmas useful for studying property of the class. If we set Ψ = {ψ ij } with ψ 11 = ψ = 1 and ψ 1 = ψ 1 = ρ, the properties of EC, Ψ, g ) ) distribution yield following theorems. From now on, we will use Ψ to denote this standard form of dispersion matrix. Lemma.1 Let W EC, Ψ, g ) ), W = W 1, W ). Then the conditional distribution of W given that W 1 = w is EC 1 α, β, g qw) ) distribution. Here α = ρw, β = 1 ρ, g qw) = g ) u + qw))/g 1) qw)), where g 1) u) = g ) r + u)dr and qw) = w. Proof: Straightforward application of the result of Branco and Dey 1, pp. 1) yields the result. Lemma. Let W EC, Ψ, g ) ), W = W 1, W ). If Z is set to equal to W 1 conditionally on a < W < b. Then the pdf of Z is f Z z) = f g 1)z) { F gqz) λ 1 b λz) F gqz) λ 1 a λz) }, for z R,.1) F g 1)b) F g 1)a) where λ = ρ/ 1 ρ, λ 1 = 1 + λ ) 1/ = 1/ 1 ρ and f g 1) ) and F g 1) ) are the pdf and the cdf of EC 1, 1, g 1) ), respectively. F gqz) is the cdf of EC 1, 1, g qz) ) with qz) = z.

3 Perturbed Distribution Associated with Truncated Elliptical Model 485 Proof: The pdf of Z can be expressed as f Z z) = P a < W < b z)f W1 z). P a < W < b) Using the property of EC, Ψ, g ) ) see, Fang et al. 199, p.43) we obtain the marginal distribution, W 1 EC 1, 1, g 1) ). Further Lemma.1 gives W W 1 = z EC 1 α, β, g qz) ) and hence P a < W < b z) = a α P < W α < b α ) z β β β = F gqz) λ 1 b λz) F gqz) λ 1 a λz). Noticing that f W1 z) = f g 1)z) and P a < W < b) = F g 1)b) F g 1)a), we have the result. 3. Perturbed Symmetric Distribution A generalization of Lemma. yields the following result. Let X = X, Y ) and let X EC θ, Σ, g ) ) with θ = θ 1, θ ), Σ = {σ ij }, σ ii = σi and σ 1 = ρσ 1 σ. Then, the pdf of Z [X a < Y < b] variable is fzz) = f g 1)u 1z)) { F λ gq z) 1ub) λu 1 z)) F λ gq z) 1ua) λu 1 z)) } { σ 1 Fg1) ub)) F g1) ua)) }, 3.1) for z R, where u 1 z) = z θ 1 )/σ 1, ua) = a θ )/σ, ub) = b θ )/σ, F gq z) is the cdf of EC 1, 1, g q z)) with q z) = u 1 z). The density of Z variable is obtained from.1) using the transformation relations X = σ 1 W 1 + θ 1 and Y = σ W + θ. From now on, we use the same notations without redefining. Definition 3.1 If a random variable Z has density function 3.1), then we say that Z is a perturbed symmetric random variable with parameters θ, Σ and a non-negative perturb function { Fgq z) λ 1ub) λu 1 z)) F λ gq z) 1ua) λu 1 z)) } pz; θ, σ, ρ) = For brevity we shall also say that Z is PS a,b) θ, Σ, g ) ). { Fg1) ub)) F. 3.) g 1) ua))} We see, from 3.1), that the perturbed distribution arise when the symmetric density f g 1)u 1 z))/σ 1 of potential observation z gets distorted so that it is multiplied by some non-negative perturb function pz; θ, σ, ρ). From Lemma.1 and 3.1), we can get an alternative and convenient expression for the pdf of PS a,b) θ, Σ, g ) ) distribution as f Zz) = ub) [ {r g ) ρu1 z)} ] ua) 1 ρ + u 1 z) dr σ { }, z R. 3.3) 1 1 ρ F g1) ub)) F g1) ua))

4 486 Hea-Jung Kim There are many subclasses of bivariate elliptical distributions, such as bivariate normals, bivariate t-distributions, symmetric bivariate Pearson types VII, bivariate scale mixture normal distributions and some others see, Fang et al. 199, for the others). These are the most useful subclasses from both practical and theoretical aspects. In the following examples, we see that 3.3) yields some perturbed symmetric distributions from these subclasses. Example 3.1 A Perturbed Normal Distribution) The generator function for a bivariate normal is g ) N u) = π) 1 exp u ). 