Results 1 to 4 of 4

Math Help - Challenging Estimation Question - Help needed

  1. #1
    Newbie
    Joined
    May 2008
    Posts
    21

    Challenging Estimation Question - Help needed

    To show that an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: to estimate the mean of a population with the finite variance s2, we first take a random sample of size n. Then we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2,3,, or n, we use as our estimator the mean of the random sample; otherwise, we use the estimate n2. Show that this estimation procedure is
    a) consistent;
    b) neither unbiased nor asymptotically unbiased.
    Find the variance of this estimator.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by math beginner View Post
    To show that an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: to estimate the mean of a population with the finite variance s2, we first take a random sample of size n. Then we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2,3,…, or n, we use as our estimator the mean of the random sample; otherwise, we use the estimate n2. Show that this estimation procedure is
    a) consistent;
    b) neither unbiased nor asymptotically unbiased.
    Find the variance of this estimator.
    What have you done? Or what is the specific difficulty that you have?

    You are given a sequence of estimators for the mean, for this to be a consistent sequence of estimators you need to show that it converges in probability to the population mean. You can use the consistency of sample means as a tool to do this.

    For the second part for a given sample size compute the expectation of the estimator, and look at the limit of this expectation as n becomes arbitarily large.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2008
    Posts
    21
    i have trouble with the part that says, "otherwise, we use the estimate n2 (squared)". I don't know how to factor that in the calculations. any tips?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by math beginner View Post
    i have trouble with the part that says, "otherwise, we use the estimate n2 (squared)". I don't know how to factor that in the calculations. any tips?
    The estimator t_n is \bar{x} with probability (n-1)/n and n^2 with probability 1/n.

    Hence the expectation of the estimator is:

    E(t_n)=\frac{n-1}{n} \mu + \frac{1}{n} n^2

    To be consistent requires that for all \varepsilon >0 :

    \lim_{n \to \infty} Pr(|t_n-\mu|<\varepsilon )=1

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. A Challenging PDE question
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: August 30th 2011, 05:18 AM
  2. question about Remainder Estimation Theorem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 5th 2011, 02:25 PM
  3. Statistical estimation (classical) question
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 9th 2009, 10:18 PM
  4. Challenging question?
    Posted in the Algebra Forum
    Replies: 1
    Last Post: July 1st 2008, 02:34 AM
  5. challenging proof(section idea needed)
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: November 17th 2006, 10:06 AM

Search Tags


/mathhelpforum @mathhelpforum