410Hw09ans - STAT 410 Homework#9(due Friday October 30 by...

This preview shows pages 1–4. Sign up to view the full content.

STAT 410 Homework #9 Fall 2009 (due Friday, October 30, by 3:00 p.m.) Warm-up: 4.2.3 By Chebyshev’s Inequality, P ( | W n μ | ε ) 2 2 W ε σ n = 2 ε p n b 0 as n for all ε > 0. Therefore, μ W P n . 1 – 2. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 θ θ 1 θ y k , y k ; θ 1. The income information has been collected on a random sample of n individuals: Y 1 , Y 2 , … , Y n . 1. Assume k is known. Hint: Recall that k n n n i i ln 1 Y ln θ ˆ - = = = k ln ln Y 1 - and k - = Y Y θ ~ .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
a) Show that the maximum likelihood estimator θ ˆ is a consistent estimator of θ . Hint: Find E ( ln Y ) first. E ( ln Y ) = + k dy y k y 1 θ θ 1 θ ln = - - k dy y y k 1 θ θ ln θ = ( 29 k y y y k - - - - - θ 1 θ 1 θ θ 2 θ θ ln = θ 1 ln + k . By WLLN, ( 29 θ 1 Y E Y ln 1 ln ln 1 + = = k n P n i i . a P n X , g is continuous at a ( 29 ( 29 X a g g P n Consider g ( x ) = k x ln 1 - . Then g ( x ) is continuous at θ 1 ln + = k a . θ ˆ Y ln 1 1 = = n i i n g ( θ = a g . θ θ ˆ P . θ ˆ is a consistent estimator of θ . b) Show that the method of moments estimator θ ~ is a consistent estimator of θ . ( 29 ( 29 1 θ θ θ 1 θ θ Y E θ θ 1 θ θ Y ; - = = = = - + - k dy y k dy y k y dy y f y k k . By WLLN, ( 29 1 θ θ Y E Y - = k P . a P n X , g is continuous at a ( 29 ( 29 X a g g P n
Consider g ( x ) = k x x - . Then g ( x ) is continuous at 1 θ θ - = k a .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern