## 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 296

It is desired to test H0 against all alternatives. If the hypothesis

random variable *, (X, - npi0f 1 nPio has an approximate chi-square distribution

with k — 1 degrees of freedom. Since, when

value of ...

It is desired to test H0 against all alternatives. If the hypothesis

**H0 is true**, therandom variable *, (X, - npi0f 1 nPio has an approximate chi-square distribution

with k — 1 degrees of freedom. Since, when

**H0 is true**, npm is the expectedvalue of ...

Page 397

We shall use one random value of X to test the simple hypothesis H0:9 = j against

the alternative simple hypothesis Hx : 9 = | ... The latter statement follows from the

fact that, when

We shall use one random value of X to test the simple hypothesis H0:9 = j against

the alternative simple hypothesis Hx : 9 = | ... The latter statement follows from the

fact that, when

**H0 is true**, there are but two subsets, Ax and A2, of the sample ...Page 506

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

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

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

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

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

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

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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