5. Suppose that Y₁, Y2, Y3 denote a random sample from an exponential distribution with densityfunction1f(y) == e 6, y> 0, 00Y₁ + 2Y₂Five estimators are proposed for the parameter 0, Ô₁ = Y₁, 0₂:Y₁+Y₂032=Ô₁ = Y(1)="3and Ô5 = Y(a) Compute the bias of each estimator and hence identify the unbiased estimators.(b) Find for each of the estimators, the mean square error and hence the standard error.(c) Find the efficiency of 85 relative to each of the other four estimators and comment on yourresults.

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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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There is a relation between the following variables as y = 1 / (a * x ^ b) (x ^ b means x over b) a) Calculate the correlation coefficient and interpret the degree of the relationship? b) Estimate the y-value for x = 4.3 and the x-value for y = 0.90 by obtaining the regression equation
two random independant variables X and Y with distributions: X ∼ Poisson(λ) og Y ∼ Poisson(2λ), andobservations x = 2 og y = 5 .(a) What is the log-likelihood function? b) calculate MLE for the observated samples
There is a relationship between the following variables as y = 1 / (a * x ^ b) (x ^ b: means x over b). a) Calculate the correlation coefficient and interpret the degree of the relationship? b) Estimate the y value for x = 4.3 and the x value for y = 0.90 by obtaining the regression equation. x 1.2 1.5 2.0 3.0 4.0 5.0 y 0.58 0.50 0.32 0.18 0.15 0.10 x 1.2 1.5 2.0 3.0 4.0 5.0 y 0.58 0.50 0.32 0.18 0.15 0.10
Consider the data (1,3), (2,5), (3,9.67) (a) Derive the normal equations for the best fit of the form f(x) = - - + Dx? (b) Find C, D using the normal equations derived above and given data. (c) Find a suitable Linearization for f (x) =- + Dx² (Don't find C, D)
Let Y1, Y2,..., Y, denote a random sample from the density function given by 1 yª-'e=y/®, y> 0, f(y[a, 0) = elsewhere, where a > 0 is known. a Find the MLE Ô of 0. b Find the expected value and variance of ê. c. Is the MLE ô an unbiased estimator for 0?
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Let yit denote the outcome of a random variable for the i-th individual at time and let Xit denote a covariate measure on the i-th individual at time t, t, i = 1, ..., n, t = 1,..., T. The data could be modeled using the simple linear regression model: Yit = Bo + B₁xt + Eit, Eit~N(0, t) a) Explain why using this approach may be inappropriate. b) Write a mathematical expression of the random intercept model and the random coefficients models as alternatives to this model. c) Provide an expression for the proportion of variance attributable to each of the variance components in the random coefficients models. d) Write out an expression for the likelihood, priors and posterior for the model in part (c) e) Write out a WinBUGS code for this problem.
7. Let X~ N (0,0²) and {X; : i = 1,2,..., n} be a random sample from X. (a) Formulate the log-likelihood function. (b) Find the ML estimator of o². (c) Derive the variance of the ML estimator of o2, 62. Does the variance of ô2 achieve the CR bound? (d) Derive the asymptotic distribution of √n (-o).
Please answer number 2
Let X and Y be independent random variables. X is N(1,9) and Y is uniform on the interval {--1, 1]. Lat Write down the joint density for (X,Y) o Give the mean and variance of Y c) Give the median of X I dY Give the correlation coefficient p of X and Y

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Suppose that Y₁, Y2, Y3 denote a random sample from an exponential distribution with density function 1 f(y) = =-e-³, y>0, 0>0 Y₁+Y₂ Y₁+2Y₂, ₁ = Y(1) Five estimators are proposed for the parameter 0, Ô₁ = Y₁, Ô₂ = · and Â =Ÿ 2 3 2,03 =