Assume that X1,..., Xn constitute an iid sample from a distribution with pdf Spa-PrP=1 if 0 < * < a, f (x) = otherwise for some unknown parameters p > 0 and a > 0. (a) Derive the maximum likelihood estimate for a. If you decide to use calculus methods, you have to perform a second derivative test. (b) Derive the maximum likelihood estimate for p. If you decide to use calculus methods, you have to perform a second derivative test. (c) Derive the equations that allow you to find the method of moment estimators for p and a. Your equations must explicitly contain p and a. You do NOT have to actually solve the equations.

Assume that X1,..., Xn constitute an iid sample from a distribution with pdf Spa-PrP=1 if 0 < * < a, f (x) = otherwise for some unknown parameters p > 0 and a > 0. (a) Derive the maximum likelihood estimate for a. If you decide to use calculus methods, you have to perform a second derivative test. (b) Derive the maximum likelihood estimate for p. If you decide to use calculus methods, you have to perform a second derivative test. (c) Derive the equations that allow you to find the method of moment estimators for p and a. Your equations must explicitly contain p and a. You do NOT have to actually solve the equations.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

Let x1, X2, ..., Xn be a random sample from a normal population with unknown mean u and an unknown variance o?. At a given significance level a, derive the generalized likelihood ratio test for Ho : o 5. versus Completely specify the test when a = population: 0.05 and n = 26. Does the power function of the test depend on the mean of the The following `answers" have been proposed. %3B 26 (a) For a = 0.05 and n = 26 the test rejects Ho when E(X; – X„)² > 941.25. The power function of the test indirectly depends on the population mean u via = 7. (b) For a = 0.05 and n = 26 the test rejects Ho when E(X; – Xn)² > 941.25. The power function of the test does not depend on the population mean µ. (c) For a = 26 ,26 0.05 and n = 26 the test rejects Ho when E(X; – Xn)² > 188.25. The power function of the test does not depend on the population mean µ. (d) For a = 0.05 and n = 26 the test rejects Ho when E(X; – X„)² > 188.25. The power function of the test indirectly depends on the population mean…
Define limited information maximum likelihood (LIML) estimator? Explain how it minimizes the homoskedasticity?
For each of the following distributions, compute the maximum likelihood estimator based on 7 i.i.d. observations X₁,..., X and the Fisher information, if defined. If it is not, enter DNE in each applicable input box. (a) (Enter barX_n for the sample average X Maximum likelihood estimator = X~ Ber (p), p = (0,1) Xn = 1-7 n T X₁. Hint: Use the definition of Fisher information that leads to the shorter computation. (If the Fisher information is not defined, enter DNE.) Fisher information I (p) = Use Fisher Information to find the asymptotic variance V (p) of the MLE >>. V (B) = 9 Save
3) Explain what is meant by constructing a confidence interval for an unknown parameter 0 from a given sample x, ..., N. Let a family of PDFS fS(x: 0). -o 0, S(x: 0) = 0. x 0. Explain how to construct a 95% confidence interval for a from a sample x, . , x. Justify the claims about the distributions you use in your construction.
A random sample of 70 observations produced a mean of 27.5 and a standard deviation of 3.26. Find a 90% confidence interval for < U < (b) Find a 95% confidence interval for < U < (c) Find a 99% confidence interval for < U <
Suppose that X₁,..., Xn is a random sample from a distribution with probability den- sity function 2 ) = { +/- - - fx(x) = √3/₂e-x/0, 0 0. You are provided the information that the maximum likelihood estimator of is Ô i=1 = 2n (You do not need to derive this mle.) (a) Verify that Fisher's information in the random sample is given by 2n In (0) 02
please include all the steps. thank you
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 X equal the length of life of a 60-watt light bulb marketed by a certain manufacturer. We do not know the distribution of X except that Var(X) = 1158. Let u be the mean of X. Suppose a random sample of n = 25 bulbs is tested until they burn out, yielding a sample mean of = 1368 hours. (i) Compute an approximate 95% confidence interval for u. (ii) Compute an approximate 95% one-sided confidence interval for that provides a lower bound for u, i.e. compute & such that ɛ P(X - ≤ μµ)≈ 0.95.
Using the t-table, find the critical value for a left-tailed test with an α = 0.05 for a sample size of 23.
Suppose that X₁ = 1; X₂ = 1; X3 = 0; X₁ = 1; X5 = 1, X6 = 1; X₂ = 0; X = 1; X9 = 0, X10 = 0, represents a random sample. Each of these X's comes from the same population and has a density of fx₁ = 0¹-* (1 - 0)*; x = 0,1 First determine the form of the maximum likelihood estimator for 0, then use the MLE formula and the data provided to find an estimate for 0. Round your answer to 3 decimal places. (Answer is 0.4)
Parametric Confidence Interval