© 2025 IJRTI | Volume 10, Issue 1 January 2025 | ISSN: 2456-3315
An Efficient Generalized Logarithmic Type
Estimator Using Probability Sampling
1
Poonam Devi1st, 2Sangeeta Malik2nd,
1
Research Scholar 2 Professor
1
Dept. of Mathematics, Baba Mastnath University, Rohtak (Haryana)
2
Dept. of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Email:
*Corresponding author: Email:
Abstract - In this research paper, we have proposed logarithmic type estimator to estimate the population parameter using
auxiliary information under simple random sampling. To determine the bias and mean square error of the suggested
estimator have been presented up to the first degree of approximation. The estimated categories that have been suggested
perform more effectively than comparable estimates for each unit when compared to other existing estimates. We have
conducted an empirical demonstration to support the usefulness of the suggested estimators.”
Keywords - Bias and Mean Square Error, Auxiliary Information, Simple Random Sampling.
1. Introduction
The fact is generally accepted, when using studies with large populations, the effectiveness of the estimators is increased when
multiple auxiliary variables are used. The following technique can be used to select some samples from the population or at any
point during the estimator creation process. In this work, the auxiliary variables were effectively used. It is well known that
when the research variable and choice possibilities are considerably related, employing simple random sampling estimators of
the population mean for equal probability sampling yields a large increase in efficiency when compared to the usual estimator
simple mean. To estimate the population characteristics of the primary variable under investigation, a straightforward random
sampling strategy is frequently employed. In sampling theory, it is typically beneficial to incorporate supplementary data in
order to get precise estimates of various population parameters. However, the most important component is that there should be
a correlation between the variables being studied and the other components. The use of auxiliary data in the form of product
type and exponential ratio estimators was initially presented by Bahl and Malik [5]., Singh et al. [3] and some significant current
research also Murthy [2] include who recommended a log type estimator using the knowledge of auxiliary information presented
in a set of variables. Practically and empirically, authors evaluated the intended types of estimators to a few previous estimators,
and when supplementary data existed, they claimed that the new estimators performed better. They also came to the conclusion
from their investigation that the updated estimators performed better than the current estimators. According to a study, the
suggested types of estimators outperformed standard estimators of regression, ratios, and product types.
Methods of Sampling and Notations
Consider a population of N units, where Ụ= ϻ1, ϻ2,……….., ϻN. Let X and Y used for the auxiliary variables and the research
variables, respectively. Let (yi, xi) represent the N pairs of sample observations for the auxiliary variables and study variables,
respectively, for the 𝑖𝑡ℎ unit that were acquired using simple random sampling without replacement from the population size of
N. Let the population means of the auxiliary and study variables be represented by 𝑋̅ and 𝑌̅ respectively and the sample means
by 𝑥̅ and 𝑦̅.
Let's clarify,
sy2 = Sy2 (ey + 1) , sx2 = Sx2 (ex + 1) , sz2 = Sz2 (ez + 1)
E(ex ) = E(ey ) = E(ez ) = 0
1 1 1
E(e2y ) = n (∆400 − 1), E(e2x ) = n (∆040 − 1), E(e2z ) = n (∆004 − 1)
1 1 1
E(ex ey ) = n (∆220 − 1), E(ex ez ) = n (∆022 − 1), E(ez ey ) = n (∆202 − 1)
2
stx = (1 + g)Sx2 − gsx2 , sty
2
= (1 + g)Sy2 − gsy2 , 2
stz = (1 + g)Sz2 − gsz2
uabc 𝑛
Where ∆abc = a⁄ b⁄ c⁄ , g = 𝑁−𝑛
2 u 2u 2
u200 020 002
1
uabc = N ∑N ̅ a ̅ b ̅ c
i=1(yi − Y) (xi − X) (zi − Z) , where a, b and c being non -negative integers
IJRTI2501118 International Journal for Research Trends and Innovation (www.ijrti.org) a978
An Efficient Generalized Logarithmic Type
Estimator Using Probability Sampling
1
Poonam Devi1st, 2Sangeeta Malik2nd,
1
Research Scholar 2 Professor
1
Dept. of Mathematics, Baba Mastnath University, Rohtak (Haryana)
2
Dept. of Mathematics, Baba Mastnath University, Rohtak (Haryana)
Email:
*Corresponding author: Email:
Abstract - In this research paper, we have proposed logarithmic type estimator to estimate the population parameter using
auxiliary information under simple random sampling. To determine the bias and mean square error of the suggested
estimator have been presented up to the first degree of approximation. The estimated categories that have been suggested
perform more effectively than comparable estimates for each unit when compared to other existing estimates. We have
conducted an empirical demonstration to support the usefulness of the suggested estimators.”
