Let x1,x2,dots,xn be a random sample from f(x;alpha,beta)=begincasesdfrac1beta-alpha; & alpha<x<beta 0; & otherwise.endcases Then the MLEs of alpha and beta are respectively given by:

(C) hatalpha=x(1) and hatbeta=x(n)

Q38Section I (Objective (Q1–Q80))38 of 80

MLEs of the endpoints of a uniform distribution

Let $x_1,x_2,\dots,x_n$ be a random sample from

$f(x;\alpha,\beta)=\begin{cases}\dfrac1{\beta-\alpha}; & \alpha<x<\beta\\ 0; & \text{otherwise.}\end{cases}$

Then the MLEs of $\alpha$ and $\beta$ are respectively given by:

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