Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum v
Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum v
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, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0
1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N).
2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ.
3. Give the definition of a consistent estimator.
4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ.
5. Show that the estimator θbn is a minimum variance estimator of θ. (Hint: use the Cramer-Rao inequality given by var(θb) ≥ 1 nE ∂ ln(f(X;θ) ∂θ 2
Transcribed Image Text:Let X1,.., Xn be a random sample from a population with 0 unknown and given by the
density
1
if x > 0
f(x; 0)
= { 20e-
20 x
if x <0
1.
Show that E(X) = 20² and E(VX)= 0 (Hint: you may use that e=%za-1dz =
(α-1) ! for every α Ε Ν ).
2.
Show that the statistic
(1)
n
i=D1
Text
is an unbiased estimator of 0.
3.
Give the definition of a consistent estimator.
4.
Show that the estimator 0, given in relation (1) is a consistent estimator of 0.
n
5.
Show that the estimator 0n is a minimum variance estimator of 0. (Hint: use the
Cramer-Rao inequality given by
1
var(@) >
´Ə ln(f(X;8)`
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