## Introduction to Mathematical StatisticsAn exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform. |

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Page 335

That is, Yx is a

statistic is Y\ = ]T K(Xx). i If we can see how to form a function of Yu say q>(Yx), so

that ^l<P(^i)] = 9, then the statistic q>{Yx) is unique and is the unbiased minimum

...

That is, Yx is a

**complete sufficient statistic**for 9. ... inspection that the sufficientstatistic is Y\ = ]T K(Xx). i If we can see how to form a function of Yu say q>(Yx), so

that ^l<P(^i)] = 9, then the statistic q>{Yx) is unique and is the unbiased minimum

...

Page 359

Let Y\ < Y2 < Y3 < Y4 denote the order statistics of a random sample of size n = 4

from a distribution having p.d.f. /(x; 8) — 1/9, 0 < x < 9, zero elsewhere, where 0 <

8 < oo . Argue that the

Let Y\ < Y2 < Y3 < Y4 denote the order statistics of a random sample of size n = 4

from a distribution having p.d.f. /(x; 8) — 1/9, 0 < x < 9, zero elsewhere, where 0 <

8 < oo . Argue that the

**complete sufficient statistic**Y4 for 8 is independent of ...Page 537

*2, • • . , r») 1 1 Thus, in general, we would hope to be able to find a selector Q

that is a function of the

so that it is independent of the test statistics. It is particularly interesting to note

that ...

*2, • • . , r») 1 1 Thus, in general, we would hope to be able to find a selector Q

that is a function of the

**complete sufficient statistics**for the parameters, under H0,so that it is independent of the test statistics. It is particularly interesting to note

that ...

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### Common terms and phrases

Accordingly approximate best critical region bivariate normal distribution chi-square distribution complete sufficient statistic conditional p.d.f. conditional probability confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random depend upon 9 discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters equation estimator of 9 Example Exercise gamma distribution given H0 is true hypothesis H0 independent random variables integral joint p.d.f. Let the random Let Xu X2 likelihood function limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Xu Poisson distribution positive integer probability density functions probability set function quadratic form random experiment random sample random variables Xx reject H0 respectively sample space Section Show significance level simple hypothesis statistic for 9 testing H0 theorem u(Xu X2 unbiased estimator variance a2 XuX2 Xx and X2 Yu Y2 zero elsewhere