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 332
... sufficient statistic for 0 , and let the family { g1 ( y1 ; 0 ) : 0 € N } of probability density functions be complete . If there is a function of Y , that is an unbiased estimator of 0 , then this function of Y1 is the unique unbiased ...
... sufficient statistic for 0 , and let the family { g1 ( y1 ; 0 ) : 0 € N } of probability density functions be complete . If there is a function of Y , that is an unbiased estimator of 0 , then this function of Y1 is the unique unbiased ...
Page 336
... complete sufficient statistic for 0 . Example 2. Consider a Poisson distribution with parameter 0 , 0 < 0 < ∞ . The p.d.f. of this distribution is 0xe - 0 f ( x ; 0 ) = x ! = exp [ ( ln 0 ) x - In ( x ! ) - 0 ] , x = 0 , 1 , 2 ...
... complete sufficient statistic for 0 . Example 2. Consider a Poisson distribution with parameter 0 , 0 < 0 < ∞ . The p.d.f. of this distribution is 0xe - 0 f ( x ; 0 ) = x ! = exp [ ( ln 0 ) x - In ( x ! ) - 0 ] , x = 0 , 1 , 2 ...
Page 354
... sufficient statistic for 0 , and let the family { g1 ( y1 ; 0 ) : 0 € N } of probability density functions of Y , be complete . Let u ( X1 , X2 , . . . , Xn ) be any other statistic ( not a function of Y , alone ) . If the distribution ...
... sufficient statistic for 0 , and let the family { g1 ( y1 ; 0 ) : 0 € N } of probability density functions of Y , be complete . Let u ( X1 , X2 , . . . , Xn ) be any other statistic ( not a function of Y , alone ) . If the distribution ...
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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. g₁(y₁ gamma distribution given Hint hypothesis H₁ independent random variables integral 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 W₁ X₁ X₂ Y₁ Y₂ zero elsewhere μ₁ μ₂ Σ Σ σ²