Question 4 (24 marks) (a) Let X1,..., X, be a random sample from Uniform(-6 - 1,20...
1. (20 points) Let X1....X be a random sample from a uniform distribution over [0,0]. (a) (4 points) Find the maximum likelihood estimator (MLE) 0 MLE for 0. (b) (3 points) Is the MLE ONLE unbiased for 0? If yes, prove it: If not, construct an unbiased estimator 0, based on the MLE. (c) (4 points) Find the method of moment estimator (MME) OM ME for 8. (d) (3 points) Is the MME OMME tnbiased for 6? If yes, prove...
Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.
. (8pts) X1, X,X x. . X, is a random sample with a common probability density function, (), f(x) 6Ax3c"/?, x > 0 and ? > 0. 604 Note that E(X-4? and Var(X-4g2. a) (4pts) Find a maximum likelihood estimator of ?, say ?MLE. (b) (4pts) Find the mean squared error of ?MLE-
Q2: ALL STUDENTS (10 Marks] Let X1, ..., Xn be a random sample from the pdf f(x|0) = 0x-?, O<O<O<0. (a) (3 marks) What is a sufficient statistic for 0? (b) (4 marks) Find the MLE of 0. (c) (3 marks) Find the method of moments estimator of 0.
2. Let X1, ..., Xn be a random sample from a Poisson random variable X with parameter 1 (a) Derive the method of moments estimator of 1 (4 marks) (b) Derive the maximum likelihood estimator of 1 (4 marks) (c) Derive the least squares estimator of 1 (4 marks) (d) Propose an estimator for VAR[X] (4 marks) (e) Propose an estimator for E[X²] (4 marks)
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0, x > 0, N = (0,00). (a) Find the maximum likelihood estimator of 0. (b) Find the method of moments estimator of 0. (c) Are the estimators in a) and b) unbiased? (d) What is the variance of the estimators in a) and b)? (e) Suppose the observed sample is 2.26, 0.31, 3.75, 6.92, 9.10, 7.57, 4.79, 1.41, 2.49, 0.59. Find the maximum likelihood...
Question 5. [10 Marks] Suppose . . . ,X, be an SRS from a uniform distribution between θ and 0. a) Į1 Mark] Find the moments estimator (ME) θί of θ. b) [1 Markl Let Y- min(X1,... ,Xn) and its pdf is as follows. -ny"-1 for ye(θ,0); for y E (6,0), -, 0, -, fy(y) otherwise. Show that the maximum likelihood estimator (MLE) θ2 = ntly of θ is unbiased. c) [4 Marks] which one of θ1 and θ2 is...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
1. Let X1, ..., Xn be a random sample from a distribution with cumulative dist: 10, <<0 F(x) = (/), 0<x<B | 1, >B > (a) For this part, assume that is known and B is unknown. Find the method of moments estimator Boom of B. (b) For this part, assume that both 6 and B are unknown. Find the maximum likelihood estimators of 8 and B.
5. Let X1,...,Xn be a random sample from the pdf f(\) = 6x-2 where 0 <O<< 0. (a) Find the MLE of e. You need to justify it is a local maximum. (b) Find the method of moments estimator of 0.