## 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

Show that the sum of the observations of a random sample of size n from a

gamma distribution that has p.d.f. f(x; 9) = (\/9)e~xie, 0 < x < oo, 0 < 9 < oo, zero

elsewhere, is a

sample of ...

Show that the sum of the observations of a random sample of size n from a

gamma distribution that has p.d.f. f(x; 9) = (\/9)e~xie, 0 < x < oo, 0 < 9 < oo, zero

elsewhere, is a

**sufficient statistic**for 9. 7.15. Let Xu X2, . . . , X„ be a randomsample of ...

Page 326

We now consider a result of Rao and Blackwell from which we see that we need

consider only functions of the

estimates of parameters. In showing this, we can refer back to a result of Section

2.2: ...

We now consider a result of Rao and Blackwell from which we see that we need

consider only functions of the

**sufficient statistic**in finding the unbiased pointestimates of parameters. In showing this, we can refer back to a result of Section

2.2: ...

Page 349

There is an observation that helps us observe that almost all the

# of 9 is a function of one or more

Suppose that ...

There is an observation that helps us observe that almost all the

**sufficient****statistics**that we have studied thus far are minimal. We have noted that the m.l.e.# of 9 is a function of one or more

**sufficient statistics**, when the latter exist.Suppose that ...

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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 hypothesis H0 independent random variables integral joint p.d.f. Let the random Let Xu X2 limiting distribution marginal p.d.f. matrix moment-generating function order statistics p.d.f. of Xu 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 simple hypothesis statistic for 9 sufficient statistic testing H0 theorem unbiased estimator variance a2 Xx and X2 Yu Y2 zero elsewhere