## 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 286

That is. the probability of reiecting_J/0 when

of rejecting Hn when Hn is false is 0.31. Since the significance level of this test (or

the size of the critical region) is the power of the test when

That is. the probability of reiecting_J/0 when

**H0 is true**is 0 05, and the probabilityof rejecting Hn when Hn is false is 0.31. Since the significance level of this test (or

the size of the critical region) is the power of the test when

**H0 is true**, the ...Page 397

We shall use one random value of X to test the simple hypothesis

the alternative simple hypothesis Hx :9 ... we have the intolerable situation that

the probability of rejecting

We shall use one random value of X to test the simple hypothesis

**H0**:9 = \ againstthe alternative simple hypothesis Hx :9 ... we have the intolerable situation that

the probability of rejecting

**H0**when //, is**true**(**H0**is false) is much less than the ...Page 506

The original hypothesis was then replaced by the less restrictive hypothesis H0:

Pr(XeA,)=p,0, i = 1, 2, . . . , k; and a ... Then, if

whereas if H0 is false, Fis b[n, p = F{^)] whatever be the distribution function F(x).

The original hypothesis was then replaced by the less restrictive hypothesis H0:

Pr(XeA,)=p,0, i = 1, 2, . . . , k; and a ... Then, if

**H0 is true**, Y is b[n, p0 = F(£)];whereas if H0 is false, Fis b[n, p = F{^)] whatever be the distribution function F(x).

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### Common terms and phrases

Accordingly approximate best critical region 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 F-distribution gamma distribution given H0 is true 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 Yx percent confidence interval 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 statistic for 9 subset testing H0 theorem u(Xu X2 unbiased estimator XuX2 Xx and X2 Yu Y2 zero elsewhere