APPROXIMATE METHODS FOR DETERMINATION OF OPTIMUM STRATIFICATION BOUNDARY POINTS FOR BETA, GAMMA, AND PARETO DISTRIBUTIONS
APPROXIMATE METHODS FOR DETERMINATION OF OPTIMUM STRATIFICATION BOUNDARY POINTS FOR BETA, GAMMA, AND PARETO DISTRIBUTIONS
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Date
1986-08
Authors
BELLO, RAZAQ
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Abstract
Stratification is the division of a heterogeneous
population into relatively homogeneous sub-populations.
Using Stratification we can usually improve the efficiency
of the estimate of mean of the population. Stratification
may be inevitable in terms of administrative convenience.
Stratification of Hospitals may be by bed capacity,
of schools by enrolment, of business concern by volume of
sales, of industrial plants by employment and of voluntary
organisation by membership. te finite nopulations
encounte in practice are all discontinuous. If the
populations are sufficiently large, they can very often
be approximated by some standard continuous distributions.
Thus tables of strata boundaries for these standard
distributions can profitably be utilized for the stratification
of an. actual population. Under Tchunrow-Neyman
allocation, Dalenius (1950) derived equations for the
strata boundaries yhwhich minimize the variance of th
estimated population mean. But these equations are
troublesome to solve.
A number of approximate methods that could quickly
solve the Dalenius equations have been proposed by
several authors.
These approximate methods are:
(i) Cum /f Method
(ii) Durbin's Method
(i i i) Ekman's Method
(iv) Equal Range Method (v) Equal Sized Strata Method
(vi) Equalization of Strata Total Method.
The author has developed computational procedures for
Parens
the determination of s t r a t a boundaries for Beta and Gamma
*
Distributions with various values of the parameters using
the above methods except (iii).
The methods are compared using the variance of the
stratified sample mean. The best boundaries are those
that give the least variance of the stratified sample mean.
All the methods construct strata boundaries quickly fox-
Beta Distribution. For Gamma Distribution, only the cum gammamma method constructs strata boundaries quickly. For ParetO
distribution with parameter 0 all the methods consider
lead to a very substantial reduction in variance of the
estimate of the mean of population except the equal ran
method, and all the method construct the strata boundaries
quickly.
We conclude that the cum gamma of f method is the best
method, of constructing strata boundaries among those
considered in this study.
A critical examination of the behavior of the variance
as a function of the parameters p and a of the Beta distribution
and the number of strata k has shown that for Beta
distribution the variance decreases by about 10%, for a
unit increase in a when 1 _< p -< 4 and k are fixed; while
the variance increases and then decreases when p _> 5. We
have the sane conclusion when p and q are interchanged. For the Gamma distribution with parameters beta and p it
is observed that for fixed k the variances increase as one
of the numbers v or 8 is increased while the other remains
constant.
Description
A thesis submitted to the Postgraduate School, Ahmadu
Bello University, Zaria, in partial fulfilment of the
requirements for the depree of Doctor of Philosophy
in Statistics.
Keywords
APPROXIMATE METHODS,, DETERMINATION,, OPTIMUM,, STRATIFICATION,, BOUNDARY POINTS,, BETA, GAMMA, PARETO DISTRIBUTIONS