Introduction to Mathematical Statistics |
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Page 205
... statistic whose mathematical expectation is equal to a parameter 0 is called an unbiased statistic for the parameter 0 . Otherwise the statistic is said to be biased . = = Now it would seem that if Y1 = u1 ( X1 , X2 , ... , xn ) is to ...
... statistic whose mathematical expectation is equal to a parameter 0 is called an unbiased statistic for the parameter 0 . Otherwise the statistic is said to be biased . = = Now it would seem that if Y1 = u1 ( X1 , X2 , ... , xn ) is to ...
Page 218
... statistic for the parameter 0. Let Y2 = u2 ( X1 , X2 , Xn ) be another statistic ( but not a function of Y1 alone ) which is an unbiased statistic for 0 ; that is , E ( Y2 ) = 0. Consider E ( Y2y1 ) . This expectation is a function of ...
... statistic for the parameter 0. Let Y2 = u2 ( X1 , X2 , Xn ) be another statistic ( but not a function of Y1 alone ) which is an unbiased statistic for 0 ; that is , E ( Y2 ) = 0. Consider E ( Y2y1 ) . This expectation is a function of ...
Page 222
... unbiased statistic for 0 . That is , the statistic ( Y1 ) is the best statistic for 0. This fact is stated in the following theorem . Theorem 5. Let X1 , X2 , ... , Xn , n a fixed positive integer , denote a random sample from a ...
... unbiased statistic for 0 . That is , the statistic ( Y1 ) is the best statistic for 0. This fact is stated in the following theorem . Theorem 5. Let X1 , X2 , ... , Xn , n a fixed positive integer , denote a random sample from a ...
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A₁ A₂ Accordingly best critical region c₁ cent confidence interval chi-square distribution complete sufficient statistic compute conditional p.d.f. confidence interval Consider continuous type critical region decision function defined degrees of freedom denote a random discrete type distribution function distribution having p.d.f. Equation Example EXERCISES function F(x given H₁ hypothesis H independent random variables integral joint p.d.f. k₁ Let X1 Let Y₁ limiting distribution marginal p.d.f. matrix maximum likelihood moment-generating function mutually stochastically independent noncentral normal distribution order statistics p.d.f. of Y₁ P(A₁ p₁ Poisson distribution positive integer probability density functions quadratic form random experiment random interval random sample random variables X1 respectively sample space Show significance level simple hypothesis statistic Y₁ stochastically independent random sufficient statistic theorem unbiased statistic variance oČ w₁ X₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