## Introduction to mathematical statistics |

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Page 205

That is, our sample arises from a distribution that has the p.d.f. f(x; 8); 8e Q. Our

problem is that of defining a statistic Y1 ... Any statistic whose mathematical

expectation is equal to a parameter 8 is called an

parameter 8.

That is, our sample arises from a distribution that has the p.d.f. f(x; 8); 8e Q. Our

problem is that of defining a statistic Y1 ... Any statistic whose mathematical

expectation is equal to a parameter 8 is called an

**unbiased statistic**for theparameter 8.

Page 218

Let Y2 = u2(X1, X2, . . . , Xn) be another statistic (but not a function of Y1 alone)

which is an

expectation is a function of y1, say, yiV1). Since Y1 is a sufficient statistic for 8, the

...

Let Y2 = u2(X1, X2, . . . , Xn) be another statistic (but not a function of Y1 alone)

which is an

**unbiased statistic**for 0; that is, E(Y2) = 9. Consider E(Y2\yi). Thisexpectation is a function of y1, say, yiV1). Since Y1 is a sufficient statistic for 8, the

...

Page 222

E\^Yj)] = 8 for all values of 8, 8 e Q. Let ^(Y^ be another continuous function of the

sufficient statistic Y1 alone so that we have ... That is, at all points of nonzero

probability density, we have, for every continuous

...

E\^Yj)] = 8 for all values of 8, 8 e Q. Let ^(Y^ be another continuous function of the

sufficient statistic Y1 alone so that we have ... That is, at all points of nonzero

probability density, we have, for every continuous

**unbiased statistic**</>(Yj), <p(yi)...

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### Common terms and phrases

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