Page 281 - Fundamentals of Probability and Statistics for Engineers

264 Fundamentals of Probability and Statistics for Engineers

Following similar procedures as given above, we can show that

9
EfM k gˆ k ;
=
1 9:15†
varfM k gˆ 2k †;
k
2 ;
n
where k is the kth moment of population X.

9.1.4 ORDER STATISTICS

A sample X 1 , X 2 ,..., X n can be ranked in order of increasing numerical mag-
nitude. Let X (1) , X (2) ,..., X (n) be such a rearranged sample, where X (1) is the
smallest and X (n) the largest. Then X (k) is called the kth-order statistic. Extreme
values X (1) and X (n) are of particular importance in applications, and their
properties have been discussed in Section 7.6.
In terms of the probability distribution function (PDF) of population X,
F X (x), it follows from Equations (7.89) and (7.91) that the PDFs of X (1) and
X (n) are

n
x†ˆ 1 ‰1 F X x†Š ; 9:16†
F X 1†
n
x†ˆ F x†: 9:17†
X
F X n†
If X is continuous, the pdfs of X (1) and X (n) are of the form [see Equations (7.90)
and (7.92)]

x†ˆ n‰1 F X x†Š n 1 f X x†; 9:18†
f X 1†
x†ˆ nF n 1 x† f X x†: 9:19†
X
f X n†
The means and variances of order statistics can be obtained through integration,
but they are not expressible as simple functions of the moments of population X.

9.2 QUALITY CRITERIA FOR ESTIMATES

We are now in a position to propose a number of criteria under which the
quality of an estimate can be evaluated. These criteria define generally desirable
properties for an estimate to have as well as provide a guide by which the
quality of one estimate can be compared with that of another.

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