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Q64Section ST (Statistics (Q11–Q65))64 of 65

Unbiased estimator through the maximum

Let X1,X2X_1, X_2 be a random sample from a distribution having the population density function f(x)={1/θ,0<x<θ 0,otherwisef(x) = \begin{cases} 1/\theta, & 0 < x < \theta \ 0, & \text{otherwise} \end{cases}, where θ∈(0,∞)\theta \in (0, \infty). Let X(2)=max⁡{X1,X2}X_{(2)} = \max\{X_1, X_2\} and ψ(θ)=Pθ(X1+X2<1)\psi(\theta) = P_\theta(X_1 + X_2 < 1), θ>0\theta > 0. Let δ(X(2))\delta(X_{(2)}) be an unbiased estimator of ψ(θ)\psi(\theta) that depends on observations X1X_1 and X2X_2 only through X(2)X_{(2)}. If δ(t)\delta(t) is a continuous function on (0,∞)(0, \infty), then the value of 18δ(3/4)18 \delta(3/4) equals ________ (answer in integer).

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