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 240
... converges in probability to a random variable X if , for every € > 0 , or equivalently , lim Pr ( X , – X ] < € ) = 1 , ∞18 lim Pr ( X ,, - X ] ≥ € ) = 0 . ∞18 Statisticians are usually interested in this convergence when the random ...
... converges in probability to a random variable X if , for every € > 0 , or equivalently , lim Pr ( X , – X ] < € ) = 1 , ∞18 lim Pr ( X ,, - X ] ≥ € ) = 0 . ∞18 Statisticians are usually interested in this convergence when the random ...
Page 255
... probability to p and 1 - p , respectively ; thus ( Y / n ) ( 1 − Y „ / n ) converges in probability to p ( 1 − p ) . Then , by Theorem 4 , ( Yn / n ) ( 1 − Yn / n ) / [ p ( 1 − p ) ] converges in probability to 1 , and Theorem 5 ...
... probability to p and 1 - p , respectively ; thus ( Y / n ) ( 1 − Y „ / n ) converges in probability to p ( 1 − p ) . Then , by Theorem 4 , ( Yn / n ) ( 1 − Yn / n ) / [ p ( 1 − p ) ] converges in probability to 1 , and Theorem 5 ...
Page 256
... converges in probability to cd . ( c ) If d0 , the ratio U / V , converges in probability to c / d . 5.37 . Let U , converge in probability to c . If h ( u ) is a continuous function at u = c , prove that h ( Un ) converges in probability ...
... converges in probability to cd . ( c ) If d0 , the ratio U / V , converges in probability to c / d . 5.37 . Let U , converge in probability to c . If h ( u ) is a continuous function at u = c , prove that h ( Un ) converges in probability ...
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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