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A Bayesian-like estimator of the process capability index by Pearn W. L., Lin G. H. PDF

By Pearn W. L., Lin G. H.

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23) we get Eκ(p) T = k 1−p 1−p and Vκ(p) (T ) = k 2 . p p In the previous examples we have already studied different ways of parametrizing an exponential family. However, among all parametrizations there is one in particular, not mentioned so far, that has a special meaning. This is the so-called mean value parametrization which is considered at the conclusion of this section. , Brown (1986) and Witting (1985)). 22. 24) is a diffeomorphism of Δ0 onto the open set γm (Δ0 ). Proof. 19 that K is strictly convex.

A convex function v is strictly convex in (a, b) if and only if v(αx + (1 − α)y) < αv(x) + (1 − α)v(y), a < x, y < b, x = y, 0 < α < 1. 62). ∗ Let v : (0, ∞) → R be convex. 62) is nonnegative and v0 (1) = 0. It holds D+ v0 (x) = D+ v(x) − D+ v(1). The function v0 is nonincreasing in (0, 1] and nondecreasing in (1, ∞). The function v is strictly convex at 1 if and only if at least one of the following two cases holds: v0 (x) > 0, x ∈ (0, 1), or v0 (x) > 0, x ∈ (1, ∞). For later purposes we define the ∗ -conjugate function by 1 v∗ (x) = xv( ), x x > 0.

Find the Lebesgue density of N(μ, σ 2 ), where both μ and σ 2 are unknown, in the mean value parametrization. When inspecting the structure of the density fθ of an exponential family one might get the impression that the factor exp{−K(θ)} plays only a subordinate role as a normalizing factor. 10) reveals the meaning of exp{−K(θ)}. Only this function and the underlying measure ν vary from one family to another. Therefore exp{−K(θ)} carries the full structure of the exponential family. Furthermore, we show that K(θ) determines the measure ν = μ ◦ T −1 uniquely.

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A Bayesian-like estimator of the process capability index Cpmk by Pearn W. L., Lin G. H.


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