## 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 322

Here, of course, i M(x) dx = Q(9) 7.13. Let Xu X2, . . . , X„ be a random sample of

size n from a geometric distribution that has p.d.f. f(x; 9) = (1 - 9)% x = 0, 1, 2, . . . ,

0 < 6 < 1, „ zero elsewhere. Show that £ A", is a sufficient

Here, of course, i M(x) dx = Q(9) 7.13. Let Xu X2, . . . , X„ be a random sample of

size n from a geometric distribution that has p.d.f. f(x; 9) = (1 - 9)% x = 0, 1, 2, . . . ,

0 < 6 < 1, „ zero elsewhere. Show that £ A", is a sufficient

**statistic for 9**. i 7.14.Page 335

The points of positive probability density and the function R(yx) do not depend

upon 9. ... fixed positive integer) is a random sample from a distribution with „

p.d.f. f(x; 9), the statistic Yx = £ K(Xx) is a sufficient

gx(y\', ...

The points of positive probability density and the function R(yx) do not depend

upon 9. ... fixed positive integer) is a random sample from a distribution with „

p.d.f. f(x; 9), the statistic Yx = £ K(Xx) is a sufficient

**statistic for 9**and i the family {gx(y\', ...

Page 352

for— oo<</<oo,0<c<oo, has a distribution that does not depend upon

Again there are many examples: [max (Xi) - min (X,)]/S, "£ (Xi+ , - ^)2/52, {X, ...

for— oo<</<oo,0<c<oo, has a distribution that does not depend upon

**9**, and 92.**Statistics**like this Z = u(Xu X2, . . . , X„) are location- and-scale-invariant**statistics**.Again there are many examples: [max (Xi) - min (X,)]/S, "£ (Xi+ , - ^)2/52, {X, ...

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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