Greener Journal of Science Engineering and Technological Research
ISSN: 2276-7835
Vol. 2 (1), pp. 001-006, January 2012.
Research Article
Equal and Unequal Allocation Models for Ranked Set Sampling with Normal Distribution Parameters. 1
Badmus, Nofiu Idowu, 2Akingbade, Akeem Adekunle, 3Ayoola, Femi Joshua, 4Bamidele, Fatai Olawale
1
Department of Statistics, Abraham Adesanya Campus, Ijebu-Igbo. Ogun State Institute of Technology, Igbesa. 2 Department of Statistics, Yaba College of Technology, Yaba, Lagos 3 Department of Statistics, University of Ibadan, Ibadan, Nigeria 4 Department of Business Administration, Abraham Adesanya Campus, Ijebu-Igbo. Ogun State Institute of Technology, Igbesa Corresponding Author. E-mail:
[email protected], Phone: 08033676685
Abstract Ranked Set Sampling (RSS) performs better than Simple Random Sampling (SRS) when units corresponding to each rank are allocated equally. The performance of ranked set sampling further improves when appropriate unequal allocation is implemented instead of equal allocation. (Amarjot Kaur, G.P. Patil, and Charles Taillie 1997) came about the performance of ranked set sampling further improves when appropriate unequal allocation is implemented instead of equal allocation. They attempt to a “near” optimal allocation by exploiting knowledge of other, more easily available, characteristics of the population, like skewness, kurtosis and coefficient of variation. They also considered two right tail allocation models that assign more quantification to (i) the largest order statistic and (ii) the two largest order statistics. An attempt is made in this paper review to derive variance of equal and unequal allocations models for ranked set sampling and it is shown that the RSS sample mean is an unbiased estimator of the population mean through the method of likelihood function using Normal Distribution. It is observed that the variances of order statistics increase as the rank orders increase. We then compare the variance of RSS with variance of SRS and we got to know that the variance of ranked set sampling (RSS) is smaller than that of simple random sampling (SRS). However, the relative precision of the RSS is determine to SRS, Neyman’s optimal allocation relative to SRS and Neyman allocation all with equal allocation and with the same quantification by numerical example using water pollution data (secondary data). Keywords: Normal Distribution, Order Statistics, Ranked Set Sampling, Simple Random Sampling
1.
Introduction
The potential for observational economy was first recognized by (McIntyre 1952) who proposed ranked set sampling (RSS) to estimate mean pasture and forage yields. The RSS is useful and cost-effective technique for improving estimate of the mean when actual measurements are difficult and ranking is relatively easy. (Kaur, A., Patil, G.P., and Taillie C., 1994), studied the unequal allocation models for ranked set sampling with skew distribution. Also, (Kaur, A., G.P. Patil, and C. Taillie 1997) studied various equal and unequal allocation schemes. They studied the role played by the skewness, kurtosis, and the co-efficient of variation for obtaining the allocation factor(s) and in devising rules of Thumb. (Min Zhu and You-Gan Wang 2005) considered quantile estimation when ranking is perfect. They propose a new quantile estimator which is motivated by the fact that the observations from RSS are not identically distributed. In other words, observations selected by different ranks should contribute differently because they follow different distribution. It is proven analytically that the new estimator has a smaller asymptotic variance in general including the balanced design. They also considered how to select the allocation sequence ( ) to minimize the asymptotic variance. Jorgen Lauridsen (2005) investigated on n conceptual random samples each of size n from sample obtained the ℎ ordered value, denoted by () . (Ammar and Sarhen 2009) considered to estimate the population mean as well as the three-parameter distribution called modified weibull distribution (MWD) which is a general form of some well-known distributions such as exponential, Rayleigh, linear failure rate and Weibull distribution. The investigated this distribution and studied some essential properties of this distribution. (Al-Hadharami, S.A., 2010) investigated ranked set sampling to estimate the population mean as well as the three-parameters of the modified Weibull distribution. Maximum likelihood estimators (MLE for the parameters of modified weibull distribution (MWD) are also studied numerically since no closed form is available for the estimators. www.gjournals.org
1
Greener Journal of Science Engineering and Technological Research
Vol. 2 (1), pp. 001-006, January 2012.
ISSN: 2276-7835
In this paper, we derive variance of equal and unequal allocation models through the method of likelihood functions with normal distribution. Section 2 contains a general outline of the RSS with (a) equal allocation model and (b) unequal allocation model. It is shown that RSS under equal and unequal allocation models sample means are unbiased estimator of the population mean regardless of whether ranking is perfect or not. The relative precision of RSS under equal allocation is shown and the optimal allocation under unequal allocation is shown. In section 3, we present the numerical illustrative example on RSS and SRS using water pollution data (secondary data) under equal allocation, it then shown that the sample mean using RSS has a smaller variance than the sample mean using SRS when the number of observations is the same. This is to show that RSS is more efficient than SRS. Section 4 contains discussion and conclusion.
2a.
RSS estimator of Mean ( ) and Variance ( ) under equal allocation. ~ (, ) ( ) =
# $
"
%$ √!
2a.1
Where, = = ) * | , ] =
&' " (
√!
(see Jorgen Lauridsen 2005)
exp {− ∑67
(&'3 4 )$ $
}
2a.2
taking likelihood in equation (2a.2) and differentiating with respect to and also, equating the resulting expression to zero, we get 8
8( 8
9:* ; |