3.4) Thus the density fz z) of Z PS a,b)θ, Σ, g ) N ) variable is obtained from 3.3) and it is fzz) = φu 1z)) {Φλ 1 ub) λu 1 z)) Φλ 1 ua) λu 1 z))}, z R. 3.5) σ 1 {Φub)) Φua))} This agrees with the density given in Kim 7). When a =, b =, θ = and Σ = Ψ, we see that 3.5) reduces to the density of skew-normal distribution SN ) by Azzalini 1985). Example 3. A Perturbed t ν Distribution) The generator function for a bivariate t-distribution with the degrees of freedom ν is g ) ν u) = C 1 ν + u) ν+, 3.6) where C 1 = ν ν/ Γ{ν + )/}/π Γ{ν/}). It follows from 3.3) that the pdf fz z) of PS a,b) θ, Σ, g ν ) ) distribution is fzz) = f νu 1 z)){f ν+1 v z)) F ν+1 v 1 z))}, z R, 3.7) σ 1 {F ν ub)) F ν ua))} where f ν ) and F ν ) denote the pdf and the cdf of a univariate standard t ν distribution, while F ν+1 ) is the cdf of a univariate standard t ν+1 distribution. Example 3.3 A Perturbed Pearson Type VII Distribution) A 1 random vector X is said to have a symmetric bivariate Pearson Type VII distribution if it has a density generator function g ) V II u) = C 1 + u m) N, N > 1, m > 3.8) and C = πm) 1 ΓN)/ΓN 1). Some analytic algebra plugging 3.8) in 3.3) leads to the pdf of PS a,b) θ, Σ, g ) V II ) distribution given by f Zz) = δσ 1 ) 1 f N 1,m u 1 z)){f N,m+1 w z)) F N,m+1 w 1 z))}, z R, 3.1) where δ = F N 1/,m ub)) F N 1/,m ua)), w 1 z) = {λ 1ua) λu 1 z)} m + 1 m + u1 z) and w z) = {λ 1ub) λu 1 z)} m + 1 m + u1 z).

5 Perturbed Distribution Associated with Truncated Elliptical Model 487 Here f N 1/,m ) and F N 1/,m ) denote the pdf and the cdf of a univariate standard Pearson type VII distribution with parameters N 1/ and m, while F N,m+1 ) is the cdf of a univariate standard Pearson type VII distribution with parameters N and m +1. The perturbed Pearson type VII distribution includes a number of important distributions such as the perturbed t distributionfor N=m/+1) and the perturbed Cauchy distributionfor m=1 and N=3/). Example 3.4 A Perturbed Scale Mixture of Normal Distribution) The generator function for a bivariate scale mixture of normal is { g ) H,K u) = {πkη)} 1 exp u } dhη), 3.11) Kη) where η is a mixing variable with the cdf Hη) and Kη) is a weight function. From 3.3), the pdf fz z) of the perturbed scale mixture of normal distribution, written PS a,b) θ, Σ, g ) H,K ), is given as 1 πδ 1 Kη)σ 1 1 ρ ub) ua) exp [ {r ρu 1z)} Kη)1 ρ ) + u 1z) ] r Hη). Kη) Transforming {r ρu 1 z)}/ Kη)1 ρ ) to w yields ) fzz) 1 u = 1 z) φ {Φu 1z)) Φu z))}dhη), 3.1) δ 1 σ 1 Kη) Kη) for z R, where δ 1 = ) ) ub) ua) Φ Φ dhη), Kη) Kη) u 1z) = λ 1ub) λu 1 z) Kη) and u z) = λ 1ua) λu 1 z) Kη). Note that 3.1) reduces to 3.5) when Hη) is degenerate with Kη) = 1. Also note that 3.1) is reduced to 3.7) when Kη) = 1/η with Hη) as the cdf of a gamma distribution, i.e., η Gν/, /ν) so that νη χ ν. Other combinations of Hη) and Kη) functions for the scale mixture of normal distribution are considered by Chen and Dey 1998) and Branco and Dey 1), among other. Those combinations can be applied to 3.1) to coming at various classes of perturbed symmetric distributions. For example, through the scale mixtures by Chen and Dey 1998), a class of perturbed stable densities, perturbed logistic densities and that of perturbed symmetric power densities can also be obtained from 3.1). 4. Property of the Distribution 4.1. Distributional properties Now we will state some interesting properties for the PS a,b) θ, Σ, g ) ) distribution as well as the associate examples. Some properties are focused on those of PS a,b) θ, Σ, g ) K,H )

