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 355
... complete sufficient statistic for the parameter μ , - ∞ < μ < ∞∞ . Consider the statistic — S2 = 1 n Σ ( x X ) 2 , i = 1 which is location - invariant . Thus S2 must have a distribution that does not depend upon μ ; and hence , by ...
... complete sufficient statistic for the parameter μ , - ∞ < μ < ∞∞ . Consider the statistic — S2 = 1 n Σ ( x X ) 2 , i = 1 which is location - invariant . Thus S2 must have a distribution that does not depend upon μ ; and hence , by ...
Page 359
... complete sufficient statistic for 0 , is independent of Z. n 7.59 . Let X1 , X2 , . . . , X „ be a random sample from the normal distribution N ( 0 , σ2 ) , - ∞ << ∞ . Prove that a necessary and sufficient condition that the statistics ...
... complete sufficient statistic for 0 , is independent of Z. n 7.59 . Let X1 , X2 , . . . , X „ be a random sample from the normal distribution N ( 0 , σ2 ) , - ∞ << ∞ . Prove that a necessary and sufficient condition that the statistics ...
Page 537
... complete sufficient statistics for the parameters , under H。, so that it is independent of the test statistics . It is particularly interesting to note that it is relatively easy to use this technique in nonparametric methods by using ...
... complete sufficient statistics for the parameters , under H。, so that it is independent of the test statistics . It is particularly interesting to note that it is relatively easy to use this technique in nonparametric methods by using ...
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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