1 Inference about a multivariate normal dispersion matrix

Multivariate Normal Distribution and Related Inference: VI

Dr. Sumitra Purkayastha, Professor

Applied Statistics Unit, Indian Statistical Institute, Kolkata sumitra@isical.ac.in

1 Inference about a multivariate normal dispersion matrix

Objectives of the lecture We consider independent and identically distributed observationsX1, . . . ,Xn

having a common N_p(µ,Σ)distribution, whereµis anunknownp×1vector. Our goals:

• To develop likelihood ratio test for

H₀ :Σ=σ²Ip against H₁:H₀ is f alse.

(a) Expression for maximum likelihood estimator of(µ, σ)under H0and also without any restric- tion

(b) Expression for the likelihood ratio test statisticΛ (c) Asymptotic distribution of−2 log Λunder H0

• To develop likelihood ratio test for independence of two disjoint partitions ofX1.

(a) Expressions for maximum likelihood estimator of(µ, σ)under H0and also without any restric- tion

(b) Expression for the likelihood ratio test statisticΛ

(c) Exact null distribution of−2 log Λunder H0, leading to a distribution theoretic question, and asymptotic null distribution as well

1.1 Test for sphericity

Distributional assumption and formulation of the problem

We assume thatX1, . . . ,Xnare independent and identically distributed observations having a com- mon N_p(µ,Σ)distribution, whereµis anunknownp×1vector.

We wish to test the following hypothesis:

H₀ :Σ=σ²I_p against H₁:H₀ is f alse.

WhenΣ=σ²I_p, the common density function ofX_i’s is given by C·exp(−(x−µ)^T(x−µ)/2).

Thus the (common) density function is constant on spheres. This is why the problem of testing H₀:Σ=σ²Ipis known in the multivariate analysis literature astest for sphericity.

Descriptive statistics Under assumption of normality, the following descriptive statistics are suffi- cient.

Sample mean= ¯X = 1 n

n

X

i=1

Xi

Sample dispersion matrix=S= 1 n

n

X

i=1

(X_i−X)(X¯ _i−X¯)^T.

We assume thatn≥p+ 1. This will ensurealmost sureinvertibility ofS.

We shall mention the likelihood ratio test statistic and state also its asymptotic distribution.

Expressions for maximum likelihood estimators Maximum likelihood estimator of(µ, σ²)under H0:

( ˆµ,σˆ²) = ( ¯X, p⁻¹tr(S)) Maximum likelihood estimator of(µ,Σ)without any restriction:

( ˆµ,Σ) = ( ¯ˆ X,S).

Expression for the likelihood ratio test statisticΛand its asymptotic distribution The expression for the log likelihood ratio statistic is given by

−2 log Λ =np(loga₀−logg₀) =nplog(a₀/g₀), where

a0 :=arithmetic mean of the eigenvalues of S=trace(S)/p, and

g0 :=geometric mean of the eigenvalues of S= (det(S))^1/p. We reject H0ifa0/g0is too large.

The asymptotic distribution of−2 log Λunder H0 isχ²_f, wheref := (p−1)(p+ 2)/2.

1.2 Test for independence

Distributional assumption and formulation of the problem

We assume thatX₁, . . . ,X_nare independent and identically distributed observations having a com- mon Np(µ,Σ)distribution, whereµis anunknownp×1vector.

Suppose we partition eachX_ias follows:

X_i =

Xi,1

X_i,2

.

The (random) vectorXi,1(Xi,2) is of sizep1×1(p2×1), wherep1+p2=p.

The corresponding partitions inµandΣare µ=

µ₁ µ₂

, Σ=

Σ₁₁ Σ₁₂ Σ₂₁ Σ₂₂

.

The mean vector ofX_i,1(X_i,2) isµ₁(µ₂). The dispersion matrix ofX_i,1(X_i,2) isΣ₁₁(Σ₂₂). The covariance matrix ofXi,1 andXi,2isΣ12.

Distributional assumption and formulation of the problem

We wish to test if X1,1 andX1,2 are independent. In terms of the partition ofΣit can be stated equivalently as

H₀ :Σ=0.

Descriptive statistics Under assumption of normality, the following descriptive statistics are suffi- cient.

Sample mean= ¯X = 1 n

n

X

i=1

Xi

Sample dispersion matrix=S= 1 n

n

X

i=1

(X_i−X)(X¯ _i−X¯)^T.

We consider the partitions ofX¯ andS, analogous to those consider earlier as follows:

X¯ = X¯1

X¯₂

, S=

S11 S12

S₂₁ S₂₂

.

We assume thatn≥p+ 1. This will ensurealmost sureinvertibility ofS.and hence of each ofS₁₁ andS22.

We shall mention the likelihood ratio test statistic and state also its asymptotic distribution.

Expressions for maximum likelihood estimators Maximum likelihood estimator of(µ,Σ)under H₀:

( ¯X,

S₁₁ S₁₂ S21 S22

)

Maximum likelihood estimator of(µ,Σ)without any restriction:

( ˆµ,Σ) = ( ¯ˆ X,S).

Expression for the likelihood ratio test statisticΛand its null distribution Likelihood ratio test statistic is given by

−2 log Λ =−nlog

|S₂₂−S₂₁S⁻¹₁₁S₁₂|

|S₂₂|

=−nlog |I_p−S⁻¹₂₂S₂₁S⁻¹₁₁S₁₂| .

The exact null distribution ofΛ^2/nisΛ(p₂, n−1−p₁, p₁),Wilks’ lambda distribution with parameters p₂, n−1−p₁, p₁.

Question:What is Wilks’ lambda (Λ(p, m, n)) distribution?

The asymptotic null distribution of−2 log Λisχ²_p1p2. 1.3

Contents of the lectureWe considered independent and identically distributed observationsX1, . . . ,Xn

having a common N_p(µ,Σ)distribution, whereµis anunknownp×1vector. Our goals have been the following:

• To develop likelihood ratio test for

H₀ :Σ=σ²I_p against H₁:H₀ is f alse.

(a) Maximum likelihood estimator ofµunder H0and also without any restriction (b) Expression for the log-likelihood ratio test statistic−2Λ

(c) Asymptotic distribution of−2 log Λunder H0

• To develop likelihood ratio test for independence of two disjoint partitions ofX₁.

(a) Expressions for maximum likelihood estimator of(µ, σ)under H₀and also without any restriction

(b) Expression for the likelihood ratio test statisticΛ

(c) Exact null distribution of−2 log Λunder H0, leading to a distribution theoretic question, and asymptotic null distribution as well