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 308
... unbiased estimator of 0 . = ... For illustration , let X1 , X2 , ... , X , denote a random sample from a distribution that is N ( 0 , 1 ) , ∞ << ∞o . Since the statistic X ( X1 + X 2 + ··· + X , ) / 9 is N ( 0 , 4 ) , X is an unbiased ...
... unbiased estimator of 0 . = ... For illustration , let X1 , X2 , ... , X , denote a random sample from a distribution that is N ( 0 , 1 ) , ∞ << ∞o . Since the statistic X ( X1 + X 2 + ··· + X , ) / 9 is N ( 0 , 4 ) , X is an unbiased ...
Page 327
Robert V. Hogg, Allen Thornton Craig. first some unbiased estimator Y2 in their search for ø ( Y1 ) , an unbiased estimator of @ based upon the sufficient statistic Y1 . This is not the case at all , and Theorem 3 simply convinces us ...
Robert V. Hogg, Allen Thornton Craig. first some unbiased estimator Y2 in their search for ø ( Y1 ) , an unbiased estimator of @ based upon the sufficient statistic Y1 . This is not the case at all , and Theorem 3 simply convinces us ...
Page 340
... estimator of e - 20 that is unbiased and has minimum variance . Consider Y = ( -1 ) . We have ∞ ( -0 ) 3e - 0 x ! E ( Y ) = E [ ( − 1 ) ' ] = Σ x = 0 = -20 Accordingly , ( -1 ) is the unbiased minimum variance estimator of e - 20 ...
... estimator of e - 20 that is unbiased and has minimum variance . Consider Y = ( -1 ) . We have ∞ ( -0 ) 3e - 0 x ! E ( Y ) = E [ ( − 1 ) ' ] = Σ x = 0 = -20 Accordingly , ( -1 ) is the unbiased minimum variance estimator of e - 20 ...
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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