Modified
Regression-Cum-Dual Mean Imputation Schemes for Estimating Population Mean
Under Two-Phase Simple Random Sampling
By
B.B.Bayedo1,
K.E.Lasisi2, A.Ahmed3 & A.A.Issa4
1,2,3&4Department
of Statistics, Abubakar Tafawa Balewa University, Bauchi, Nigeria.
Corresponding
Author email: bbb.edu79@gmail.com.
ABSTRACT
In this research,
It has been paramount for many researchers under sample survey to use auxiliary
information incorporation with the study variables in an estimation stage to
modify estimator in order to increase the precision of the estimated population
mean. This research
work proposed
modified regression-cum-dual mean imputation schemes
for estimating population mean under two-phase simple random sampling.it concluded that the modified estimators
demonstrated a high level of efficiency over the existing estimators considered
for both case one and two..
Keywords: Auxiliary
variable, Population
Mean, Mean Squared Errors (MSE) and Bias
1.0 Introduction
Sampling theory is the field of statistics that is concerned
with the collection, analysis and interpretation of data gathered from sampling
the population under study. The application of sampling theory is concerned not
only with the proper selection of observations from the auxiliary variable. Many prominent authors developed several modified and
improved ratio, regression, and exponential type estimators by using the
population information of the auxiliary variable x. However, the information
about the population mean of the auxiliary variable is not always available. In
the environment mentioned above, the most popular sampling scheme is the
two-phase sampling scheme which was first established by Neyman (1938) to
accumulate information on sampling. It is customarily acquired when the
accumulation of information on a study variable is very costly but relatively
cheaper to accumulate information on auxiliary variables that are correlated
with the study variables. Due to these reasons, the two-phase sampling becomes
a powerful and cost-effective scheme for obtaining the authentic estimate in
the one-phase sample for the unknown parameters of the auxiliary variable.
Authors like Kumar and Bahl (2006), Singh and Vishwakarma (2007), Singh (2011),
Ozgul and Cingi (2014), Kalita et al. (2016), Noor-Al-Amin et al. (2016), Bazad and Bazad (2019), Bhushan and Gupta (2019),
Adamu et al (2019) and Bhushan et al. (2023) have extensively worked
under two-phase sampling. However, the applicability of the aforementioned estimators
depends on the complete availability of sample information measured on both the
study and auxiliary variables. Reduction in the sizes of sample information due
to non-response decreases the efficiency of these estimators. In literature,
Sukhatme (1962), Choudhury and Singh (2015), Audu and Adewara (2017a,b), Adamu et al. (2019) investigated the classical
ratio estimator in two-phase sampling. Following Srivenkataraman (1980), Kumar
and Bahl (2006) envisaged a class of dual to exponential type ratio estimators
using two phases. Singh and Vishwakarma (2007) investigated Bahl and Tuteja's
(1991) exponential ratio and product estimators of population mean under
two-phase sampling. Singh (2011), Ozgul and Cingi (2014) developed a class of
exponential regression cum ratio estimator in two-phase sampling. Kalita et al. (2016) suggested exponential
ratio-cum-exponential dual-to-ratio estimators using two-phase sampling.
Following Kumar and Bahl (2006) and Kalita et
al. (2016), Bazad and Bazad (2019) developed some classes of dual-to-ratio
exponential-type estimators. Bhushan and Gupta (2019) provided some log-type
estimators of population mean using two-phase sampling. Zaman and Kadilar
(2021a) introduced a new class of exponential estimators for finite population
mean in two-phase sampling whereas Zaman and Kadilar (2021b) examined an
exponential ratio and product estimators of population mean under two-phase
sampling. Bhushan et al. (2021a)
suggested some efficient classes of estimators. Bhushan et al. (2021b) developed some efficient classes of estimators under
two-phase sampling. Recently, Bhushan and Kumar (2023) suggested a new
efficient class of estimators of population mean using two-phase sampling.
2.0 LITERATURE REVIEW
2.1 Some Existing Estimators under SRSWOR
The contributions of Cochran (1942), Deming (1956),
Hurvitz (1952) and others helped in laying the foundation of modern sampling
theory. The work done during these periods made important contribution to the
modern sampling theory by suggesting methods of utilizing the auxiliary
information for the purpose of estimation of the population mean in order to
increase the precision of the estimates.
To discuss some of the developed estimators in
literature based on auxiliary variables, the following descriptions about the
population and sample units were considered.
Let 
be a population of size 
and
be two real valued
functions having values
on the 
unit of 
.. Let 
and 
be the population means of 
and 
respectively with 
and 
as coefficients of
variation of and
.
Let a pair 
of simple random
sample of size 
be drawn without
replacement from the population and 
, 
be sample means based on the sample drawn.
The usual
sample mean
, traditional ratio estimator
(Cochran, 1942), product estimator 
(Murthy, 1964) and Dual to ratio estimator
(Strivenkataramana, 1980) with their
respective bias and variance/mean square error are defined as
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
Where
, 
,
, 
,
, 
,
.
