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For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population following the exponential distribution with density function given by 3 f(x)= 0, -e-%¸ if x > 0 Otherwise. Five possible estimators of are Ô‚ = X₁, ô₂ = ¹/(X₁ + X₂), Ô‚ = — (X₁ + 2X₂), Ô₁ = X, and ô₂ = -—(X₁ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0²2. (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?

For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population following the exponential distribution with density function given by 3 f(x)= 0, -e-%¸ if x > 0 Otherwise. Five possible estimators of are Ô‚ = X₁, ô₂ = ¹/(X₁ + X₂), Ô‚ = — (X₁ + 2X₂), Ô₁ = X, and ô₂ = -—(X₁ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0²2. (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?

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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For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?
Transcribed Image Text:For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?
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