## 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 318

Since this ratio does not depend upon 9, the sum Yx is a sufficient

Example 3. Let Yx < Y2 < . . . < Y„ denote the order statistics of a random sample

of size n from the distribution with p.d.f. Here we use the indicator function of set A

...

Since this ratio does not depend upon 9, the sum Yx is a sufficient

**statistic for 9**.Example 3. Let Yx < Y2 < . . . < Y„ denote the order statistics of a random sample

of size n from the distribution with p.d.f. Here we use the indicator function of set A

...

Page 321

x\x2 xn Since k2(xx, x2, . . . , x„) does not depend upon 9, the product X\X2- . . X„

is a sufficient

incorrectly the factorization theorem in those instances in which the domain of

positive ...

x\x2 xn Since k2(xx, x2, . . . , x„) does not depend upon 9, the product X\X2- . . X„

is a sufficient

**statistic for 9**. There is a tendency for some readers to applyincorrectly the factorization theorem in those instances in which the domain of

positive ...

Page 349

We have noted that the m.l.e. 9~ of 9 is a function of one or more sufficient

statistics, when the latter exist. Suppose that this m.l.e. 9~ is also sufficient. Since

this sufficient

minimal.

We have noted that the m.l.e. 9~ of 9 is a function of one or more sufficient

statistics, when the latter exist. Suppose that this m.l.e. 9~ is also sufficient. Since

this sufficient

**statistic 9**~ is a function of the other sufficient statistics, it must beminimal.

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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