# Question: With reference to the uniform population of Example 10 4 use

With reference to the uniform population of Example 10.4, use the definition of consistency to show that Yn, the nth order statistic, is a consistent estimator of the parameter β.

Example 10.4

If X1, X2, . . . , Xn constitute a random sample from a uniform population with α = 0, show that the largest sample value (that is, the nth order statistic, Yn) is a biased estimator of the parameter β. Also, modify this estimator of β to make it unbiased

Example 10.4

If X1, X2, . . . , Xn constitute a random sample from a uniform population with α = 0, show that the largest sample value (that is, the nth order statistic, Yn) is a biased estimator of the parameter β. Also, modify this estimator of β to make it unbiased

## Answer to relevant Questions

Substituting “asymptotically unbiased” for “ unbiased” in Theorem 10.3, show that X + 1 / n+ 2 is a consistent estimator of the binomial parameter θ. If X1 and X2 are independent random variables having binomial distributions with the parameters θ and n1 and θ and n2, show that X1 + X2 / n1 + n2 is a sufficient estimator of θ. Given a random sample of size n from a population that has the known mean µ and the finite variance σ2, show that X1, X2, . . . , Xn constitute a random sample of size n from a gamma population with α = 2, use the method of maximum likelihood to find a formula for estimating β. Show that X + 1 / n + 2 is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?Post your question