Introduction to Mathematical Statistics |
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Page 351
... symmetric matrix is one , the matrix is idempotent , that is , A2 = A , and con- versely ( see Exercise 13.7 ) . Accordingly , if X'AX / σ2 has a chi - square distribution , then A2 = A and the random variable is x2 ( r ) , where r is ...
... symmetric matrix is one , the matrix is idempotent , that is , A2 = A , and con- versely ( see Exercise 13.7 ) . Accordingly , if X'AX / σ2 has a chi - square distribution , then A2 = A and the random variable is x2 ( r ) , where r is ...
Page 352
... symmetric matrix . Prove that each of the nonzero characteristic numbers of A is equal to one if and only if A2 = A. Hint . Let L be an orthogonal matrix such that L'AL note that A is idempotent if and only if L'AL is idempotent ...
... symmetric matrix . Prove that each of the nonzero characteristic numbers of A is equal to one if and only if A2 = A. Hint . Let L be an orthogonal matrix such that L'AL note that A is idempotent if and only if L'AL is idempotent ...
Page 358
... matrix , and let A be a real symmetric matrix of order n . Accept the fact that the linear form b'X and the quadratic form X'AX are stochastically independent if and only if b'A = 0. Use this fact to prove that b'X and X'AX are ...
... matrix , and let A be a real symmetric matrix of order n . Accept the fact that the linear form b'X and the quadratic form X'AX are stochastically independent if and only if b'A = 0. Use this fact to prove that b'X and X'AX are ...
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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 o² w₁ X₁ X₁ and X2 X₂ Y₂ Z₁ zero elsewhere μ₁ μ₂ Σ Σ