10
Mean Vectors and Covariance Matrices
Generalizations for univariate inference procedures
1 Inference procedures for covariance matrices
Univariate test - Test Ho:(2=[pic] vs. Ha:(2([pic] for some positive constant [pic].
Multivariate test - Test Ho:(=(0 vs. Ha:(((0 for some known matrix (0.
To be discussed later.
2 Inference procedures for a mean vector
Univariate test - Test Ho:(=(0 vs. Ha:(((0 for some constant (0.
Multivariate test - Test Ho:(=(0 vs. Ha:(((0 for some known p(1 vector (0.
Hotelling’s T2 statistic
Let x1,…,xN ~ i.i.d. Np((,() where ( and ( are unknown.
Let [pic]. When p=1, [pic], i.e., the square of Student’s t statistic used for the univariate test.
[pic] ~ Fp, N-p
Using Hotelling’s T2 and its distributional properties, hypothesis tests and confidence regions for ( can be constructed.
Reject Ho:(=(0 if [pic] > F1-(,p, N-p
where F1-(,p, N-p is the (1-() quantile of a F-distribution.
A (1-()100% confidence region for ( is the set of ( such that
T2 ( [pic]F1-(,p, N-p
Let [pic] and [pic] be the eigenvalues and corresponding eigenvectors of [pic], respectively.
A (1-()100% confidence region for ( is a p-dimensional ellipsoid centered at [pic]. The longest axis has length of
[pic]
and is in the direction of [pic]. The second longest axis has length of
[pic]
and is in the direction of [pic]. The third, fourth,… have similar axes. See p.236-7 of Johnson and Wichern (1998) for more information about how to construct the ellipsoid.
Example: Bivariate normal from Chapter 4 (biv_normal_ch10.R)
x~[pic]
Suppose 20 observations are generated from this distribution. Below is R code used to calculate the Hotelling’s T2 for the test of Ho:(=[15, 20]( vs. Ha:(([15, 20]( and its corresponding output. Note that Ho would really be true here!
> mu sigma N p library(mvtnorm) #Need for rmvnorm() function
>
> #Use the same seed number
> set.seed(508)
> x meanx1 meanx2 mu.hat mu.hat
[,1]
meanx1 14.74184
meanx2 19.84626
> sigma.hat sigma.hat
[,1] [,2]
[1,] 1.11524274 0.04275627
[2,] 0.04275627 0.57296045
> #construct the Hotelling's T^2 for the hypothesis Ho:mu=[15,20], Ha:mu[15,20]
> mu.Ho
> T.sq p.val cat("T^2 is", round(T.sq,2), "and the p-value is", round(p.val,4), "\n")
T^2 is 1.92 and the p-value is 0.4206
[pic]
#sequence of numbers from 10-25 by 0.1
x1 ................
................
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