# Applied Statistical Decision Theory by Howard Raiffa

By Howard Raiffa

"In the sphere of statistical choice concept, Raiffa and Schlaifer have sought to advance new analytic concepts wherein the fashionable conception of application and subjective likelihood can really be utilized to the industrial research of general sampling problems."

—From the foreword to their vintage paintings *Applied Statistical selection Theory*. First released within the Sixties via Harvard collage and MIT Press, the publication is now provided in a brand new paperback version from Wiley

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Moreover, Iµ (x) = sup ξx − Λµ (ξ) : ξ ≥ 0 for x ∈ [m, ∞) Iµ (x) = sup ξx − Λµ (ξ) : ξ ≤ 0 for x ∈ (−∞, m]. and Finally, if α = inf x ∈ R : µ (−∞, x] > 0 and β = sup x ∈ R : µ [x, ∞) > 0 , then Iµ is smooth on (α, β) and identically +∞ off of [α, β]. In fact, either µ({m}) = 1 and α = m = β or m ∈ (α, β), in which case Λµ is a smooth, strictly increasing mapping from R onto (α, β), Iµ (x) = Ξµ (x) x − Λµ Ξµ (x) , x ∈ (α, β), where Ξµ = Λµ −1 is the inverse of Λµ , µ({α}) = e−Iµ (α) if α > −∞, and µ({β}) = e−Iµ (β) if β < ∞.

12) |α − m| ≤ 2Var(Y ) for all α ∈ med(Y ). 13 (L´ evy’s Reflection Principle). Let Xn : n ∈ Z+ be a sequence of P-independent random variables, and, for k ≤ , choose α ,k ∈ med S − Sk . 14) P max Sn + αN,n ≥ ≤ 2P SN ≥ , max Sn + αN,n ≥ ≤ 2P |SN | ≥ . 14) to both the sequences Xn : n ≥ 1} and {−Xn : n ≥ 1} and then adding the two results. 14), set A1 = S1 + αN,1 ≥ An+1 = max S + αN, and and Sn+1 + αN,n+1 ≥ < 1≤ ≤n 41 for 1 ≤ n < N . Obviously, the An ’s are mutually disjoint and N An = n=1 max Sn + αN,n ≥ 1≤n≤N .

Concentrate at one point). In particular, because Λ (ξ) is the variance of the νξ , we know that Λ > 0 everywhere. Hence, Λ is strictly increasing and therefore admits a smooth inverse Ξ on its image. Furthermore, because Λ (ξ) is the mean of νξ , it is clear that the image of Λ is contained in (α, β). At the same time, given an x ∈ (α, β), note that e−ξx eξy µ(dy) −→ ∞ as |ξ| → ∞, R and therefore ξ ξx − Λ(ξ) achieves a maximum at some point ξx ∈ R. In addition, by the first derivative test, Λ (ξx ) = x, and so ξx = Ξ−1 (x).