Page 300 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 283

Example 9.12. Suppose that population X has a uniform distribution over the

range (0, ) and we wish to estimate parameter from a sample of size n.

The density function of X is
1
8
< ; for 0 x ;
f x; †ˆ 9:73†
0; elsewhere;
:
and the first moment is

1 ˆ : 9:74†
2
It follows from the method of moments that, on letting 1 ˆ X, we obtain
n
2 X
^
ˆ 2X ˆ X j : 9:75†
n
jˆ1
Upon little reflection, the validity of this estimator is somewhat questionable
because, by definition, all values assumed by X are supposed to lie within

interval (0, ). However, we see from Equation (9.75) that it is possible that
^
some of the samples are greater than . Intuitively, a better estimator might be
^
ˆ X n† ; 9:76†
where X (n) is the nth-order statistic. As we will see, this would be the outcome
following the method of maximum likelihood, to be discussed in the next
section.
Since the method of moments requires only i , the moments of population X,
the knowledge of its pdf is not necessary. This advantage is demonstrated in
Example 9.13.

Example 9.13. Problem: consider measuring the length r of an object with use

of a sensing instrument. Owing to inherent inaccuracies in the instrument, what
is actually measured is X, as shown in Figure 9.3, where X 1 and X 2 are
identically and normally distributed with mean zero and unknown variance
^
2
2 . Determine a moment estimator for ˆ r on the basis of a sample of size
n from X.
Answer: now, random variable X is
2
2 1=2
X ˆ‰ r ‡ X 1 † ‡ X Š : 9:77†
2
The pdf of X with unknown parameters and 2 can be found by using
techniques developed in Chapter 5. It is, however, unnecessary here since some

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