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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

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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  1. Let X1,...,Xn ∼ iid Pois(θ), and we wish to estimate φ = Pθ(X = 0) = e−θ.

    1. (a)  Find the MLE φˆ.

    2. (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.

    3. (c)  Consider Yi = I (Xi = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution

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