6 488 Hea-Jung Kim Figure 3.1: Various Shapes of perturbed symmetric about zero densities with ρ =.8 a) the pdf of SN λ) PE, ), Ψ, g ) N ) distribution, where λ = ρ/ 1 ρ ; b) the pdf of PE.5,), Ψ, g ) N ) distribution; c) the pdf of PE.5,), Ψ, g ν ) ) distribution with ν = 3; d) the pdf of PE.5,), Ψ, g ) V II ) distribution with m = N = 7) which incudes several perturbed distributions obtained from well-known symmetric distributions. Property 4.1. If Z PS a,b) θ, Σ, g ) ), then Z PS a,b) θ, Σ, g ) ), where θ and Σ are obtained from θ and Σ by changing their elements θ 1 and ρ with θ 1 and ρ, respectively. Property 4.. Let UZ) = Z θ 1 )/σ 1, where Z PS a,b) θ, Σ, g ) ). Then Property 4.3. If Z PS θ, )θ, Σ, g ) ), its pdf is UZ) PS ua),ub)), Ψ, g ) ). 4.1) f Z z) = σ 1 f g 1)u 1 z))f gq z) λu 1 z)), for z R, 4.) where F gq z) ) is the cdf of EC 1, 1, g q z)) with q z) = u 1 z). The density 4.) does not depend on the values of θ and σ and it is equivalent to the univariate skew-elliptical distribution introduced by Branco and Dey 1). In

7 Perturbed Distribution Associated with Truncated Elliptical Model 489 this sense, Definition 3.1 extends the result of Branco and Dey 1) where they studied only for the case of θ =, σ = 1) and a =, b = ). When the relationship between λ and ρ in 4.) is considered, we see λ has the sign of ρ, ranges from to and is equal to zero if and only if ρ =. This directly gives the following property. Property 4.4. Suppose X EC θ, Σ, g ) ), where X = X, Y ), θ = θ 1, θ ), Σ = {σ ij } and σ 1 = ρσ 1 σ. If ρ = then [X Y > θ ] [X Y θ ] EC 1 θ 1, σ 11, g 1) ). 4.3) One can give a probabilistic representation of the distribution PS a,b) θ, Σ, g ) H,K ) in terms of a scale mixture of independent normal and truncated normal laws. The scale mixture scheme directly gives the following theorem. Theorem 4.1. Conditionally on η having the cdf Hη), let V Nµ 1, Kη)τ1 ) and W Nµ, Kη)τ ) be independent random variables. Then, for any real values a 1 and a with a 1, Z = a 1 V + a W Ia,b) PS a,b) θ, Σ, g ) H,K ), 4.4) where W Ia,b) denotes a truncated Nµ, Kη)τ ) variable and a and b are lower and upper truncation points, respectively. θ = θ 1, θ ) with θ 1 = i=1 a iµ i and θ = µ ; Σ = {σ ij }, σ ii = σi, with σ 11 = Kη) i=1 a i τ i, σ = Kη)τ and ρ = a σ /σ 1. Proof: Let U = a 1 V + a W. Then U, W ) is a bivariate normal variable with N θ, Σ) distribution. Conditionally on η, the distribution of U conditionally on a < W < b is PS a,b) θ, Σ, g ) N ), by Example 1. Given η, V and W in U = a 1V + a W are independent normal variables and hence the distribution of U given that a < W < b equals that of a 1 V + a W Ia,b) figuring in the statement of Theorem 4.1. Thus the scale mixture distribution 4.4) is directly obtained by using Example 4. When a =, Z is a scale mixture of normal distribution. Thus the representation in 4.4) reveals the structure of the class of PS a,b) θ, Σ, g ) H,K ) distributions and indicates the kind of departure from the symmetric distribution. Furthermore, the representation provides one-for-one method of generating a random variable Z with density 3.1). For generating the truncated normal variable W Ia,b), the one-for-one method by Devroye 1986) may be used. Corollary 4.1 Conditionally on η, let V and W be independent N, Kη)) random variables and let X, Y ) N, Kη)Ψ), where η is a random variable with the cdf Hη) and weight function Kη). Then 1 V + λ W 1 + λ 1 + λ Ia,b) [X a < Y < b] PS a,b), Ψ, g ) H,K ), 4.5) where λ = ρ/ 1 ρ. Proof: Setting µ 1 = µ = and τ 1 = τ = 1, we have the result from Theorem 4.1.

