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The two sample problem

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The two sample problem

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= myvs

HA: mx≠ my

The appropriate test is the t test:

The test statistic:

Reject H0 if |t| > ta/2 d.f. = n + m -2

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

if H0 is true than

has an F distribution with n1= p and

n2= n +m – p - 1

We reject

- Hotelling’s T2 statistic can be shown to have been derived by Roy’s Union-Intersection principle

Hence

form 1 – a simultaneous confidence intervals for

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.

The data are given in the following table:

Hotelling’s T2 test

A graphical explanation

is the test statistic for testing:

Hotelling’s T2 test

X2

Popn A

Popn B

X1

Univariate test for X1

X2

Popn A

Popn B

X1

Univariate test for X2

X2

Popn A

Popn B

X1

Univariate test for a1X1+ a2X2

X2

Popn A

Popn B

X1

Mahalanobis distance

A graphical explanation

Euclidean distance

Mahalanobis distance: S, a covariance matrix

Case I

X2

Popn A

Popn B

X1

Case II

X2

Popn A

Popn B

X1

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