1 Let Ymin be the smallest order statistic in a random sampl

1. Let Ymin be the smallest order statistic in a random sample of size n drawn from the uniform pdf, f sub Y (y; ?) = 1/?, 0 ? y ? ?. Find an unbiased estimator for ? based on Ymin.

2. We showed in Example 5.4.4 that

Solution

1. The definition of an unbiased estimator is that the expected value of the estimator is the same as what you want to estimate. So E(theta hat) = theta is the condition that you want to satisfy.

Also, you\'re dealing with an order statistic, which has the density function (for the rth order statistic):
f(x) = [(n!)/((r-1)!(n-r)!)] f(x)[F(x)]^(r-1)(1-F(x))^(n-r)
where f(x) is the density function and F(x) is the probability distribution.

Here, since r=1 (you\'re dealing with Y min):
f(x) = nf(x)(1-F(x))^(n-1)

Now, find the expected value of this by evaluating the integral, and if it doesn\'t equal to theta, perform a conversion so that it does. For instance, if the expected value you obtain is theta/n, then your unbiased estimator would be n(Ymin)

If you want to check your answer, it should be (n+1)Ymin.

source- Mathematical Statistics with Applications by Wackerly, Mendenhall and Scheaffer