Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Successive sampling using a product estimate
E. Artes, M. Rueda & A. Arcos
Department of Statistics and Applied Mathematics,
University of Almeria, Spain
EMail: [email protected]
Department of Statistics and Operative Research,
University of Granada, Spain
EMail: [email protected]
Abstract
The problem of estimation of afinitepopulation mean for the current occasion based on the samples selected over two occasions has been considered. For the case when the auxiliary variables are negatively correlated, a
double-sampling product estimate from the matched portion of the sample
is presented. Expressions for optimum estimator and its variance have been
derived. The gain in efficiency of the combined estimate over the direct
estimate using no information gathered on thefirstoccasion is computed.
1
Introduction
Study of marine ecology has been strongly motivated by the development and conservation offisheriesand the longstanding value to
society offish as food. The fishermen themselves sample the resource,
using a variety of gear, and were the first to observe the inherent
variability of the success of such encounters. They designed their encounters to maximize the catch per unit of fishing time and effort.
The desire to better define the fluctuation in resource abundance
and the reasons for it inevitably led to the conduct of sampling by the
researchers themselves. The development of biometrics in agriculture
was adopted as the basis for designing and analysis of the surveys.
However, it is common that classical theory of sampling cannot
Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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Applied Sciences ami (he Environment
be directly applied in some situations calling for quantification of environmental resources. If a population is subject to change, a survey
carried out on a single occasion cannot of itself give any information
on the nature or rate of such change. In certain types of population
the information on ancillary characteristics from the previous season
may be available, and can be taken into consideration, other than
the characteristic whose estimate is sought, to further increase the
efficiency.
Successive sampling has been extensively used in applied sciences
and the environment. Mainly, it is suitable in studies which involves
species in extinction. The problem of sampling on two successive
occasions with a partial replacement of sampling units was first considered by Jessen[l] in the analysis of a survey which collected farm
data. Later, the Departmen of the Environment of Ontario (Canada)
applied this sampling plan with success in designing a mail survey in
Ontario of waterfowl hunters who hunted successively during 196768 and 1968-69 (5en[6]). An estimate was developed for the current
season (1968-69), based on the relationship between the value of a
characteristic during the current season and its value during the previous season, that yielded more precise estimates of the kill of waterfowl than the usual estimates based on simple random sampling and
using the hunter's current season's performance only.
Successive sampling has also been discussed in some detail by
Patterson[4], Yates[7] and others. Their discussions have, however,
been confined to combining ratio or regression estimates from the
matched portion of the sample with a mean per unit estimate based
on the current occasion.
Frequently, the study of environmental issues involves negatively
correlated characteristics. So, the product method of estimation is
relevant to these cases.
Because in many agricultural surveys computations involving
product estimates become relatively complex, we propose to investigate in this paper some theory of successive sampling using a product
estimate and examine the efficiency over the direct estimate based exclusively on the sampling units for the current occasion.
Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Applied Sciences and the Environment
2
87
Development of product method of estimating the mean on the second occasion
2.1
Selection of the sample
Let a simple random sample of size n be selected on thefirstoccasion
from a universe of size TV and let its mean be x. Let a simple random
sample of size ra be subsampled from the n units. Let the mean on
thefirstoccasion of the sample of size m be x^ and on the second occasion i/m- In addition, let a simple random sample of size u be taken
on the second occasion from the universe TV — m left after omitting
the m units. Let this mean be y^ (unmatched portion). Also let the
total number of those sampled.on.the.second occasion.be..ra.rf.w..— n.
We assume that simple random sampling is used and the finite
population correction factor is ignored.
2.2
Notation used
Let
n = total sample size on thefirstand second occasion
m — sample size of those questioned on both occasions (matched
sample)
u — n — m (unmatched sample)
x(y) = total sample mean on thefirst(second) occasion estimating
X(Y)
Xmdjm) — matched sample mean on thefirst(second) occasion estimating f (?)
A =#
%%_
Y
P — population correlation coefficient
Z = A (2p - A)
P ~ *~n ' tbc matching fraction.
2.3
The product method of estimation
The unmatched and matched portions of the second occasion sample
provide independent estimates (y^ and ?/%) of the population mean
on the second occasion Y.
Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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Applied Sciences and the Environment
For the matched portion an estimate improved of Y may be
obtained using a double sampling product estimate
m
_
— -- 1
m
n
ran
^ mn
An estimate of the variance may be obtained by replacing Sy
and Z in (1) by their appropriate sample estimates.
We construct an estimate, y^p, of the mean, F, of the population
on the second occasion by combining the two independent estimates,
y'm y Vu with weights w and (1 —a;). Thus
i/2p = wi/^ 4- (1 - w)^
(2)
The best estimate of the mean Y on the second occasion is
obtained by using the values of w in (2) that would minimize V (yip)Now
Differentiating (3) with respect to LU and equating to zero gives
the value of uj which minimizes V (yip] whence
Also, V (yu) is given by
Hence, substituting from (1) and (5) into (4) we have
g,nu
u ^ ^y mn
Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
Applied Sciences and the Environment
89
Again substituting from (6), (5) and (1) into (3), we obtain the
minimum variance of the combined estimator
y
If, however, the estimate y of the mean Y on the current
occasion were based exclusively on the n sampling units for the second
occasion its variance would be
n
The gain in precision, G , of y^p over y is given by
ft
V(y:IP)
Necessarily p < 1. If p — 1 or p = 0, the gain is zero. For
other p there will be positive gain if p < — y and loss if p > — y, i.e.
Z < 0 will give positive gain.
Table 1 gives percent gain in precision for various values of p,
p (matching fraction) and A. G increases with increasing p absolute
value. In biological data it often happens that A remains near one,
i.e. the coefficient of variation is the same from one occasion to the
next, even if large changes do occur.
Table 1
PERCENT GAIN IN PRECISION OF A PRODUCT ESTIMATE
yt OVER A MEAN PER UNIT ESTIMATE y
A = 0.75
p 4-0.7
-0.8
-0 9
P-+
0.3
11.7
21.7
38.2
0.5
12.5
21 A
33 .3
0.3
15 .5
24.2
36 .8
0.5
16.1
23.4
32.5
A = 1.25
0.3
4.5
13.2
27.8
0.5
5.2
14
26.2
Transactions on Ecology and the Environment vol 23, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541
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Applied Sciences and the Environment
References
[1] Jessen, R. J., Statistical Investigation of a Sample Survey for Obtaining Farm Facts, Iowa Agricultural Experiment Statistical Research Bulletin, 304 , Iowa, 1942.
[2] Gandge, S.N., Varghese, T. & Prabhu-Ajgaonkar, S.G., A note
on Modified Product Estimator, Pakistan Journal of Statistics,
9(3)B, pp. 31-36, 1993.
[3] Okafor, F. C. , The theory and application of sampling over two
occasions for the estimation of current population ratio, Statistica,
1, pp. 137-147, 1992.
[4] Patterson, H.D. , Sampling on Successive Occasions with Partial
Replacement of Units, Journal of the Royal Statistical Society,
B12, pp. 241-255, 1950.
[5] Sen, A. R., A pilot survey of the characteristics of waterfowl
hunters in Ontario 1968-69, Journal of the American Statistical
Association, 65, 1,039, 1970.
[6] Sen, A.R. , Increased precision in Canadian waterfowl harvest
survey through successive sampling, J. Wildl. Manag., 33, 1971.
[7] Yates, F. , Sampling Methods for Censuses and Surveys, (4 ed.),
Griffin, London, 1981.