1. The two sample problem

2. Univariate Inference
Let x1, x2, … , xn denote a sample of n from the normal distribution with mean mx and variance s2.
Let y1, y2, … , ym denote a sample of n from the normal distribution with mean my and variance s2.
Suppose we want to test
H0: mx = my vs
HA: mx ≠ my

3. The appropriate test is the t test:
The test statistic:
Reject H0 if |t| > ta/2 d.f. = n + m -2

4. The multivariate Test
Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S.
Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S.
Suppose we want to test

5. Hotelling’s T2 statistic for the two sample problem
if H0 is true than
has an F distribution with n1 = p and
n2 = n +m – p - 1

6. Thus Hotelling’s T2 test
We reject

7. Simultaneous inference for the two-sample problem
Hotelling’s T2 statistic can be shown to have been derived by Roy’s Union-Intersection principle

8. Thus

9. Thus

10. Thus
Hence

11. Thus
form 1 – a simultaneous confidence intervals for

12. Example Annual financial data are collected for firms approximately 2 years prior to bankruptcy and for financially sound firms at about the same point in time. The data on the four variables
x1 = CF/TD = (cash flow)/(total debt),
x2 = NI/TA = (net income)/(Total assets),
x3 = CA/CL = (current assets)/(current liabilties, and
x4 = CA/NS = (current assets)/(net sales) are given in the following table.

13. The data are given in the following table:

14. A graphical explanation
Hotelling’s T2 test
A graphical explanation

15. Hotelling’s T2 statistic for the two sample problem

16. is the test statistic for testing:

17. Hotelling’s T2 test X2
Popn A
Popn B X1

18. Univariate test for X1 X2
Popn A
Popn B X1

19. Univariate test for X2 X2
Popn A
Popn B X1

20. Univariate test for a1X1 + a2X2
Popn A
Popn B X1

21. A graphical explanation
Mahalanobis distance
A graphical explanation

22. Euclidean distance

23. Mahalanobis distance: S, a covariance matrix

24. Hotelling’s T2 statistic for the two sample problem

25. Case I X2
Popn A
Popn B X1

26. Case II X2
Popn A
Popn B X1

27. In Case I the Mahalanobis distance between the mean vectors is larger than in Case II, even though the Euclidean distance is smaller. In Case I there is more separation between the two bivariate normal distributions