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 nota 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.

Answered: Let X1,...,Xn ∼ iid Pois(θ), and we…

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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Let X₁,...,X_n ∼ iid Pois(θ), and we wish to estimate φ = P_θ(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^−tis not
  
  a linear function, the inequality this result is strictly, implying that φˆ is a biased estimator.
3. (c) Consider Y_i = I (X_i = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution.

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In Step 2, am I supposed to take a second derivative of the log likelihood function and make sure it's < 0?

Also in Step 3, I don't understand how we proved φˆ is a biased estimator. I thought we proved that E φˆ = φˆ and if so it should be an unbiased estimator

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