8 49 Hea-Jung Kim For Kη) = 1, we see that 4.5) is equivalent to the probabilistic representation by Kim 7). In the case a = and b =, if we set Kη) = 1 and Kη) = 1/η, PS a,b), Ψ, g ) H,K ) distributions reduce to SN λ) by Azzalini 1985) and the skew-t ν distribution, respectively. Their probabilistic representations obtained from 4.5) agree with those of SN λ) given in Henze 1986) and skew-t ν derived by Kim ). 4.. Moments of the perturbed distributions In this section we derive a general expression for the moment generating functionmgf) for the perturbed scale mixture of normal distribution in 3.1). To compute the moments of Z PS a,b) θ, Σ, g ) H,K ), it suffices to compute the moments of UZ) = Z θ 1)/σ 1. We see from 3.1) that UZ) has the density 1 guz)) = δ 1 Kη) ) uz) φ {Φu 1z)) Φu z))}dhη), 4.6) Kη) for z R. Some algebra gives that the moment generating function of UZ) = Z θ 1 )/σ 1 is M UZ) t) = 1 δ 1 E η [{Φ u t b)) Φ u t a))}e Kη) t ], 4.7) for t R, where E η denotes that the expectation is taken with respect to the distribution of η variate having the density dhη)/dη and u t a) = ua) ρ Kη) t, u t b) = ub) ρ Kη) t. Kη) Kη) Naturally, the moments of UZ) = Z θ 1 )/σ 1 can be obtained by using the moment generating function differentiation. For example: E[UZ)] = M UZ) t) t= = ρ [ ] E η Kη) {φu b)) φu a))}, 4.8) δ 1 where u a) = ua)/ Kη) and u b) = ub)/ Kη). Unfortunately, for higher moments this rapidly becomes tedious. An alternative procedure makes use of the fact that d dx [xk+1 φx)] = k + 1)x k φx) x k+ φx), 4.9) for k = 1,, 1,, 3,..., yields the following result. Let W = UZ)/ Kη). Under the distribution 4.6), the relation 4.9) and integrating by parts gives E[k + 1)W k W k+ ], that is ρ δ 1 λ k+1 1 [ φu b))e [V + λu b)] k+1 φu a))e [V + λu a)] k+1] dhη), 4.1) for k = 1,, 1,..., where V is a N, 1) variate.

