SOME MODEL-BASED ESTIMATOR
Danutė Krapavickaitė1 and Vilma Nekrašaitė2
1 Institute
of Mathematics and Informatics, Lithuania; Statistics Lithuania, Lithuania
e-mail: krapav@ktl.mii.lt
2
Vilnius Gediminas Technical University, Lithuania; Statistics Lithuania, Lithuania
e-mail: vilma.nekrasaite@stat.gov.lt
Abstract
A design-based and model-based estimator of a total in a finite population (Valliant et. al 2000)
and estimation of the variance of the model-based estimator (Valliant 1985) is discussed. The
study variable having zero and positive values is considered. Some econometric models for this
variable are being suggested to use. Some simulation results are given.
1 Design-based and model-based approaches in survey sampling
In the main branches of statistics data is considered as realisation of some random
variables. The aim of a statistician is to make inference about the probability laws of these
random variables. The survey sampling is historically isolated from the mainstream of
statistics. The data in this field is considered as fixed and randomness is introduced by
statistician when selecting data for observation from the whole amount of data. The aim of
the survey statistician is to get a statistical estimate of some fixed parameter describing the
whole amount of data. The main branches of statistics would investigate a probability law of
the random parameter of the randomly generated data.
The main statistical approach to the survey sampling is taken in the book of Valliant
et al. (2000) and prediction of the population parameters is investigated. The values of a
study variable y : y1 , y2 ,..., y N in the finite population U  {1,2,..., N } are considered to be
random, generated by some statistical model. The population total
N
t y   yk
k 1
is also random. Given a probability sample s from U , s  U , the value of the total t y can
be predicted (denoted by tˆy ) after the individual values of y k , k U \ s are predicted (let us
denote them by ŷ k ):
tˆy   yk 
ks
 yˆ
kU \ s
k
(1)
Properties of linear (Valliant et al. 2000) and nonlinear (Valliant, 1985) models of y are
being studied, accuracy of the predictor tˆy for a total t y of study variable y is investigated.
2 Models for some skew distributed study variable
We will discuss prediction of a finite population total for a special case of a study
variable y , which obtains zero values in some cases and positive values in other cases. Such
situation arises when investigating plots under the crops, which are being grown up only in
some parts of the country; when investigating expenditure of the enterprises to the protection
of the environment.
Let us denote two vectors of the auxiliary variables x (1) , x ( 2 ) with the values
x1(1) ,..., x (N1) and x1( 2 ) ,..., x (N2 ) , and two unobserved variables y (1) , y ( 2 ) with the values
y1(1) ,..., y N(1) and y1( 2) ,..., y N( 2) satisfying the models
y (1)  1 ' x(1)  u1 ,
y ( 2 )   2 ' x ( 2 )  u2
(2)
with the vectors of constants 1 ,  2 and random errors u1 , u2 distributed according to a
normal law with zero mean (the indices denoting the values of the variables are omitted here).
Let us suppose that a study variable y satisfies condition
 y (1) if y (1)  y ( 2) ,
y
0 otherwise.
(3)
This is a censored regression model with unobserved stochastic threshold - a case of
Heckman model (Maddala, 1983). In the case when y ( 2)  0 instead of (2) the variable y in
(3) satisfies the conditions of the censored regression (tobit) model (Greene (2002), Maddala
(1983)).
The estimator (1) for the model (3) will be investigated in the lecture. The results of
simulations show that the accuracy of the tobit-model based estimator of total in some cases
can be much better than that of design-based estimator under a simple random sampling.
References
Greene W.H. (2002) Econometric Analysis. Prentice Hall, Upper Saddle River.
Maddala G.S. (1983) Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press,
Cambridge.
Valliant, R. (1985) Nonlinear Prediction Theory and the Estimation of Proportion in a Finite Population. Journal
of American Statistical Association, 80, 631-641.
Valliant R.,, Dorfman A.H. and Royall M. (2000) Finite Population Sampling and Inference: a Prediction
Approach. John Wiley & Sons, New York.