alexa Comparison of Horvitz and Thompson Estimator with that of Rao, Hartley and Cochran Estimator in PPS without Replacement Scheme | OMICS International
ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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Comparison of Horvitz and Thompson Estimator with that of Rao, Hartley and Cochran Estimator in PPS without Replacement Scheme

Anieting AE*

Department of Mathematics and Statistics, University of Uyo. P.M.B 1017, Uyo. Nigeria

*Corresponding Author:
Anieting AE
Department of Mathematics and Statistics
University of Uyo. P.M.B 1017, Uyo. Nigeria
E-mail: [email protected]

Received date: April 21, 2015; Accepted date: May 22, 2015; Published date: May 29, 2015

Citation: Anieting AE (2015) Comparison of Horvitz and Thompson Estimator with that of Rao, Hartley and Cochran Estimator in PPS without Replacement Scheme. J Biom Biostat 6:222. doi: 10.4172/2155-6180.1000222

Copyright: © 2015 Anieting AE. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are are credited.

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Abstract

This paper was focused on the comparison of Horvitz and Thompson estimator of population total with that of Rao, Hartley and Cochran estimator in PPS without replacement scheme when a sample of size six is taken from the same finite population. The data used were from Nigerian Bureau of Statistics bulletin. The result showed that the variances of both estimators gave positive values but the variance of the estimator by Rao, Hartley and Cochran was smaller making it a better estimator.

Keywords

PPS without replacement; Rao, Hartley and Cochran estimator; Horvitz and Thompson estimator

Introduction

Unequal probability sampling scheme or proportional to size (PPS) sampling scheme ensures that the possibility of selecting each unit into the sample is in proportion to the size or measure of size of the unit provided that the sizes of the individual units in the population are known and are correlated with the variable of interest, Horvitz and Thompson (1952) were the first to give theoretical frame work of PPS without replacement [1]. The sample variance of the population total estimator given by Horvitz and Thompson (1952) has the possibility of giving negative values for some samples. This problem has made statistician to work on this scheme. They include Yates and Grundy(1953), Sen (1953), Durbin(1953), Rao, Hartley and Cochran (1962) etc. Rao, Hartley and Cochran (1962) gave a modified estimator based on PPS without replacement scheme. The variance of the estimator for population total under Rao, Hartley and Cochran is always positive. But how do the estimators given by Horvitz and Thompson , and that given by Rao,Hartley and Cochran fare when the sample size is more than two, say six, assuming that the variance of the population total estimator by Horvitz and Thompson gives a positive value. The data for empirical verification of this study is from Nigerian Bureau of Statistics bulletin [1].

The two procedures

Horvitz and thompson estimation (procedure): First, the cumulative totals and ranges of the measure of size are formed [2]. Then to select a sample of size n without replacement, a random start between 1 and k inclusive (K=X/n) is selected using a table of random numbers. If the number selected is r then the units in the sample are those in whose ranges the umbers r,r+k,r+2k,….,r+(n-1)k fall. The probability of selecting the unit Ui in the sample of size n is

image

If any unit in the population has its size greater than or equal to k such unit is removed before sampling and taken into the sample with probability unity. The probability that any pair of units (Ui, Uj) is together in the sample is

image

Where qij is the number of the random numbers between 1 and k inclusive, which will select Ui and Uj simultaneously in the sample?

An unbiased estimator of the population total for PPS without replacement as given by Horvitz and Thompson is

image

While the sample estimator of image is given as

image

Rao, hartley and cochran estimation (procedure): Divide a population of N units into n groups at random with group g containing Ng units ( g=1,2,….,n) such that N1+…. + Nn = N. Thereafter select one unit independently from each group [3]. This will give a total of n units selected in the sample with PPS without replacement. The probability of selecting Ui in the sample in gth group is image

Xi’s are the auxiliary variables and Pg is the sum of the initial probabilities in gth group.

The Rao, Hartley and Cochran estimator of the population total is

image

Where ygi is the value of the study variate for the ith unit in gth group and image

The estimator of the variance is given as

image

Analysis

The correlation coefficient r between the auxiliary variable xi (the number of local government in each state) and the variable of interest yi (the number of police stations in Nigeria in each state) is 0.60. Other results of the analysis are summarized in Tables 1-3.

S/N States Xi Range Yi
1 Abia 18 1-18 22
2 Adamawa 21 19-39 41
3 AkwaIbom 31 40-70 65
4 Anambra 20 71-91 52
5 Bauchi 8 92-111 38
6 Bayelsa 23 112-119 37
7 Benue 27 120-142 27
8 Borno 18 143-169 42
9 Cross River 25 170-187 25
10 Delta 12 188-212 47
11 Ebonyi 18 213-224 29
12 Edo 16 225-242 31
13 Ekiti 16 243-258 21
14 Enugu 11 259-274 21
15 Gombe 27 275-285 14
16 Imo 27 286-312 63
17 Jigawa 27 313-339 65
18 Kaduna 23 340-362 59
19 Kano 45 363-407 123
20 Katsina 34 408-442 38
21 Kebbi 20 443-461 9
22 Kogi 21 462-482 40
23 Kwara 16 483-498 7
24 Lagos 20 499-518 43
25 Nassarawa 13 519-531 60
26 Niger 26 532-557 39
27 Ogun 20 558-577 25
28 Ondo 18 578-595 36
29 Osun 30 596-625 50
30 Oyo 33 626-658 47
31 Plateau 17 659-675 20
32 Rivers 23 676-698 34
33 Sokoto 23 699-721 3
34 Taraba 16 722-737 71
35 Yobe 17 738-754 19
36 Zamfara 14 755-768 35

Table 1: The correlation coefficient Are between the auxiliary variable xi and the variable of interest yi.

s/n sample πi yi yii
1 Bauchi 20/128 38 243.2
2 Edo 18/128 31 220.4
3 Kaduna 23/128 59 328.3
4 Kwara 16/128 7 56
5 Osun 30/128 50 213.3
6 Yobe 17/128 19 143.1
        1204.3

Table 2: The population total estimate using Horvitz and Thompson estimator.

Selected states P*i Pi Pg ygi ygi/P*i
Bornu 27/137 27/768 137/768 42 213.11
Benue 23/126 23/768 126/768 27 147.91
Edo 18/133 18/768 133/768 31 229.06
Taraba 16/101 16/768 101/768 71 448.19
Delta 25/113 25/768 113/768 47 212.44
Ogun 20/158 20/768 158/768 25 197.5
          1448.21

Table 3: Using eqn 2.1.1., the variance gives 690317.62 also using 2.2.1 the variance gives 102892.4.

Results and discussions

Based on a sample size of six from a population of 36 units, Tables 2 and 3 show that the population total estimate using Horvitz and Thompson estimator is 1204 which is smaller than 1448, the population total estimate using Rao,Hartley and Cochran estimator. Also the variance of the estimate of the population total by Horvitz and Thompson is 690317.62 which is bigger than 102892.4, the variance of the estimate of the population total by Rao, Hartley and Cochran [4].

Conclusion

Since the variance of the estimator given by Rao, Hartley and Cochran gives a value smaller than that given by Horvitz and Thompson with respect to sample size of six,it shows that the estimator given by Rao,Hartley and Cochran is better than that given by Horvitz and Thompson.

References

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