Let X1,...,Xn ∼ iid Pois(θ), and we wish to estimate φ = Pθ(X = 0) = e−θ. (a) Find the MLE φˆ. (b) Use Jensen’s inequality to derive a result comparing E(φˆ) to φ. In fact, since g(t) = e−t is not a linear function, the inequality this result is strictly, implying that φˆ is a biased estimator. (c) Consider Yi = I (Xi = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution

Let X1,...,Xn ∼ iid Pois(θ), and we wish to estimate φ = Pθ(X = 0) = e−θ. (a) Find the MLE φˆ. (b) Use Jensen’s inequality to derive a result comparing E(φˆ) to φ. In fact, since g(t) = e−t is not a linear function, the inequality this result is strictly, implying that φˆ is a biased estimator. (c) Consider Yi = I (Xi = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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