Introduction to Mathematical StatisticsThe fifth edition of text offers a careful presentation of the probability needed for mathematical statistics and the mathematics of statistical inference. Offering a background for those who wish to go on to study statistical applications or more advanced theory, this text presents a thorough treatment of the mathematics of statistics. |
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Page 381
... Equation ( 1 ) is limiting N ( 0 , 1 ) by the central limit theorem . Moreover , the mean a2 ln f ( X ;; 0 ) 202 converges in probability to its expected value , namely I ( 0 ) . So the denominator of the right - hand member of Equation ...
... Equation ( 1 ) is limiting N ( 0 , 1 ) by the central limit theorem . Moreover , the mean a2 ln f ( X ;; 0 ) 202 converges in probability to its expected value , namely I ( 0 ) . So the denominator of the right - hand member of Equation ...
Page 391
... Equation ( 1 ) . One such scheme , Newton's method , is described . Let Ô , be a first estimate of 0 , such as = median ( x ; ) . Approximate the left - hand member of Equation ( 1 ) by the first two terms of Taylor's expansion about to ...
... Equation ( 1 ) . One such scheme , Newton's method , is described . Let Ô , be a first estimate of 0 , such as = median ( x ; ) . Approximate the left - hand member of Equation ( 1 ) by the first two terms of Taylor's expansion about to ...
Page 491
... equation | B , - i ― i - ― 2I = 0. Since B1 = I A1 , this equation can be written as | I A , 21 = 0. Thus we have | A , − ( 1 − 2 ) I | = 0. But each root of the last equation is one minus a characteristic number of A. Since B , has ...
... equation | B , - i ― i - ― 2I = 0. Since B1 = I A1 , this equation can be written as | I A , 21 = 0. Thus we have | A , − ( 1 − 2 ) I | = 0. But each root of the last equation is one minus a characteristic number of A. Since B , has ...
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A₁ A₂ Accordingly approximate best critical region C₁ C₂ chi-square distribution complete sufficient statistic conditional p.d.f. confidence interval Consider continuous type converges in probability correlation coefficient critical region defined degrees of freedom denote a random discrete type distribution function F(x distribution with mean distribution with p.d.f. distribution with parameters dx₁ equation Example Exercise g₁(y₁ gamma distribution given H₁ Hint hypothesis H independent random variables integral Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix mean µ moment-generating function order statistics p.d.f. of Y₁ P(C₁ p₁ percent confidence interval Poisson distribution positive integer probability density functions probability set function r₁ random experiment random sample respectively sample space Section Show significance level simple hypothesis sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ Σ Σ σ²