The two sample problem
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The two sample problem. Univariate Inference. Let x 1 , x 2 , … , x n denote a sample of n from the normal distribution with mean m x and variance s 2 . Let y 1 , y 2 , … , y m denote a sample of n from the normal distribution with mean m y and variance s 2 .

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The two sample problem

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The two sample problem

The two sample problem


Univariate inference

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= myvs

HA: mx≠ my


The two sample problem

The appropriate test is the t test:

The test statistic:

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


The multivariate test

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


Hotelling s t 2 statistic for the two sample problem

Hotelling’s T2 statisticfor the two sample problem

if H0 is true than

has an F distribution with n1= p and

n2= n +m – p - 1


Thus hotelling s t 2 test

ThusHotelling’s T2 test

We reject


Simultaneous inference for the two sample problem

Simultaneous inference for the two-sample problem

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


The two sample problem

Thus


The two sample problem

Thus


The two sample problem

Thus

Hence


The two sample problem

Thus

form 1 – a simultaneous confidence intervals for


The two sample problem

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 two sample problem

The data are given in the following table:


Hotelling s t 2 test

Hotelling’s T2 test

A graphical explanation


Hotelling s t 2 statistic for the two sample problem1

Hotelling’s T2 statisticfor the two sample problem


The two sample problem

is the test statistic for testing:


The two sample problem

Hotelling’s T2 test

X2

Popn A

Popn B

X1


The two sample problem

Univariate test for X1

X2

Popn A

Popn B

X1


The two sample problem

Univariate test for X2

X2

Popn A

Popn B

X1


The two sample problem

Univariate test for a1X1+ a2X2

X2

Popn A

Popn B

X1


Mahalanobis distance

Mahalanobis distance

A graphical explanation


The two sample problem

Euclidean distance


The two sample problem

Mahalanobis distance: S, a covariance matrix


Hotelling s t 2 statistic for the two sample problem2

Hotelling’s T2 statisticfor the two sample problem


The two sample problem

Case I

X2

Popn A

Popn B

X1


The two sample problem

Case II

X2

Popn A

Popn B

X1


The two sample problem

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


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