By D.R. Cox

This ebook might be of curiosity to senior undergraduate and postgraduate scholars of utilized facts.

**Read Online or Download Applied Statistics: Principles and Examples (Chapman & Hall CRC Texts in Statistical Science) PDF**

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G. Hamedani and ˆ (1 + E12 ) , σ = σ/ where µˆ and σˆ are MVLUEs of µ and σ and ˆ Var[ˆµ] Cov (ˆµ, σ) ˆ Var[σ] ˆ Cov (ˆµ, σ) = σ2 E11 E12 . E21 E22 The mean squared errors of these estimators are 2 (1 + E22 )−1 , MSE[µ] = σ2 E11 − E12 and MSE[σ] = σ2 E22 (1 + E22 )−1 . We have E [(µ − µ) (σ − σ)] = σ2 E12 (1 + E22 )−1 . 2), we obtain µ = (m + 1) XU(1) − XU(m) /m, σ = XU(m) − XU(1) /m, Var [µ] = σ2 m2 + m − 1 /m, and Var [σ] = σ2 (m − 1) /m2 . 5. Prediction of Record Values We will predict the sth upper value based on the first m record values for s > m.

Xn−r2,n . Suppose that X has an absolutely continuous cd f F of the form −∞ < µ < ∞, F ((x − µ) /σ) , σ > 0. Further, we assume that E [Xr,n ] = µ + αr σ, Var [Xr,n ] = vrr σ2 , Cov (Xr,n , Xs,n ) = vrs σ , 2 r1 + 1 ≤ r ≤ n − r 2 , r1 + 1 ≤ r, s ≤ n − r2 . Let X = (Xr1 +1,n , . , Xn−r2,n ), then we can write α, E X = µ11 + σα with 1 = (1, 1, . , 1) , α = (αr1 +1 , . , αn−r2 ) , and Var X = σ2 ∑, where ∑ is an (n − r2 − r1 ) × (n − r2 − r1 ) matrix with elements vrs , r1 < r, s ≤ n − r2 . 13) 2 .

N−r2 ) , where αr = E [(Xr,n − µ) /σ] = r 1 ∑ n− j+1. j=1 Simple calculations show that ∑ α αr1+1 − (n − r1 − 1) , 1, 1, . , 1, r2 + 1 , cr1+1 α2 −1 α ∑ α = r1 +1 + (n − r1 − r2 − 1) , cr1 +1 −1 = ∑ 1 = αr +1/cr +1, −1 1 ∑ 1 = 1/cr +1, −1 1 ∑ α = αr +1 /cr +1 , −1 α 1 1 1 1 1 αr1 +1 −1 −1 1 ∑ 1α ∑ = − (n − r1 − 1) , 1, 1, . , r2 + 1 , cr1+1 cr1 +1 1 αr1 +1 −1 −1 , 0, 0, . , 0 , 1 ∑ α1 ∑ = cr1+1 cr1 +1 1 ∆= α ∑ −1 α 1 ∑ −1 1 − α∑ −1 1 2 = (n − r1 − r2 − 1) /cr1 +1 . Upon simplification, we obtain σˆ∗ = 1 1 ∆ ∑ −1 = 1α ∑ −1 −11 1 n − r2 − r 1 − 1 ∑ −1 α1 ∑ −1 X n−r2 ∑ X j,n − (n − r1 )Xr1 +1,n + r2 Xn−r2,n .