Hotelling Sample Statistic

appendix C

Hotelling sample statistic

This appendix derives the one sample Hotelling tests, whose much more complete treatments are in [Mardia et al., 1979].

Suppose that is a random sample from a population with p.d.f , where is a parameter vector. The likelihood function of sample is

and the corresponding log likelihood function is

As we know, the likelihood (log likelihood) function is central to the theory of statistical inference.

Example C.1[p97, Mardia et al., 1979]:Suppose is a random sample from , where means -variate normal distribution with mean, , and covariance matrix , then,

Take , , then,

Writing , we have,

So,

PropositionC.1[p103, Mardia et al., 1979]:Suppose is a random sample from , , maximum likelihood estimation (MLE) of andare

, .

Proof:

Using new parameters,, then the log likelihood function becomes,

is symmetric positive definite, from matrix differentiation we have,

, .

So, , if we denote . Then let , we get , i.e., .

■

DefinitionC.1[p123, Mardia etal., 1979]:Suppose that is a random sample from a distribution with parameter , and and are any two hypotheses, then the likelihood ratio (LR) statistic for testing against is defined as

where is the largest value which the likelihood function takes in region , . Equivalently, we can use the statistic,

, with , .

DefinitionC.2[p66, Mardia et al., 1979]:If can be written as , where is a data matrix whose each column is i.i.d. , then is said to have a Wishart distribution with scale matrix and degrees of freedom parameter ; we write .

PropositionC.2 [p69, Mardia et al., 1979]: Suppose is a random sample from , , then the sample mean, , and the sample covariance, , are independent, and

, .

Proof:

Please refer to [p69, Mardia, Kent and Bibby, 1979].

■

DefinitionC.3 [p74, Mardia et al., 1979]: If can be written as where and are independently distributed as and , then we say that has the Hotelling distribution with parameters and . We write .

PropositionC.3 [p74, Mardia et al., 1979]: If and are independently distributed as and , then

Proof:

Let and , then , satisfy the requirements in Definition C.3.

■

PropositionC.4 [p74, Mardia et al., 1979]: If and are the sample mean and sample covariance matrix of a sample of size from , then

Proof:

Take , , then . From PropositionC.3, we got

■

PropositionC.5 [p74, Mardia et al., 1979]

Proof:

Please refer to [p74, Mardia et al., 1979].

■

PropositionC.6 [p75, Mardia et al., 1979]: If and are the sample mean and sample covariance matrix of a sample of size from , then

Proof:

This result directly follows from Proposition C.5.

■

PropositionC.7[p125, Mardia et al., 1979]:Suppose is a random sample from , is fixed but unknown, then the LRstatistic for testing against is known as one sample Hotelling statistic.

Proof:

Under , and , with; under , and . So,

Obviously, , so

From Results4 and 6, we got that,

or .

■