Example 17-23. Prove that under certain general conditions of regularity to be statedclearly the mean square deviation E(0 – O)² of an estimator 0 of the parameter 0, can never fallbelow a positive limit depending only on the density function f (x, 0), the size of the sample andthe bias of the estimate.

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Answered: Example 17-23. Prove that under certain…

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, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.
(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1
3. Suppose Y is a random sample of size 1 from a population with density function Oyº-1 0 0.5. (b) The hypotheses are Ho : 0 = 0, versus Ha : 0 > 00. What is the value of 00 if the signficance level of the test specified in (a) is 0.10? (c) Suppose a random sample of n = 7 observations is taken from this population to test the hypotheses Ho : 0 = 2 versus Ha : 0 > 2. As a decision rule, the experimenter plans to record X, the number of that exceed 0.9, and reject if X > 4. That is, the experimenter plans to use the test: ys 1 X >4 $(X) = { 0 x < 4 What is the probability of making a Type I error? Need Your Help with Part - (c)
A random variable Y have a distribution with parameter a > 0 and y, > 0 if its density function is given by: ays if y > yo fG) = ya+1 (0, elsewhere i) Derive the mean of Y. Show all necessary steps. (7) Show whether or not that the variance of Y is ays (а-2)(а-1)2" ii) (8)
Let Y1, Y2,..., Y, denote a random sample from the density function given by 1 yª-'e=y/®, y> 0, f(y[a, 0) = elsewhere, where a > 0 is known. a Find the MLE Ô of 0. b Find the expected value and variance of ê. c. Is the MLE ô an unbiased estimator for 0?
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
5. Suppose that Y₁, Y2, Y3 denote a random sample from an exponential distribution with density function 1 f(y) == e 6, y> 0, 00 Y₁ + 2Y₂ Five estimators are proposed for the parameter 0, Ô₁ = Y₁, 0₂: Y₁+Y₂ 03 2 = Ô₁ = Y(1) = " 3 and Ô5 = Y (a) Compute the bias of each estimator and hence identify the unbiased estimators. (b) Find for each of the estimators, the mean square error and hence the standard error. (c) Find the efficiency of 85 relative to each of the other four estimators and comment on your results.
Page 263, 4.4.8
3
2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.
7. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).

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