Problem 4
Define f(x) as follows
θ2 -1<=x<0
1-θ2 0<=x>1
0 otherwise
Let X1, … Xn be iid random variables with density f for some unknown θ (0,1),
Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b.
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ...
Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Let X1, . . . , Xn ∼ iid Unif(θ − 1/2 , θ + 1/2 ) for θ unknown. Find an asymptotic confidence interval for θ.
For z e R and θ (0, 1), define otherwise. Let X1 , . . . , X" be i..d. random variables with density f, for some unknown θ E (0, 1) 1 point possible (graded, results hidden) To prepare, sketch the pdf f, (z) for different values of θ E (0,1) Which of the following properties of fo (z) guarantee that it is a probability density? (Check all that apply) Note (added May 3) Note that you are not...
Let X1, X2, ..., Xn be iid with pdf f(x|θ) = θ*x(θ-1). a) Find the Maximum Likelihood Estimator of θ, and b) show that its variance converges to 0 as n approaches infinity. I have no problem with part a, finding the MLE of θ. However, I'm having some trouble with finding the variance. The professor walked us through part b generally, but I need help with univariate transformation for sigma(-ln(xi)) (see picture below - the professor used Y=sigma(-ln(x)), and...
Let Xi iid∼ N(0, θ) for i = 1, ..., n. a) Find the MLE for θ. Call it b) Is biased? c) Is consistent? d) Find the variance of (e) What is the asymptotic distribution of ?
Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why...
Question 2 Let X1,...,X, be iid Geometric random variables with parameter and probability mass function f(T; 7) = (1 - 7)" for 1 = 0,1,2,... and 0 <I<1. We wish to test: HT=0.50 HT70.50 (a) Find the three asymptotic x1) test statistics (Likelihood Ratio, Wald, and Score) for this setting. versus
Let X1, · · · ,Xn be iid from Uniform(−θ,θ), where θ > 0. Let X(1) < X(2) < ... < X(n) denotes the order statistics. (a) Find a minimal sufficient statistics for θ (d) Find the UMVUE for θ. (e) Find the UMVUE for τ(θ) = P(X1 > k).
Let Xi iid∼ Unif(0, θ) for i = 1, 2, 3 and = 4Y(1) = 2Y(2) θ* = (4/3)Y(3) (a) For each estimator compute the bias? (b) For each estimator compute the variance? (c) Which estimator is best and why?