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1. Suppose X₁,..., Xn is a random sample. (a) If X, are iid Bernoulli (0) with € (0, 1), show that the likelihood function can be written in the canonical form of the exponential family L(0) = h(x) exp {nT(x) − A* (n)} for X (X₁,..., Xn), x, E {0, 1}. Identify a sufficient statistic T for 0, and find E(T) and Var(T) using A*. = (b) Using the same argument, show that if X, are iid N(0, 1), then T = Σ₁₁ Xi is a sufficient statistic for 0. Identify E(T) and Var(T). (c) Using the same argument, if X; are iid N(0, 0), find a sufficient statistic for 0.

1. Suppose X₁,..., Xn is a random sample. (a) If X, are iid Bernoulli (0) with € (0, 1), show that the likelihood function can be written in the canonical form of the exponential family L(0) = h(x) exp {nT(x) − A* (n)} for X (X₁,..., Xn), x, E {0, 1}. Identify a sufficient statistic T for 0, and find E(T) and Var(T) using A*. = (b) Using the same argument, show that if X, are iid N(0, 1), then T = Σ₁₁ Xi is a sufficient statistic for 0. Identify E(T) and Var(T). (c) Using the same argument, if X; are iid N(0, 0), find a sufficient statistic for 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. Suppose X₁,. Xn is a random sample. 9 (a) If X₁ are iid Bernoulli(0) with € (0,1), show that the likelihood function can be written in the canonical form of the exponential family L(0) = h(x) exp {nT(x) — A*(n)} for X = (X₁,..., Xn), X; = {0, 1}. Identify a sufficient statistic T for 0, and find E(T) and Var(T) using A*. (b) i Using the same argument, show that if X, are iid N(0, 1), then T = Σ1 X₁ is a sufficient statistic for 0. Identify E(T) and Var(T). (c) Using the same argument, if X, are iid N(0, 0), find a sufficient statistic for 0.
Transcribed Image Text:1. Suppose X₁,. Xn is a random sample. 9 (a) If X₁ are iid Bernoulli(0) with € (0,1), show that the likelihood function can be written in the canonical form of the exponential family L(0) = h(x) exp {nT(x) — A*(n)} for X = (X₁,..., Xn), X; = {0, 1}. Identify a sufficient statistic T for 0, and find E(T) and Var(T) using A*. (b) i Using the same argument, show that if X, are iid N(0, 1), then T = Σ1 X₁ is a sufficient statistic for 0. Identify E(T) and Var(T). (c) Using the same argument, if X, are iid N(0, 0), find a sufficient statistic for 0.
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