9 Perturbed Distribution Associated with Truncated Elliptical Model 491 By setting k = 1,, 1, we obtain three expressions, which may be solved to yield the first three moments of UZ). Higher moments could be found similarly. One obtains, E[UZ)] = ρ [ ] E η Kη){φu b)) φu a))}, δ 1 [ ] E[UZ) ] =E η [Kη)] ρ E η Kη){u b)φu b)) u a)φu a))}, δ 1 E[UZ) 3 ] = ρ E η [Kη) 3 {3 ρ + u b)ρ )φu b)) δ 1 ] 3 ρ + u a)ρ )φu a))}. By using the Binomial expansion, one can see that the general formula for the moments of Z PE a,b) θ, Σ, g ) H,K ) is E[Z k ] = k j= ) k θ k j 1 σ j 1 j E[UZ)j ]. 4.11) When θ 1 =, E[Z k ] = σ1 k E[UZ) k ]. It is also noted that existence of the moments depends on the mixing distribution Hη). The class of perturbed scale mixture of normal distribution includes well-known skewelliptical distributions as in the following two examples. EXAMPLE 5. One member of the perturbed scale mixture normal distribution in 3.1) is the PS θ, )θ, Σ, g ) H,K ) distribution, for which H is degenerate with Kη) = 1. From 3.1), we obtain the pdf given by f Zz) = σ 1 φuz))φλu 1 z)), z R. 4.1) This distribution is equivalent to the skew-normal distribution, SN θ 1, σ 1, λ), by Azzalini 1985). From 4.11), one finds the moments, ) E[Z] = θ 1 + σ 1 ρ π, VarZ) = σ 1 1 ρ π and α 3 = ) 4 π 1 π ρ3 ) 3 1 ρ. 4.13) π These values of Z agree with those given in Arnold et al. 1993). See Azzalini 1985) and Henze 1986) for the other properties of the distribution. EXAMPLE 6. Another member of the perturbed scale mixture of normal distribution distribution is the skew-t ν distribution. When Kη) = 1/η and η Gν/, /ν), the pdf 3.1) of the PS θ, )θ, Σ, g ) H,K ) distribution reduces to fzz) = λu 1 z) ) ν + 1 f ν u 1 z))f ν+1, z R, 4.14) σ 1 ν + u1 z)

10 49 Hea-Jung Kim where f ν ) and F ν ) denote the pdf and the cdf of a univariate standard t ν distribution, while F ν+1 ) is the cdf of a univariate standard t ν+1 distribution. This distribution is equivalent to the skew-t ν distribution, ST θ 1, σ 1, ν, λ), by Kim ). From 4.11), we obtain, ν EZ) = θ 1 + σ 1 ρ π VarZ) = σ1 ν Γ {ν 1)/}, for ν > 1, Γ {ν/} ν EZ), for ν >, A α 3 =, VarZ) 3 for ν > 3, 4.15) where [ ν ) 1 )] A = σ1λ 3 3νΓ{ν 1)/} 1 π 1 + λ ) 3 ν )Γ{ν/} ν 3 + λ B, [ ] ν ) ν )Γ{ν 1)/} ρ B = + 3ν 3) 3πΓ{ν/} 1 and λ =. 1 ρ It is easily seen that the distribution does not depend on θ and σ and leads to a parametric class of distributions that have strict inclusion of t ν distribution for the case θ 1 =, σ 1 = 1 and ρ = ) and perturbed Cauchy distributionfor ν = 1). See Kim ) for the other properties and applications of the distribution 4.14). Note that B > by using a numerical evaluation for ν > 3. Thus 4.13) and 4.15) imply that the distributions in both examples are skewed to the rightthe left) when ρ > ρ < ). 5. Applications The aim of this section is to provide simple applications of the material presented so far, focusing on those of PS a,b) θ, Σ, g ) N ) and PS a,b)θ, Σ, g ν ) ) distributions Comparing elasticities in regression Suppose we have a model where quantity Q depends on price P, so that the demand function is Q = αp β u. 5.1) When we take logs of 5.1), we obtain log Q = log α + β log P + log u which is of the standard form Y = α + βx + e. 5.) In the econometric context, β is called the price elasticity of the demand Q. The price elasticity is formally defined as the relative change in quantity demanded divided by the relative change in price, i.e. β=percent change in Q)/percent change in P ). See Rudy ) for the analysis of the price elasticity to diagnose consumer expenditure. Let observational models of two quantitiesq 1 and Q ) in 5.1) be y ij = α i + β i x ij + e ij, i = 1, ; j = 1,..., n i, 5.3)

