Suppose X is a random variable with mean 0 and variance 2 . Recall that the

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Suppose X is a random variable with mean 0 and variance σ2. Recall that the function Fx,ϵ(t) is the cdf of the random variable U = I1−ϵX + [1 − I1−ϵ]W, where X, 1−ϵ, and W are independent random variables, X has cdf FX(t), W has cdf Δx(t), and I1−ϵ has a binomial(1,1 − ϵ ) distribution. Define the functional Var(FX) = Var(X) = σ2. To derive the influence function of the variance, perform the following steps:
(a) Show that E(U) = ϵx.
(b) Show that Var(U) = (1 − )σ2 + ϵx2 − ϵ2x2.
(c) Obtain the partial derivative of the right side of this last equation with respect to ϵ. This is the influence function.

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Introduction To Mathematical Statistics

ISBN: 9780321794710

7th Edition

Authors: Robert V., Joseph W. McKean, Allen T. Craig

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