International Journal of Science and Research (IJSR)
ISSN: 2319-7064
SJIF (2022): 7.942
Some useful Estimators for Estimation of
Population Mean under PPS Sampling
Sangeeta Malik1, Poonam Devi2
1Professor, Department of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Corresponding Author Email: sangeetastat[at]gmail.com
2Research Scholar, Department of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Email: poonamsharawat9[at]gmail.com
Abstract: To estimate population parameter(s) using auxiliary information, many authors have given a variety of estimation techniques.
In this research paper we have proposed estimators using trigonometric type estimator to estimate the population's parameters under PPS
sampling, which yield beneficial results. First-degree scale estimators have been shown using data from an auxiliary variable. For the
suggested measurement scales, an evaluation of bias and mean squared error is carried out. The recommended estimator types outperform
comparable estimates for each unit when compared to other affective estimates. We'll compare the outcomes with existing estimators. We
have done an empirical illustration and graphical representation is also included to justify the utility of the proposed estimators.
Keywords: PPS Sampling, Trigonometry Type estimator, Mean Square Errors
1. Introduction sampling in presence of extra auxiliary information. Sangeeta
malik and Kusum [12] have proposed a new log type estimator
It is commonly known that using several auxiliary variables in simple random sampling.
increases the estimators' efficiency in surveys with large
populations. The following method may be applied Let U represent a size M finite population. For every unit i,
throughout the estimator design procedure or when choosing let (yi,xi) represent a pair of values corresponding to the study
a sample from the population. The auxiliary variables were variable y and an auxiliary variable x.
1 𝑦𝑖 1 𝑥𝑖
successfully employed in this work. It is commonly known 𝑦̅pps= ∑𝑚 𝑖=1 and 𝑥̅ pps= ∑𝑚
𝑖=1
𝑚 𝑝𝑖 𝑚 𝑝𝑖
that using pps estimators of the population mean instead of 1 𝑦𝑖 2
the traditional estimator simple mean for equal probability MSE(𝑦̅pps) = ∑𝑚
𝑖=1 ( − 𝑌) 𝑝𝑖 and
𝑚 𝑝𝑖
sampling results in a significant efficiency gain when the 1 𝑥𝑖 2
research variable and selection probabilities are substantially MSE(𝑥̅ pps) = ∑𝑚
𝑖=1 ( − 𝑋) 𝑝𝑖
𝑚 𝑝𝑖
connected. The pps estimate is unacceptable since it depends
on multiplicity but not on order. Then an outcome, the final
estimator is not as practical as the initial pps estimator. This
2. Notations
research proposes different estimators for population mean
We take a finite population of M units for the current study.
under pps sampling. As a result, the final estimator is not as
The study variable y and auxiliary variable x the obtaining
practical as the initial pps estimator. This research proposes
different estimators for population mean under pps sampling. mean estimates 𝑦̅ and 𝑥̅ of the population mean 𝑌̅ and 𝑋̅. To
Anita and Shashi Bahl [2] proposed an alternative estimator understand the bias and MSE. Let define
𝑦 𝑥
for the population mean under the probability proportional to 𝑞𝑖 = 𝑖 , 𝑝𝑖 = 𝑖
𝑀𝑘𝑖 𝑀𝑘𝑖
size sampling. John Graunt used the ratio estimator for the 1
∑𝑚
𝑦 𝑞̅𝑚 =
𝑚 𝑖=1 𝑞𝑖 = 𝑦
̅pps
first time to determine the ratio ,where x was the estimated 1
𝑥 𝑝̅𝑚 = ∑𝑚
𝑖=1 𝑝𝑖 = 𝑥̅ pps