11 Perturbed Distribution Associated with Truncated Elliptical Model 493 iid where e ij N, τi ) and e 1j and e j are independent variables. Suppose, from the microeconomic theory, that the price elasticity of the demand for Q 1 is inelastic to price so that β 1, 1). Then the price elasticities of the demands for Q 1 and Q can be compared by the distribution of β /β 1 obtained from the Bayesian approach. We shall take a reference prior that is independently uniform in αi, β i and log τi, so that παi, β i, τi ) 1/τ i I < β 1 < 1). It is now clear that the marginal posterior distributions of β i s i = 1, ) are independent and β 1 Data T N b 1, τ 1 ) I < β 1 < 1) and β Data N b, τ ) S 11 S for known τi, i = 1,, β 1 Data T t ni ) b 1, s 1 S 11 ) I < β 1 < 1) and β Data t ni ) b, for unknown τi, where b i = S i /S ii, n i )s i = S Sy/S ii, S ii = n i j=1 x ij x i ), S i = n i j=1 y ij ȳ i )x ij x i ), S = n i j=1 y ij ȳ i ), x i = n i j=1 x ij/n i and ȳ i = ni j=1 y ij/n i. Here T Na, b)i < β 1 < 1) and T t ni 1)a, b)i < β 1 < 1) denote the normal and t ni 1) distributions truncated below at and above at 1, respectively. a and b are the location and scale parameters of each distribution. For known τ i, the posterior distribution function F c) of β /β 1 is the same as P Z ) obtained from Z PS,1) θ c, Σ c, g ) N ) by Theorem 4.1, where θ c = θ 1, θ ) with θ 1 = b cb 1 and θ = b 1 ; Σ c = {σ ij } with σ 11 = c τ1 /S 11 + τ /S, σ = τ1 /S 11, σ 1 = cτ1 /S 11 and ρ = σ 1 / σ 11 σ. Thus the distribution function Fc z) of PS,1) θ c, Σ c, g ) N ) variable in 3.5) leads to the distribution function F c) of β /β 1. That is F c) = F c ) = 1 L θ 1, θ ), ρ σ11 σ ) 1 θ Φ Φ σ L θ 1 σ11, 1 θ σ, ρ ) θ σ ), where Lα, β, ρ) denotes the orthant probability function of the standard bivariate normal variable defined in Johnson and Kotz 197). For the case τi is unknown but with large n i i = 1, ), an approximate posterior distribution function of β /β 1 can be obtained by substituting n i s i /n i ) for τi, i = 1, in the scale matrix Σ c of the PS,1) θ c, Σ c, g ) N ) distribution. 5.. Paired sample problem Let X 1, Y 1 ),..., X n, Y n ) be paired random sample from bivariate normal distribution, where CorrX i, Y i ) = ρ, X i Nµ X, σ iid iid ) and Y i Nµ Y, σ ). Then sampling distributions of n 1/ X µ X )/s p and n 1/ Ȳ µ Y )/s p are identically distributed as N, σ /s p) with s p/σ Gν/, /ν), where ν = n and { n } νs p = 1 ρ ) 1 X i X) n ρ X i X)Y n i Ȳ ) + Y i Ȳ ). i=1 i=1 i=1 s S )

12 494 Hea-Jung Kim Above result and the definition of the PS a,b) θ, Σ, g ) ν ) distributionalso see Example 6) immediately gives n 1 n 1 X µ X s p Ȳ > µy ST, 1, ν, λ), X µ X s p Ȳ < µy ST, 1, ν, λ), where λ = ρ 1 ρ. From 4.14), we obtain the unconditional pdf of Z X = n 1/ X µ X )/s p that is ) ) λz X ν + 1 λz hz X ) = f ν z X )F ν+1 X ν f ν + z ν z X )F ν+1 X ν + z X = f ν z X ), where f ν z X ) is the pdf of a standard t ν distribution. Thus, we see that irrespective of the value of ρ ρ 1), Z X t n. Similar argument gives Z Y = n 1/ Ȳ µ Y )/s p t n. These imply that, Z X and Z Y are pivotal quantities for µ X and µ Y for the bivariate normal population with common variance σ and known ρ A Kalman filtering A well-known property of the normal distribution is that, if Y is NZ, σ ) where a priori Z is a normal variable, then the a posteriori distribution Z is still normal. Kim 7) showed that analogous fact is true if a priori Z has probability density function in 3.5). Under this prior, some simple algebra shows that the a posteriori density function of Z given Y = y is still of type 3.5) with θ 1, σ 1, λ, ua), ub)) replaced by y/τ + θ 1 /σ1 1/τ + 1/σ1, 1 σ ) 1 ) 1, λ τ 1 + σ 1 τ, { λ 1 ua) + λσ } 1y θ 1 ) {1 σ1 + τ + λ τ σ1 + τ { λ 1 ub) + λσ } 1y θ 1 ) {1 σ1 + τ + λ τ σ1 + τ } 1, } 1. Note that the parameter λ shrinks towards, independently of y and that the updating formulae of the first two parameters are the same as those of the normal case. Consider now the Kalman filtering setting Z t = ρz t 1 + ε t, Y t = Z t + η t, t = 1,,..., where {ε t } is white noise N, σ ε) and {η t } is white noise N, σ η), with {ε t } independent of {η t }. If the initial prior of Z is normal, then all subsequent posterior distributions of

