## 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 262

The corresponding statistic, 6 = -j\x, = x, is called the maximum likelihood

estimator of 9. The observed value of 0, „ ' . - namely ... may be regarded as a

function of 9. When so regarded, it is called the

random sample, ...

The corresponding statistic, 6 = -j\x, = x, is called the maximum likelihood

estimator of 9. The observed value of 0, „ ' . - namely ... may be regarded as a

function of 9. When so regarded, it is called the

**likelihood function**L of therandom sample, ...

Page 264

Let the joint p.d.f. g(x, y, . . . , z; 0,, 02, . . . , 0J, (0,, 92, . . . , 9m) e Q, depend on m

parameters. This joint p.d.f., when regarded as a function of (9U 92, . • . , 0m)eQ,

is called the

Let the joint p.d.f. g(x, y, . . . , z; 0,, 02, . . . , 0J, (0,, 92, . . . , 9m) e Q, depend on m

parameters. This joint p.d.f., when regarded as a function of (9U 92, . • . , 0m)eQ,

is called the

**likelihood function**of the random variables. Then those functions ...Page 324

Robert V. Hogg. from the way the z-values were obtained, the two

using either of these

value ...

Robert V. Hogg. from the way the z-values were obtained, the two

**likelihood****functions**enjoy the property of being proportional, namely Thus, for illustration,using either of these

**likelihood functions**, the m.l.e. of 9 is j, /« because this is thevalue ...

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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