Page 294 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 277

where is the unknown parameter. We have
n x j n
Y e
; 9:54†
p X x j ; †ˆ Q
x j !
jˆ1
which can be factorized in the form of Equation (9.50) by letting
e
g 1 ˆ x j n ;
and
1
g 2 ˆ Q
x j !
It is seen that
n
X
Y ˆ X j
jˆ1

is a sufficient statistic for .

9.3 METHODS OF ESTIMATION

Based on the estimation criteria defined in Section 9.2, some estimation tech-
niques that yield ‘good’, and sometimes ‘best’, estimates of distribution param-
eters are now developed.
Two approaches to the parameter estimation problem are discussed in what
follows: point estimation and interval estimation. In point estimation, we use
certain prescribed methods to arrive at a value for ^ as a function of the
observed data that we accept as a ‘good’ estimate of – good in terms of
unbiasedness, minimum variance, etc., as defined by the estimation criteria.
In many scientific studies it is more useful to obtain information about a
parameter beyond a single number as its estimate. Interval estimation is a
procedure by which bounds on the parameter value are obtained that not only
give information on the numerical value of the parameter but also give an
indication of the level of confidence one can place on the possible numerical
value of the parameter on the basis of a sample. Point estimation will be
discussed first, followed by the development of methods of interval estimation.

9.3.1 POINT ESTIMATION

We now proceed to present two general methods of finding point estimators for
distribution parameters on the basis of a sample from a population.

TLFeBOOK