total number of births that were recorded in the identical 𝑚
𝜎𝑞2
locations during the previous year and y was the overall 𝜎𝑞2 = ∑𝑀 ̅ 2
1 𝑘𝑖 (𝑞𝑖 − 𝑌 ) , = cq2
𝑦̅ 2
population. If the correlation coefficient (ρ) is positive then 2
we use the ratio estimator. If the correlation coefficient (ρ) is 𝜎𝑝2 = ∑𝑀 ̅ 2 𝜎𝑝
1 𝑘𝑖 (𝑝𝑖 − 𝑋 ) , 𝑥̅ 2 = cp
2
negative then we cannot use the ratio estimator. In such cases 𝑀
∑𝑖=1 𝑘𝑖 {𝑞𝑖 −𝑌̅ (𝑝𝑖 −𝑋̅2 )}
Goodman has proposed another type estimator say product 𝜌𝑝𝑞 =
𝜎𝑞 𝜎𝑝
estimator. S. Bhal and Tuteja [13] have recommended a ratio 𝑞̅
𝑌̅𝑅 = 𝑚 𝑋̅
and product type estimator. C.Kadilar and H.Cingi,Ratio[6] 𝑝̅𝑚
𝑝̅ 𝑞̅
have proposed an estimator for the population variance in 𝑌̅𝑃 = 𝑚̅ 𝑚
𝑋
simple and stratified random sampling. Mishra Madhulika,
B.P. Singh, Rajesh Singh [8] have recommended an estimation
of population mean using two auxiliary variables in stratified
3. Proposed Estimators
random sampling. Nikita and Sangeeta malik [10] have
proposed a generalized logarithmic ratio and product type 𝑌̅pd1 = 𝑦̅pps + 𝜆1 sin(𝑋̅ − 𝑥̅𝑝𝑝𝑠 ) and -------(1)
𝑋̅−𝑥̅
estimators in simple random sampling. P.A. Patel and 𝑌̅pd2 = 𝑦̅pps [1 + 𝜆2 sin ( ̅ 𝑝𝑝𝑠 )] ------(2)
𝑋+𝑥̅𝑝𝑝𝑠
Shraddha Bhatt [11] have suggested some estimation of finite
population total under probability proportional to size Where 𝜆1 and 𝜆2 are optimum constant.
Volume 13 Issue 4, April 2024
Fully Refereed | Open Access | Double Blind Peer Reviewed Journal
www.ijsr.net
Paper ID: SR24425172020 DOI: https://dx.doi.org/10.21275/SR24425172020 1764
ISSN: 2319-7064
SJIF (2022): 7.942
Some useful Estimators for Estimation of
Population Mean under PPS Sampling
Sangeeta Malik1, Poonam Devi2
1Professor, Department of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Corresponding Author Email: sangeetastat[at]gmail.com
2Research Scholar, Department of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Email: poonamsharawat9[at]gmail.com
Abstract: To estimate population parameter(s) using auxiliary information, many authors have given a variety of estimation techniques.
In this research paper we have proposed estimators using trigonometric type estimator to estimate the population's parameters under PPS
sampling, which yield beneficial results. First-degree scale estimators have been shown using data from an auxiliary variable. For the
suggested measurement scales, an evaluation of bias and mean squared error is carried out. The recommended estimator types outperform
comparable estimates for each unit when compared to other affective estimates. We'll compare the outcomes with existing estimators. We
have done an empirical illustration and graphical representation is also included to justify the utility of the proposed estimators.
Keywords: PPS Sampling, Trigonometry Type estimator, Mean Square Errors
1. Introduction sampling in presence of extra auxiliary information. Sangeeta
malik and Kusum [12] have proposed a new log type estimator
It is commonly known that using several auxiliary variables in simple random sampling.
increases the estimators' efficiency in surveys with large
populations. The following method may be applied Let U represent a size M finite population. For every unit i,
throughout the estimator design procedure or when choosing let (yi,xi) represent a pair of values corresponding to the study
a sample from the population. The auxiliary variables were variable y and an auxiliary variable x.
