## Introduction to mathematical statistics |

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Page 343

This investigation assumes that the student is familiar with elementary

algebra, with real

. Henceforth the expression quadratic form means a quadratic form in a

prescribed ...

This investigation assumes that the student is familiar with elementary

**matrix**algebra, with real

**symmetric**quadratic forms, and with orthogonal transformations. Henceforth the expression quadratic form means a quadratic form in a

prescribed ...

Page 351

Let A denote the

Then Q/a2 is x2(r) if and only if A2 = A. Remark. If the normal distribution in

Theorem 1 is n(p, a2), the condition A2 = A remains a necessary and sufficient

condition ...

Let A denote the

**symmetric matrix**of Q and let r, 0 < r < n, denote the rank of A.Then Q/a2 is x2(r) if and only if A2 = A. Remark. If the normal distribution in

Theorem 1 is n(p, a2), the condition A2 = A remains a necessary and sufficient

condition ...

Page 352

Show that Q/aa does not have a chi-square distribution. Find the moment-

generating function of Q/a2. 13.7. Let A be a real

each of the nonzero characteristic numbers of A is equal to one if and only if A2 =

A. Hint.

Show that Q/aa does not have a chi-square distribution. Find the moment-

generating function of Q/a2. 13.7. Let A be a real

**symmetric matrix**. Prove thateach of the nonzero characteristic numbers of A is equal to one if and only if A2 =

A. Hint.

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Accordingly best critical region binomial distribution cent confidence interval Chapter chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges stochastically critical region decision function defined degrees of freedom denote a random discrete type distribution having p.d.f. Equation Example EXERCISES F distribution function of Y1 given H0 is true independent random variables inequality integral joint p.d.f. Let the random Let X1 limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral order statistics Poisson distribution positive integer power function Pr X1 probability density functions probability set function quadratic form random experiment random interval random sample random variables X1 reject H0 respectively sample space Show significance level simple hypothesis H0 statistic for 9 statistic Y1 stochastically independent random subset testing H0 theorem type of random unbiased statistic variance a2 X1 and X2 Xn denote zero elsewhere