Introduction to Mathematical Statistics |
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Page 117
... Jacobian ( denoted by J ) of the transformation . In most mathematical areas , J = w ' ( y ) is referred to as the Jacobian of the inverse transformation x = w ( y ) , but in this book it will be called the Jacobian of the ...
... Jacobian ( denoted by J ) of the transformation . In most mathematical areas , J = w ' ( y ) is referred to as the Jacobian of the inverse transformation x = w ( y ) , but in this book it will be called the Jacobian of the ...
Page 118
... Jacobian of the transformation and will be denoted by the symbol J. It will be assumed that these first - order partial derivatives are continuous and that the Jacobian J is not identically equal to zero in B. An illustrative example ...
... Jacobian of the transformation and will be denoted by the symbol J. It will be assumed that these first - order partial derivatives are continuous and that the Jacobian J is not identically equal to zero in B. An illustrative example ...
Page 211
... Jacobian J. Thus = = w2 ( Y1 , Yn ) , .... .... Xn = wn ( Y1 , .... yn ) and f [ wi ( y1 , ... , Yn ) ; 0 ]・・・ f ... Jacobian J , it follows that h ( y2 , ... , Yn y1 ; 0 ) does not depend upon 0 and that Y1 is a sufficient statistic ...
... Jacobian J. Thus = = w2 ( Y1 , Yn ) , .... .... Xn = wn ( Y1 , .... yn ) and f [ wi ( y1 , ... , Yn ) ; 0 ]・・・ f ... Jacobian J , it follows that h ( y2 , ... , Yn y1 ; 0 ) does not depend upon 0 and that Y1 is a sufficient statistic ...
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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