1 𝑦𝑖 1 𝑥𝑖
successfully employed in this work. It is commonly known 𝑦̅pps= ∑𝑚 𝑖=1 and 𝑥̅ pps= ∑𝑚
𝑖=1
𝑚 𝑝𝑖 𝑚 𝑝𝑖
that using pps estimators of the population mean instead of 1 𝑦𝑖 2
the traditional estimator simple mean for equal probability MSE(𝑦̅pps) = ∑𝑚
𝑖=1 ( − 𝑌) 𝑝𝑖 and
𝑚 𝑝𝑖
sampling results in a significant efficiency gain when the 1 𝑥𝑖 2
research variable and selection probabilities are substantially MSE(𝑥̅ pps) = ∑𝑚
𝑖=1 ( − 𝑋) 𝑝𝑖
𝑚 𝑝𝑖
connected. The pps estimate is unacceptable since it depends
on multiplicity but not on order. Then an outcome, the final
estimator is not as practical as the initial pps estimator. This
2. Notations
research proposes different estimators for population mean
We take a finite population of M units for the current study.
under pps sampling. As a result, the final estimator is not as
The study variable y and auxiliary variable x the obtaining
practical as the initial pps estimator. This research proposes
different estimators for population mean under pps sampling. mean estimates 𝑦̅ and 𝑥̅ of the population mean 𝑌̅ and 𝑋̅. To
Anita and Shashi Bahl [2] proposed an alternative estimator understand the bias and MSE. Let define
𝑦 𝑥
for the population mean under the probability proportional to 𝑞𝑖 = 𝑖 , 𝑝𝑖 = 𝑖
𝑀𝑘𝑖 𝑀𝑘𝑖
size sampling. John Graunt used the ratio estimator for the 1
∑𝑚
𝑦 𝑞̅𝑚 =
𝑚 𝑖=1 𝑞𝑖 = 𝑦
̅pps
first time to determine the ratio ,where x was the estimated 1
𝑥 𝑝̅𝑚 = ∑𝑚
𝑖=1 𝑝𝑖 = 𝑥̅ pps
total number of births that were recorded in the identical 𝑚
𝜎𝑞2
locations during the previous year and y was the overall 𝜎𝑞2 = ∑𝑀 ̅ 2
1 𝑘𝑖 (𝑞𝑖 − 𝑌 ) , = cq2
𝑦̅ 2
population. If the correlation coefficient (ρ) is positive then 2
we use the ratio estimator. If the correlation coefficient (ρ) is 𝜎𝑝2 = ∑𝑀 ̅ 2 𝜎𝑝
1 𝑘𝑖 (𝑝𝑖 − 𝑋 ) , 𝑥̅ 2 = cp
2
negative then we cannot use the ratio estimator. In such cases 𝑀
∑𝑖=1 𝑘𝑖 {𝑞𝑖 −𝑌̅ (𝑝𝑖 −𝑋̅2 )}
Goodman has proposed another type estimator say product 𝜌𝑝𝑞 =
𝜎𝑞 𝜎𝑝
estimator. S. Bhal and Tuteja [13] have recommended a ratio 𝑞̅
𝑌̅𝑅 = 𝑚 𝑋̅
and product type estimator. C.Kadilar and H.Cingi,Ratio[6] 𝑝̅𝑚
𝑝̅ 𝑞̅
have proposed an estimator for the population variance in 𝑌̅𝑃 = 𝑚̅ 𝑚
𝑋
simple and stratified random sampling. Mishra Madhulika,
B.P. Singh, Rajesh Singh [8] have recommended an estimation
of population mean using two auxiliary variables in stratified
3. Proposed Estimators
random sampling. Nikita and Sangeeta malik [10] have
proposed a generalized logarithmic ratio and product type 𝑌̅pd1 = 𝑦̅pps + 𝜆1 sin(𝑋̅ − 𝑥̅𝑝𝑝𝑠 ) and -------(1)
𝑋̅−𝑥̅
estimators in simple random sampling. P.A. Patel and 𝑌̅pd2 = 𝑦̅pps [1 + 𝜆2 sin ( ̅ 𝑝𝑝𝑠 )] ------(2)
𝑋+𝑥̅𝑝𝑝𝑠
Shraddha Bhatt [11] have suggested some estimation of finite
population total under probability proportional to size Where 𝜆1 and 𝜆2 are optimum constant.
Volume 13 Issue 4, April 2024
Fully Refereed | Open Access | Double Blind Peer Reviewed Journal
www.ijsr.net
Paper ID: SR24425172020 DOI: https://dx.doi.org/10.21275/SR24425172020 1764
