(c) What is the asymptotic distribution of \(\sqrt{n}(\hat{\theta} - \theta)\)?6.2.9. If \(X_1, X_2, \ldots, X_n\) is a random sample from a distribution with pdf\[f(x; \theta) = \begin{cases} \frac{3\theta^3}{(x+\theta)^4} & 0 < x < \infty, 0 < \theta < \infty \\0 & \text{elsewhere}\end{cases}\]Show that \(Y = 2\bar{X}\) is an unbiased estimator of \(\theta\) and determine its efficiency.

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Answered: 6.2.9. If X₁, X2,..., Xn is a random…

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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= 7.1.7. Let X1, X2,..., Xn denote a random sample from a distribution that is N(μ,0), 0 < 0 <∞o, where u is unknown. Let Y E(Xi-X)2/n and let L[0,8(y)] = [0-8(y)]². If we consider decision functions of the form 8(y) = by, where b does not depend upon y, show that R(0, 8) = (0²/n²) [(n² - 1)6² -2n(n-1)b+n²]. Show that b = n/(n+1) yields a minimum risk decision function of this form. Note that nY/(n+1) is not an unbiased estimator of 0. With 8(y) ny/(n+1) and 0<0<∞, determine max, R(0,8) if it exists. =
1.9.18. Find the mean and the variance of the distribution that has the cdf x <0 0
1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
7.4.5. Show that the first order statistic Y₁ of a random sample of size n from the distribution having pdf f(x; 0) = e-(-0), 0 < x <∞, -∞ < 0 < ∞, zero elsewhere, is a complete sufficient statistic for 0. Find the unique function of this statistic which is the MVUE of 0. 7.4.6. Let a random sample of size n be taken from a distribution of the discrete
4. 65th Pg. 59, 1. 7. 19. * Find the of the distribution having pot F(x) = 1x1/4₂ Percentile where -a, < x < 2, f(x) = 0, elsewhere.

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6.2.9. If X₁, X2,..., Xn is a random sample from a distribution with pdf 303 f(x; 0) (x+0)4 = -{ 0