Transcribed TextTranscribed Text

(1) Suppose that Y 1 , ,Y n n is a random sample where the density of each random variable Y is fy (y)=202y-3, y 0 0 for some parameter 0>1. Let 0 III min {Y1 Yn}. (a) Calculate B(ô), the bias of 0. (b) Determine the value of the constant c for which 0 =cô is unbiased. (c) Calculate the mean-square error of 01. (2) Suppose that are independent and identically distributed Exp(8) random variables. (a) Let X1=Y(+...+Y,.Use moment generating functions to show that X 1 has a Gamma(n,6 distribution. (b) Let that X 2 has an Exp of distribution. n (c) Verify that III is an unbiased estimator for 0. (d) Verify that 0 =nX2 is an unbiased estimator for 0. (e) Which estimator, 0 or O2 is preferred for the estimation of O? (4) Suppose that Y is normally distributed with mean 0 and unknown variance of Y 2 (a) Show that the quantity is a pivotal quantity. 2 (b) Use the pivotal quantity 3/6 to construct a 95% confidence interval for of y² (5) Let Y1,Y2, and Y3 be a random sample from a normal distribution with mean u and variance o2 , where both u and 62 are unknown. Consider the following estimators for u: = - 1 + - 2 + - 3 4 2 4 1 1 1 A2 = - 3 Y1 + - 3 Y2 + - 3 Y3 (a) Show that A1 and A2 are unbiased estimators for u. (b) Find the variances of A and A2. (c) Find the efficiency of A2 relative to A1. , and tell me which is more efficient estimator for u. (6) Let Y1, , Y2 Yn n denote a random sample from the probability density function 0 y8-1 , 00 f(y/0)= , elsewhere (a) Show that On =Y is unbiased estimator for 0 0+1 (b) Find the variance of On. (c) Show that On is a consistent estimator for = 0 0+1

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available General Statistics Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats