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 103
... Accordingly , X , and X2 are independent . If we now refer to Example 1 , we see that the joint p.d.f. f ( x1 , x2 ) = x1 + x2 , = = 0 0 < x < 1 , 0 < x2 < 1 , elsewhere , cannot be written as the product of a nonnegative function of x ...
... Accordingly , X , and X2 are independent . If we now refer to Example 1 , we see that the joint p.d.f. f ( x1 , x2 ) = x1 + x2 , = = 0 0 < x < 1 , 0 < x2 < 1 , elsewhere , cannot be written as the product of a nonnegative function of x ...
Page 150
... Accordingly , if we set t = t1 + t2p ( σ2 / σ1 ) , we see that M ( t1 , t2 ) is given by ཨ་ རི་ པ་ 02 σι μι + μ1 t1 + t2p 02 σι + σ2 ( ** ? ) 2 or , equivalently , M ( t1 , t2 ) = exp μ1t1 + μ2tz + expμ1t1 σ } t2 + 2μ¤ ̧σ2t , t1⁄2 + σ ...
... Accordingly , if we set t = t1 + t2p ( σ2 / σ1 ) , we see that M ( t1 , t2 ) is given by ཨ་ རི་ པ་ 02 σι μι + μ1 t1 + t2p 02 σι + σ2 ( ** ? ) 2 or , equivalently , M ( t1 , t2 ) = exp μ1t1 + μ2tz + expμ1t1 σ } t2 + 2μ¤ ̧σ2t , t1⁄2 + σ ...
Page 378
... Accordingly , E [ ( 0 ́d ln f ( X ; 0 ) до E ( X - 0 ) 2 02 = = 02 02 018 = 1 0 The Rao Cramér lower bound in this ... Accordingly , [ a2 In f ( X ; 0 ) 0 1 1 - E = 002 03 202 202 - - Thus the Rao - Cramér lower bound is 202 / n . Now ...
... Accordingly , E [ ( 0 ́d ln f ( X ; 0 ) до E ( X - 0 ) 2 02 = = 02 02 018 = 1 0 The Rao Cramér lower bound in this ... Accordingly , [ a2 In f ( X ; 0 ) 0 1 1 - E = 002 03 202 202 - - Thus the Rao - Cramér lower bound is 202 / n . Now ...
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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 Find the p.d.f. gamma distribution given Hint hypothesis H₁ independent random variables integral joint p.d.f. Let the random Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix moment-generating function order statistics 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 subset sufficient statistic t-distribution t₂ theorem unbiased estimator variance o² X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ μ₂ σ²