Introduction to Mathematical Statistics |
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Page 249
... decision functions is better than the other for some values of and the other decision function is better for other values of 0 . If , however , we had restricted our consideration to decision functions w such that E [ w ( Y ) ] = 0 for ...
... decision functions is better than the other for some values of and the other decision function is better for other values of 0 . If , however , we had restricted our consideration to decision functions w such that E [ w ( Y ) ] = 0 for ...
Page 250
... decision functions of the form w ( y ) = b + y / n , where b does not depend upon y , show that R ( 0 , w ) b2 + 0 / n . What decision function of this form yields a uniformly smaller risk than every other decision function of this form ...
... decision functions of the form w ( y ) = b + y / n , where b does not depend upon y , show that R ( 0 , w ) b2 + 0 / n . What decision function of this form yields a uniformly smaller risk than every other decision function of this form ...
Page 279
... decision function pro- cedure , we need a function w of the observed values of X1 , ... , Xn ( or , of the observed value of a statistic Y ) that decides which of the two values of 0 , 0 ' or 0 " , to accept . That is , the function w ...
... decision function pro- cedure , we need a function w of the observed values of X1 , ... , Xn ( or , of the observed value of a statistic Y ) that decides which of the two values of 0 , 0 ' or 0 " , to accept . That is , the function w ...
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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 F distribution function F(x given hypothesis H₁ 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 Poisson distribution positive integer probability density functions quadratic form random experiment random interval random sample random variables X1 respectively Show significance level simple hypothesis statistic Y₁ stochastically independent random sufficient statistic theorem unbiased statistic variables X₁ variance o² W₁ X₁ X₁ and X2 X₂ x²(n Y₂ Z₁ zero elsewhere μ₁ σ²