Sequential interval estimation for the exponential hazard rate when the loss function is strictly convex
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Let X1,X∈, ㆍ ㆍ ㆍ ,Xn be independent and identically distributed random variables having common exponential density withunknown mean μ. In the sequential con dence interval estimationfor the exponential hazard rate θ = 1=μ, when the loss function isstrictly convex, the following stopping rule is proposed with the halflength d of prescribed con dence interval In for the parameter θ, τ = smallest integer n such that n ≥ x2/2θ2/d2+2,where θ = (n - 1)Xn-1/=n is the minimum risk estimator for θand zα/2 is defined by P(｜Z｜≤α=∈) = 1 - α (α ∈ (0, 1)) withZ ~ N(0, 1). For the confidence intervals In which is required tosatisfy P(θ ∈ In) ≥ 1 - α. These estimated intervals Iτ have theasymptotic consistency of the sequential procedure,limd→0 P(θ ∈ Iτ ) = 1 - α, where α ∈ (0, 1) is given.
Abstract
Let X1,X∈, ㆍ ㆍ ㆍ ,Xn be independent and identically distributed random variables having common exponential density withunknown mean μ. In the sequential con dence interval estimationfor the exponential hazard rate θ = 1=μ, when the loss function isstrictly convex, the following stopping rule is proposed with the halflength d of prescribed con dence interval In for the parameter θ, τ = smallest integer n such that n ≥ x2/2θ2/d2+2,where θ = (n - 1)Xn-1/=n is the minimum risk estimator for θand zα/2 is defined by P(｜Z｜≤α=∈) = 1 - α (α ∈ (0, 1)) withZ ~ N(0, 1). For the confidence intervals In which is required tosatisfy P(θ ∈ In) ≥ 1 - α. These estimated intervals Iτ have theasymptotic consistency of the sequential procedure,limd→0 P(θ ∈ Iτ ) = 1 - α, where α ∈ (0, 1) is given.
참고문헌 (12)
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