Let X 1 ,...,X n be a random sample from U (0, θ) distribution. Let Y n = max{X 1 ,...,X n }. We know (from Example XXXXXXXXXXthat θˆ 1 = Y n is a maximum likelihood estimator of θ. (a) Show that θˆ 2...

Let X₁,...,X_n
be a random sample from U (0, θ) distribution. Let Y_n
= max{X₁,...,X_n}.

We know (from Example 5.3.4) that θˆ₁
= Y_n
is a maximum likelihood estimator of θ.

(a) Show that θˆ₂
= 2X is a method of moments estimator.

(b) Show that θˆ₁
is a biased estimator, and θˆ₂
is an unbiased estimator of θ.

(c) Show that θ^ˆ
₃
=
 θˆ₁
is an unbiased estimator of θ.

Example 5.3.4

Let X₁,...,X_n
be a random sample from U(0, θ), θ > 0. Find the MLE of θ.