Introduction to Probability and Statistics for Engineers and Scientists

270 Chapter 7: Parameter Estimation

EXAMPLE 7.7c LetX 1 ,...,Xndenote a sample from a uniform (0,θ) distribution, where
θis assumed unknown. Since

E[Xi]=

θ
2
a “natural” estimator to consider is the unbiased estimator

d 1 =d 1 (X)=

2

∑n

i= 1

Xi

n

SinceE[d 1 ]=θ, it follows that

r(d 1 ,θ)=Var(d 1 )

=

4

n

Var(Xi)

=

4

n

θ^2

12

since Var(Xi)=

θ^2

12

=

θ^2
3 n
A second possible estimator ofθis the maximum likelihood estimator, which, as shown
in Example 7.2d, is given by

d 2 =d 2 (X)=max

i

Xi

To compute the mean square error ofd 2 as an estimator ofθ, we need to first compute
its mean (so as to determine its bias) and variance. To do so, note that the distribution
function ofd 2 is as follows:

F 2 (x)≡P{d 2 (X)≤x}

=P{max

i

Xi≤x}

=P{Xi≤x for alli=1,...,n}

=

∏n

i= 1

P{Xi≤x} by independence

=

(x

θ

)n

x≤θ

Hence, upon differentiating, we obtain that the density function ofd 2 ,is

f 2 (x)=

nxn−^1

θn

,x≤θ