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sample average. Suppose we want to estimate the variance g(p) = p(1 – p). 1. Consider estimator g(n) = In(1 – In). Is it consistent for g(p)? Why 2. Assuming p + 1/2, find the distribution of n(g(Jn) – g(p)) as n →

sample average. Suppose we want to estimate the variance g(p) = p(1 – p). 1. Consider estimator g(n) = In(1 – In). Is it consistent for g(p)? Why 2. Assuming p + 1/2, find the distribution of n(g(Jn) – g(p)) as n →

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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Question

Solve both parts of the problem attached

**Problem Statement:** Suppose we observe \( X_1, X_2, \ldots, X_n \sim \text{iid } Ber(p) \) and let \( \bar{X}_n \) be the sample average. Suppose we want to estimate the variance \( g(p) = p(1-p) \). 1. Consider estimator \( g(\bar{y}_n) = \bar{y}_n (1 - \bar{y}_n) \). Is it consistent for \( g(p) \)? Why? 2. Assuming \( p \neq 1/2 \), find the distribution of \( \sqrt{n} \left( g(\bar{y}_n) - g(p) \right) \) as \( n \to \infty \).
Transcribed Image Text:**Problem Statement:** Suppose we observe \( X_1, X_2, \ldots, X_n \sim \text{iid } Ber(p) \) and let \( \bar{X}_n \) be the sample average. Suppose we want to estimate the variance \( g(p) = p(1-p) \). 1. Consider estimator \( g(\bar{y}_n) = \bar{y}_n (1 - \bar{y}_n) \). Is it consistent for \( g(p) \)? Why? 2. Assuming \( p \neq 1/2 \), find the distribution of \( \sqrt{n} \left( g(\bar{y}_n) - g(p) \right) \) as \( n \to \infty \).
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