## 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

Mxx) = Yjf(

^2)/2|l(^2kl)^2 /•o0 /*00 £[«(*„ jr2)] = x2)f(

laxa2 = [E(XxX2) - nxn2\/axa2 M(tu t2) = E(e'^ + W), ^%^ = £(*, jy Independence ...

Mxx) = Yjf(

**xux2**) or f(**xux2**)dx2 x2 J-oo fix x ) . /•» /2|l(x2lxl)= 77 ^ , £[«(A'2)|jc1]= «(^2)/2|l(^2kl)^2 /•o0 /*00 £[«(*„ jr2)] = x2)f(

**xux2**)dxx dx2 p = E[(Xx - ni)(X2 - n2)]laxa2 = [E(XxX2) - nxn2\/axa2 M(tu t2) = E(e'^ + W), ^%^ = £(*, jy Independence ...

Page 173

h(xx , x2) dxx dx2 . We wish now to change variables of integration by writing yx =

ux(

analysis that this change of variables requires h(xu x2) dxx dx2 = Thus, for every

set ...

h(xx , x2) dxx dx2 . We wish now to change variables of integration by writing yx =

ux(

**xux2**),y2 = u2(**xux2**),orxx = wx{yuy2\x2 = w2(yu j2). It has been proved inanalysis that this change of variables requires h(xu x2) dxx dx2 = Thus, for every

set ...

Page 262

That is, L[u(xu x2, . . . , x„);

every 6 e Q. Then the statistic u{ Xu X2, . . . , X„) will be called a maximum

likelihood estimator (hereafter abbreviated m.l.e.) of 6 and will be denoted by the

...

That is, L[u(xu x2, . . . , x„);

**xux2**, . . . , x„] is at least as great as L(0;**xux2**, . . . ,x„) forevery 6 e Q. Then the statistic u{ Xu X2, . . . , X„) will be called a maximum

likelihood estimator (hereafter abbreviated m.l.e.) of 6 and will be denoted by the

...

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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