By Bradley Efron, R.J. Tibshirani
Statistics is a topic of many makes use of and unusually few potent practitioners. the conventional highway to statistical wisdom is blocked, for many, by way of a powerful wall of arithmetic. The technique in An creation to the Bootstrap avoids that wall. It palms scientists and engineers, in addition to statisticians, with the computational innovations they should examine and comprehend complex information units.
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6) and simplifying, we obtain, when n = m + 1 that q 1 q+1 p+q+1 p E XU(m) XU(m+1) − E XU(m) q+1 , 1 p q+1 p q+1 E XU(m) XU(n) − E XU(m) XU(n−1) q+1 . 5) follow readily when the above two equations are rewritten. 4. 4), we obtain, for n ≥ m + 1, that p q+1 n p+q+1 E XU(m)XU(n) = E XU(m) + (q + 1) ∑ j=m+1 p q XU(m) XU( j) . 5. For n ≥ m + 1, Cov XU(m) , XU(n) = Var XU(m) . Proof. 7), we obtain 2 E XU(m) XU(n) = E XU(m) + (n − m) E XU(m) . 3), we obtain E XU(n) = E XU(m) + (n − m) , n > m. 9), we get for n ≥ m + 1 Cov XU(m) , XU(n) = E XU(m) XU(n) − E XU(m) E XU(n) 2 + (n − m) E XU(m) − E XU(m) = E XU(m) 2 − (n − m) E XU(m) = Var XU(m) .
2) can be used in a simple way to compute all the simple moments of all the record values. Once again, using property that f (x) = 1 − F (x), we can derive some simple recurrence relations for the product moments of record values. 3. For m ≥ 1 and p, q = 0, 1, 2, . . 4) 40 M. G. Hamedani and for 1 ≤ m ≤ n − 2, p, q = 0, 1, 2, . . 5) Proof. Let us consider 1 ≤ m < n and p, q = 0, 1, 2, . . 6) 0 where ∞ yq [R (y) − R (x)]n−m−1 f (y) dy I (x) = x ∞ = yq [R (y) − R (x)]n−m−1 (1 − F (y))dy, since f (y) = 1 − F (y) .
G. Hamedani But Xˆ U(s) depends on the unknown parameter σ. If we substitute the minimum variance unbiased estimate σˆ for σ, then Xˆ U(s) becomes equal to XˆU(s) . Now E[Xˆ U(s) ] = µ + sσ = E XU(s) ,Var[Xˆ U(s) ] = mσ2 and MSE[Xˆ U(s) ] = E[(Xˆ U(s) − XU(s) )2] = (s − m) σ2 . We like to mention also that by considering the mean squared errors of XˆU(s) , XU(s) and Xˆ U(s) , it can be shown that MSE[Xˆ U(s) ] = E[(Xˆ U(s) − XU(s) )2] = (s − m) σ2 . 6. Limiting Distribution of Record Values We have seen that for µ = 0 and σ = 1, E XU(n) = n and Var XU(n) = n.
An introduction to bootstrap by Bradley Efron, R.J. Tibshirani