Keywords - Bias and Mean Square Error, Auxiliary Information, Simple Random Sampling.
1. Introduction
The fact is generally accepted, when using studies with large populations, the effectiveness of the estimators is increased when
multiple auxiliary variables are used. The following technique can be used to select some samples from the population or at any
point during the estimator creation process. In this work, the auxiliary variables were effectively used. It is well known that
when the research variable and choice possibilities are considerably related, employing simple random sampling estimators of
the population mean for equal probability sampling yields a large increase in efficiency when compared to the usual estimator
simple mean. To estimate the population characteristics of the primary variable under investigation, a straightforward random
sampling strategy is frequently employed. In sampling theory, it is typically beneficial to incorporate supplementary data in
order to get precise estimates of various population parameters. However, the most important component is that there should be
a correlation between the variables being studied and the other components. The use of auxiliary data in the form of product
type and exponential ratio estimators was initially presented by Bahl and Malik [5]., Singh et al. [3] and some significant current
research also Murthy [2] include who recommended a log type estimator using the knowledge of auxiliary information presented
in a set of variables. Practically and empirically, authors evaluated the intended types of estimators to a few previous estimators,
and when supplementary data existed, they claimed that the new estimators performed better. They also came to the conclusion
from their investigation that the updated estimators performed better than the current estimators. According to a study, the
suggested types of estimators outperformed standard estimators of regression, ratios, and product types.
Methods of Sampling and Notations
Consider a population of N units, where Ụ= ϻ1, ϻ2,……….., ϻN. Let X and Y used for the auxiliary variables and the research
variables, respectively. Let (yi, xi) represent the N pairs of sample observations for the auxiliary variables and study variables,
respectively, for the 𝑖𝑡ℎ unit that were acquired using simple random sampling without replacement from the population size of
N. Let the population means of the auxiliary and study variables be represented by 𝑋̅ and 𝑌̅ respectively and the sample means
by 𝑥̅ and 𝑦̅.
Let's clarify,
sy2 = Sy2 (ey + 1) , sx2 = Sx2 (ex + 1) , sz2 = Sz2 (ez + 1)
E(ex ) = E(ey ) = E(ez ) = 0
1 1 1
E(e2y ) = n (∆400 − 1), E(e2x ) = n (∆040 − 1), E(e2z ) = n (∆004 − 1)
1 1 1
E(ex ey ) = n (∆220 − 1), E(ex ez ) = n (∆022 − 1), E(ez ey ) = n (∆202 − 1)
2
stx = (1 + g)Sx2 − gsx2 , sty
2
= (1 + g)Sy2 − gsy2 , 2
stz = (1 + g)Sz2 − gsz2
uabc 𝑛
Where ∆abc = a⁄ b⁄ c⁄ , g = 𝑁−𝑛
2 u 2u 2
u200 020 002
1
uabc = N ∑N ̅ a ̅ b ̅ c
i=1(yi − Y) (xi − X) (zi − Z) , where a, b and c being non -negative integers
IJRTI2501118 International Journal for Research Trends and Innovation (www.ijrti.org) a978
