## Introduction to mathematical statistics |

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Page 205

That is, our sample arises from a distribution that has the p.d.f. f(x; 8); 8e Q. Our

problem is that of defining a

xn are the observed experimental values of X1, X2, . . . , Xn, then the number y1 ...

That is, our sample arises from a distribution that has the p.d.f. f(x; 8); 8e Q. Our

problem is that of defining a

**statistic Y1**= u^X^ X2, . . ., Xn) so that if x1, x2, . . . ,xn are the observed experimental values of X1, X2, . . . , Xn, then the number y1 ...

Page 208

Why we use the terminology "sufficient statistic" can be explained as follows: If a

given Y1 = y1, does not depend upon the parameter 8, As a consequence, once

...

Why we use the terminology "sufficient statistic" can be explained as follows: If a

**statistic Y1**satisfies the preceding definition, then the conditional p.d.f. of Y2, say,given Y1 = y1, does not depend upon the parameter 8, As a consequence, once

...

Page 213

< Y„ denote the order statistics of a random sample Xj, X2, . . ., Xn from the

distribution that has p.d.f. f(x; 8) = e-(x-8), 9 < x < oo, -oo < 9 < oo, = 0 elsewhere.

The p.d.f. of the

joint p.d.f. ...

< Y„ denote the order statistics of a random sample Xj, X2, . . ., Xn from the

distribution that has p.d.f. f(x; 8) = e-(x-8), 9 < x < oo, -oo < 9 < oo, = 0 elsewhere.

The p.d.f. of the

**statistic Y1**is = ne~Myi-"\ 8 < y1 < oo, = 0 elsewhere. Thus thejoint p.d.f. ...

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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