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,…

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 random sample of n observations from each of the distributions: a. Poisson distribution with parameter θ. b. f(x, θ) = (1/ θ) e-x/θ , 0 < x. In each case find the Minimum Variance Estimator of θ and prove its efficiency.
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.
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Let X1, · · · , Xn and Y1, · · · , Yn be two random samples following Poisson(θ1) and Poisson(θ2) respectively. We would like to test H0 : θ1 = θ2 against H1 : θ1 6= θ2. Sample size n = 50, and sample means are ¯x = 6 and ¯y = 9 respectively. Calculate the likelihood ratio statistic and make a decision.
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|20 points] Suppose Y1,. where 0 < 0 < 1 is an unknown parameter. That is, Yn is a random sample of size n from Uniform(0, 20], 1 f(y) if 0
help with this problem please
A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = g(1-q). The experiment was repeated n times and x1,x2,..., Xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â = n n+S n where S = x₁. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁. (d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3. (i) What is the maximum likelihood estimate â? (ii) Find an approximate 95% confidence interval for q. (iii) Before seeing the data, our probability distribution for q has mean 0.4 and standard deviation 0.2. Find values of ao and Bo corresponding to this belief. What is then the…
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