7. (a) Let x1, X2, , Xn be a random sample from the uniform distribution -0, dsewhere .... with p.d.f. : { ,0 0 f(x, ) 0, elsewhere Obtain the maximum likelihood estimator for 0.
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- iid X1,..., Xn Geo(p) where 0 0; B>0]a conjugate family for samples from the Geometric distributions [Geo(p): 0 0 are known constants. C- Derive the Bayes estimator of p. d- Express the estimator in part (c) as a weighted average of the prior mean of p and the MLE of p.If we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that: P( (X-μ)/(σ/√10) < k ) = .05 P( (X-μ)/(S/√10) < k ) = .05 P( k < (X-μ)/(S/√10) ) = .052) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. If fram e ¹- {** f(x;0)=B for x > 4 otherwise "1 g) What is the maximum likelihood estimator of g (B) = 2(ß + 1). h) Is the maximum likelihood estimator of g (B) = 2(B + 1) unbiased or not?
- Suppose you have a random sample of observations from random variables Xi, i=1, 2, .., n. Assume that X;'s are identically distributed and they have the following distribution function: f(x;;0) =o* (1–ơ), x; = 1,2 , 3., 0The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 02. 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.Let X1, ..,Xn be a random sample of distribution f(x; 0)=e-(=-0), x>0, 0 elsewhere, where 0 > 0. (a) Find the mle of 0. (b) For testing Họ : 0 = 00 vs H1 : 0 > 00, find the Likelihood Ratio Test (LRT) in terms of the mle found in part (a)Assume that n independent count variables {X,,X2,...,X,} are identically distributed as X, ~ Poi(0) where i= 1,2,...,n and to estimate E(X,)= 0, consider the sample mean estimator ô = -5x,. n (b) Use the expectation of the distribution of S = nô to compute the expectation E(@) and the bias of this estimator. State whether this estimator unbiased. (c) Use the variance of the distribution of S= nô to examine var(Ô) and the standard error of this estimator. Use the variance and the bias of ê to compute the Mean Squared Error (MSE) of this estimator. Comment on the MSE behaviour in the asymptotic limit as n→o. State whether this estimator is consistent or (d) not.Let X be a random variable with a uniform distribution on the finite interval [-1,0], where > -1 is unknown. Suppose that a random sample of size n is drawn from the distribution, with observations £₁,...,n Write down the likelihood function for the parameter 0, and find the maximum likeli- hood estimate (MLE) for 0.Let X1,..., X, a random sample with distribution f(x; 0) = (0 + 1)æº, 0 -1. (a) Find the method of moments estimator of 0. (b) Show that the method of moments estimator is consistent. (c) Find the maximum likelihood estimator of 0. (d) takes value 0 = 1.2. Use the parametric bootstrap method to obtain a 95% revised bootstrap percentile confidence interval using the maximum likelihood estimate. Make sure to include a plot of the bootstrapped values and an interpretation of your confidence interval. Given a sample of size n = 500, suppose that the maximum likelihood estimate of 0b) Let X₁, X₂, X, and Y₁, Y₂,...,Y be random samples from populations with moment generating 25 functions Mx, (t) = e³t+t² and My(t) = respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂-X₂ + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|-|≥10) < 0.04?Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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