13 Perturbed Distribution Associated with Truncated Elliptical Model 495 Z t given Y 1,..., Y t ) are still normal. An analogous property holds for the distribution 3.5): if the initial prior distributions of Z is of type 3.5), then all subsequent posterior distributions of Z t given Y 1,..., Y t ) are again of type 3.5). This fact follows from the above conjugacy property and the application of Theorem 4.1. When Z is a random variable with density function 3.) and ε is an independent N, σ ε) variable, Z + ε has density function of type 3.5) by Theorem Conclusion This paper has presented a new class of perturbed elliptical distributions. To form the class of distributions, we considered a conditioning method to the bivariate elliptical distributions. This introduces yet other perturb function, inducing perturbation of the symmetry with univariate elliptical distribution, that brings additional flexibility of modeling skewed and heavy tailed distribution. The results obtained in this paper extend many properties of the bivariate elliptical distributions in a nontrivial way. Several properties of the class are studied, especially for the perturbed normal and perturbed scale mixture of normal distributions. The study shows that we have at hand a class of distributions with following properties: i) strict inclusion of the normal, skew-normal and their scale mixture densities, ii) mathematical tractability and iii) wide applicability in solving statistical problems as demonstrated by Section 5. References Arellano-Valle, R. B., Branco, M. D. and Genton, M. G. 6). A unified view on skewed distributions arising from selections, The Canadian Journal of Statistics/La revue Canadienne de Ststistique, 34, Arnold, B. C., Beaver, R. J., Groeneveld, R. A. and Meeker, W. Q. 1993). The nontruncated marginal of a truncated bivariate normal distribution, Psychometrika, 58, Azzalini, A. 1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 1, Azzalini, A. and Capitanio, A. 3). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution, Journal of the Royal Statistical Society, Ser. B, 35, Branco, M. D. and Dey, D. K. 1). A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, 79, Chen, M. H. and Dey, D. K. 1998). Bayesian modeling of correlated binary responses via scale mixture of multivariate normal link functions, Sankhyã, 6, Devroye, L. 1986). Non-Uniform Random Variate Generation, Springer Verlag, New York. Fang, K. T., Kotz, S. and Ng, K. W. 199). Symmetric Multivariate and Related Distributions, Chapman & Hall/CRC, New York. Fang, K. T. and Zhang, Y. T. 199). Generalized Multivariate Analysis, Springer- Verlag, New York.

14 496 Hea-Jung Kim Henze, N. 1986). A probabilistic representation of the Skew-normal distribution, Scandinavian Journal of Statistics, 13, Johnson, N. L. and Kotz, S. 197). Distributions in Statistics: Continuous Multivariate Distributions, John Wiley & Sons, New York. Kim, H. J. ). Binary regression with a class of skewed t link models, Communications in Statistics-Theory and Methods, 31, Kim, H. J. 7). A class of weighted normal distributions and its variants useful for inequality constrained analysis, Statistics, 41, Kim, H. J. 8). A class of weighted multivariate normal distributions and its properties, Journal of the Multivariate Analysis, In press, doi:1.116/j.jmva Ma, Y. and Genton, M. G. 4). A flexible class of skew-symmetric distributions, Scandinavian Journal of Statistics, 31, Rudy, D. A. ). Intermediate Microeconomic Theory, Digital Authoring Resources, Denver. [Received March 8, Accepted April 8]