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1. Let Y1,...,Yn * f(y; 0). For the following densities f, find: • the moment estimator of 0; • the MLE of 0; • the Fisher information I(0) = –nE ( log f(Yi; 0)); • the Cramer-Rao lower bound for an unbiased estimator of 0. (a) Poisson: f(y; 0) = 0" exp{=0} for y E {0, 1, 2, . } and 0 > 0. for y E {0, 1,2, ...} and 0 > 0.

1. Let Y1,...,Yn * f(y; 0). For the following densities f, find: • the moment estimator of 0; • the MLE of 0; • the Fisher information I(0) = –nE ( log f(Yi; 0)); • the Cramer-Rao lower bound for an unbiased estimator of 0. (a) Poisson: f(y; 0) = 0" exp{=0} for y E {0, 1, 2, . } and 0 > 0. for y E {0, 1,2, ...} and 0 > 0.

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 \( Y_1, \ldots, Y_n \overset{\text{iid}}{\sim} f(y; \theta) \). For the following densities \( f \), find: - the moment estimator of \( \theta \); - the MLE (Maximum Likelihood Estimator) of \( \theta \); - the Fisher information \( I(\theta) = -n \mathbb{E} \left( \frac{d^2}{d\theta^2} \log f(Y_i; \theta) \right) \); - the Cramer-Rao lower bound for an unbiased estimator of \( \theta \). (a) Poisson: \( f(y; \theta) = \frac{\theta^y \exp\{-\theta\}}{y!} \) for \( y \in \{0, 1, 2, \ldots\} \) and \( \theta > 0 \). (b) Bernoulli: \( f(y; \theta) = \theta^y (1 - \theta)^{1-y} \) for \( y \in \{0, 1\} \) and \( \theta \in (0, 1) \). (c) Geometric: \( f(y; \theta) = \theta (1 - \theta)^{y-1} \) for \( y \in \{1, 2, \ldots\} \) and \( \theta \in (0, 1) \). (d) Exponential: \( f(y; \theta) = \frac{1}{\theta} \exp \left\{-\frac{y}{\theta}\right\} \) for \( y > 0 \) and \( \theta > 0 \).
Transcribed Image Text:1. Let \( Y_1, \ldots, Y_n \overset{\text{iid}}{\sim} f(y; \theta) \). For the following densities \( f \), find: - the moment estimator of \( \theta \); - the MLE (Maximum Likelihood Estimator) of \( \theta \); - the Fisher information \( I(\theta) = -n \mathbb{E} \left( \frac{d^2}{d\theta^2} \log f(Y_i; \theta) \right) \); - the Cramer-Rao lower bound for an unbiased estimator of \( \theta \). (a) Poisson: \( f(y; \theta) = \frac{\theta^y \exp\{-\theta\}}{y!} \) for \( y \in \{0, 1, 2, \ldots\} \) and \( \theta > 0 \). (b) Bernoulli: \( f(y; \theta) = \theta^y (1 - \theta)^{1-y} \) for \( y \in \{0, 1\} \) and \( \theta \in (0, 1) \). (c) Geometric: \( f(y; \theta) = \theta (1 - \theta)^{y-1} \) for \( y \in \{1, 2, \ldots\} \) and \( \theta \in (0, 1) \). (d) Exponential: \( f(y; \theta) = \frac{1}{\theta} \exp \left\{-\frac{y}{\theta}\right\} \) for \( y > 0 \) and \( \theta > 0 \).
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