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6.6-4. Find the Rao-Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator 0, if the random sample X₁, X2, ..., X is taken from each of the distributions having the following pdfs: (a) f(x;0)=(1/02)xe−x/@ (b) f(x; 0) = (1/20³) x² e-x/0, (c) f(x; 0) = (1/0)x(1-0)/8 0 < x <∞0, 0 <0 <∞0. 0

6.6-4. Find the Rao-Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator 0, if the random sample X₁, X2, ..., X is taken from each of the distributions having the following pdfs: (a) f(x;0)=(1/02)xe−x/@ (b) f(x; 0) = (1/20³) x² e-x/0, (c) f(x; 0) = (1/0)x(1-0)/8 0 < x <∞0, 0 <0 <∞0. 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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**Problem 6.6-4:** Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator \(\hat{\theta}\), if the random sample \(X_1, X_2, \ldots, X_n\) is taken from each of the distributions having the following probability density functions (pdfs): **(a)** \( f(x; \theta) = \left(\frac{1}{\theta^2}\right) x e^{-x/\theta}, \quad 0 < x < \infty, \quad 0 < \theta < \infty. \) **(b)** \( f(x; \theta) = \left(\frac{1}{2\theta^3}\right) x^2 e^{-x/\theta}, \quad 0 < x < \infty, \quad 0 < \theta < \infty. \) **(c)** \( f(x; \theta) = \left(\frac{1}{\theta}\right) x^{(1-\theta)/\theta}, \quad 0 < x < 1, \quad 0 < \theta < \infty. \)
Transcribed Image Text:**Problem 6.6-4:** Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator \(\hat{\theta}\), if the random sample \(X_1, X_2, \ldots, X_n\) is taken from each of the distributions having the following probability density functions (pdfs): **(a)** \( f(x; \theta) = \left(\frac{1}{\theta^2}\right) x e^{-x/\theta}, \quad 0 < x < \infty, \quad 0 < \theta < \infty. \) **(b)** \( f(x; \theta) = \left(\frac{1}{2\theta^3}\right) x^2 e^{-x/\theta}, \quad 0 < x < \infty, \quad 0 < \theta < \infty. \) **(c)** \( f(x; \theta) = \left(\frac{1}{\theta}\right) x^{(1-\theta)/\theta}, \quad 0 < x < 1, \quad 0 < \theta < \infty. \)
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