## Introduction to mathematical statistics |

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Page 207

In general the conditional p.d.f. h(y2, . . . , yn\y1, 8) will

indicated by the notation. In certain cases, however, as will be seen presently, the

conditional p.d.f. will not

to ...

In general the conditional p.d.f. h(y2, . . . , yn\y1, 8) will

**depend**upon 8, as isindicated by the notation. In certain cases, however, as will be seen presently, the

conditional p.d.f. will not

**depend**upon 8, and these cases will be very importantto ...

Page 232

Let Z = u(X1, X2, . . ., Xn) be any other statistic (not a function of Y1 alone). If the

distribution of Z does not

the sufficient statistic Y1. Proof. We shall prove a special case of this theorem.

Let Z = u(X1, X2, . . ., Xn) be any other statistic (not a function of Y1 alone). If the

distribution of Z does not

**depend**upon 8, then Z is stochastically independent ofthe sufficient statistic Y1. Proof. We shall prove a special case of this theorem.

Page 235

Consequently J J J J exP y w — -J(M)dw1dw2dw3dwi, which clearly does not

stochastically independent of Y4, the complete sufficient statistic for 9. Example 3.

Consequently J J J J exP y w — -J(M)dw1dw2dw3dwi, which clearly does not

**depend**upon 9. Thus the distribution of Z does not**depend**upon 8, and so Z isstochastically independent of Y4, the complete sufficient statistic for 9. Example 3.

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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