B1. Suppose we have X₁, X2,... Xn, which are i.i.d. and come from the uniform distributionUniform(μ-n, μ+n), n>0;O= (µ, n)(a) Use the Method of Moments to estimate μl and n.(b) Determine whether the estimate derived by the Method of Moments is biased forμ.(c) Use the Maximum Likelihood method to estimate and n. Use the notation:X(1) = min(X₁, X2, ..., Xn) and X(n)=max(X₁, X2,..., Xn)(d) It can be derived thatfx(1)(x)n=χεμ - ημ + n],χε[μ - ημ + η]Determine whether the estimate obtained by the Method of Maximum Likelihoodis biased for .Hint: Use substitutions y = 12η(fx(n) (x)X- (μ- - n)`2ηnn-1X - (μ-n)27"n-1"x-(μ-n)2ηand y' =x-(μ-n)2η(e) Describe the criteria which can be used to choose between these two estimates.

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Answered: B1. Suppose we have X₁, X2,... Xn,…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
B1. Suppose we have X₁, X2,... Xn, which are i.i.d. and come from the uniform distribution Uniform(μn, μ+n), n>0;0= (μ, n) (a) Use the Method of Moments to estimate and n. (b) Determine whether the estimate derived by the Method of Moments is biased for μ. (c) Use the Maximum Likelihood method to estimate and n. Use the notation: X(1) = min (X₁, X2, ..., Xn) and X(n) (d) It can be derived that max(X₁, X2, ..., Xn) fx(1) (x) n 22/17 (1 - 2η fx(n)(x) = n 2n X - X- = (μ - n)` 2η (μ n) 2η n-1 n-1 χεμ - ημ + η], χε[μ - ημ + n] Determine whether the estimate obtained by the Method of Maximum Likelihood is biased for . Hint: Use substitutions y = 1 x-(μ-n) and y'= 2n x-(μ-n) 2η (e) Describe the criteria which can be used to choose between these two estimates.
Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?
2. Assume X ~ Exp(0), so that its pdf is 1 f(x; 8) x > 0,0 < 0 < o. e We have 5 samples, 1,2,3,4, and, 5, under Exp(0). Find the maximum likelihood estimator of 0 using the five samples.
Suppose we have X₁, X2, ... Xn, which are i.i.d. and come from the uniform distribution Uniform ( - n, μ + n), n > 0; 0 = (µ, n) and n. Use the Method of Moments to estimate Determine whether the estimate derived by the Method of Moments is biased for μ. Use the Maximum Likelihood method to estimate and n. Use the notation: X(1) = nin(X₁, X2,..., Xn) and X(n) = max(X₁, X2,..., Xn) It can be derived that n fx₁)(x) = (₁ 2n fx(n) (x) = x − (µ − n) 2η χε[μ - ημ + n], χε[μ - η,μ + η] Determine whether the estimate obtained by the Method of Maximum Likelihood is biased for . Hint: Use substitutions y =1 n 2η X n-1 (μ − n) 2η n-1 " 2 x-(μ-n) and y'= x-(μ-n) 2η 2η Describe the criteria which can be used to choose between these two estimates.
Consider a random sample X1, . . . , Xn from the uniform distribution U(−θ, θ) with parameter θ > 0 (1) Find the maximum likelihood estimation of θ. (2) Find the maximum likelihood estimation of the variance of the population distribution (you can use, without proof, what the mean and the variance of the uniform distribution are).
C. (i) Find the value of k which make the following function a pdf fly) = { (15/64) +(y/64) -1<= y <= 0 (3/8) + kỵ 0
A random variable Y has a uniform distribution over the interval (?1, ?2). Derive the variance of Y. Find E(Y)2 in terms of (?1, ?2). E(Y)2 = Find E(Y2) in terms of (?1, ?2). E(Y2) = Find V(Y) in terms of (?1, ?2). V(Y) =
The time to wait between each phonecall a person recives is random given f(t) = 3e-3t for t>=0. Let T1 and T2 be two independent waiting times for this distribution. Find expected time between each call and variance. (E(T) and V(T)) Find the probability for both T1 and T2 to be greater than one.

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