## 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 observed value of 9~, „ namely £ x,/«, is called the maximum likelihood

estimate of 9. For a simple i example, ... be regarded as a function of 9. When so

regarded, it is called the

L(9 ...

The observed value of 9~, „ namely £ x,/«, is called the maximum likelihood

estimate of 9. For a simple i example, ... be regarded as a function of 9. When so

regarded, it is called the

**likelihood function**L of the random sample, and we writeL(9 ...

Page 264

Let the joint p.d.f. g(x, y, . . . , z; 0,, 62, . . . , 9m), (0,, 92, . . . , 9m) e fi, depend on m

parameters. This joint p.d.f., when regarded as a function of (01, 92, . . . , 9m)eQ,

is called the

Let the joint p.d.f. g(x, y, . . . , z; 0,, 62, . . . , 9m), (0,, 92, . . . , 9m) e fi, depend on m

parameters. This joint p.d.f., when regarded as a function of (01, 92, . . . , 9m)eQ,

is called the

**likelihood function**of the random variables. Then those functions ...Page 561

Geometric mean, 336 Gini's mean difference, 203 Goal post loss function, 311

Gosset, W. S., 182 Huber, P., 390 ... 243 Liapounov, 511

416 Likelihood principle, 312, 324 240 Likelihood ratio tests, 409, 413, 422, 452,

...

Geometric mean, 336 Gini's mean difference, 203 Goal post loss function, 311

Gosset, W. S., 182 Huber, P., 390 ... 243 Liapounov, 511

**Likelihood function**, 261,416 Likelihood principle, 312, 324 240 Likelihood ratio tests, 409, 413, 422, 452,

...

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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