The
traditional ratio estimator
and Dual to ratio estimator 
have higher efficiencies when the correlation
between the study and auxiliary variables is strong and positive while product
estimator 
has higher efficiency when the correlation
between the study and auxiliary variables is strong and negative.
For estimating the population mean 
of the study variate y, Singh
and Espejo (2003) considered a linear combination of
ratio and product estimators’ type given by
(2.12) and obtained that
estimator attained optimality when 
with 
, where Cy is the coefficient of variation of y
and Cx is the coefficient of
variation of x.
Singh
and Espejo (2007), modified that when the population
mean 
of x is not known, a first-phase sample of size 
should be drawn from the population on which
only the x-characteristic is measured in order to furnish a good estimate of . Then a second-phase sample of size n is drawn on which both the
variables y and x are measured. Let 
denote the sample mean of x
based on the first-phase sample of size , Singh
and Espejo (2007) considered a ratio–product type
estimator in the two-phase sampling given by
(2.13) Choudhury and
Singh (2012) consider the work of Singh and Espejo (2007) and modified a class
chain ratio-product type estimator
for estimating
population mean 
using two auxiliary characters under two conditions in
Singh and Espejo (2007). The modified estimator, its bias and MSE are
respectively given as
(2.14)
Subramani and Prabavathy (2014) modified two estimators of
population mean based on median of study variable using auxiliary information
as:
(2.35)
` 
(2.36) where 
and 
are Median of
the study variable, Median of the auxiliary variable and Sample median of study
variable respectively.
The MSEs of modified estimators are given below;
(2.37) where 
and 
.
Singh (2015) modified an improved
class of ratio type estimator for finite population mean using unknown weight
and power transformation strategies. The particular cases of this estimator are
Walsh (1970) estimator at 
and 
, Ray et al., (1979) estimator at 
and 
, Srivenkataramana and Tracy (1979) at 
and 
, Srivenkataramana and Tracy (1979) at and 
and Srivenkataramana
(1980) at and 
. The modified estimator and its properties are given below;
(2.38)
(2.39)
(2.40)
In his study, the modified
estimators attain the optimality when 
. The empirical study of in his work revealed that the
modified estimator is more efficient than the estimators of Walsh (1979), Ray et
al., (1979), Srivenkataramana and Tracy (1979), Srivenkataramana (1980).
3.0 MATERIALS AND METHODS
3.1 Robust
Outlier-Free Measure to be used in the Study
The
study will utilize robust, outlier-resistant parameter estimation techniques to
minimize or eliminate the influence of extreme values in the sample data. These
methods include:
i.
Gini’s Mean Difference method
ii.
Downton’s Method
iii.
The Method of Probability-Weighted Moments
3.1.1 Gini’s Mean Method
Let
, where 
is the order statistics so that 
is the distance
between adjacent observations. Then Gini’s mean method proposed by Nair
(1936) is given as
(3.1)
3.1.2 Downton’s Method
Let 
, be a random sample from a normal distribution with
mean 
and variance 
; that is, 
Let 
denote the
corresponding order statistics. The Downton estimator, 
proposed by Downton (1966)
,
is given as
(3.2)
Where
for a normal distribution and does not depend on the sample
size n.
3.1.3 Method
of probability weighted moments
Let , be a random sample from a normal distribution with
mean and variance ; that is, 
Let 
denote the corresponding order statistics. The PWMs is defined by Greenwood et al. (1979) and is given as
(3.3)
4.0 Data Analysis and Discussion of
result
For empirical validation, four simulated population
datasets with varying sampling conditions were generated to assess the
performance of the proposed estimators within Survey Sampling. Their efficiency
was evaluated against existing estimators using Bias, Mean Square Error (MSE),
and Percentage Relative Efficiency (PRE), which measure accuracy, variability,
and relative performance in estimating the population mean. Bias and MSE
assessed closeness to true parameters and estimation precision, while PRE enabled
comparative efficiency analysis, providing a systematic framework that
demonstrates the effectiveness and potential superiority of the proposed
estimators within Statistics.
4.1
Populations used for Simulation Study for positive relationship
Table 1: Biases, MSEs and PREs of
Existing Estimators and Proposed Estimators of the proposed schemes 1, 2 and 3
using N=2000, 
=800, n=300, r=200
(Case 1 & 2)
|
Estimators |
Case 1 |
Case 2 |
||||
|
Biases |
MSEs |
PREs |
Biases |
MSEs |
PREs |
|
|
Sample
mean T0 |
0.00974 |
0.029 |
100.0000 |
0.00016 |
0.0298 |
100 |
|
Lee
et al. (1994) T1 |
-0.00107 |
0.0102 |
284.2900 |
-0.00471 |
0.01019 |
292.3500 |
|
Singh
& Horn (2000) T2 |
-0.00265 |
0.00994 |
291.8400 |
-0.00543 |
0.01088 |
273.8100 |
|
Kadilar
& Cingi (2008) T3 |
0.01267 |
0.02634 |
110.1000 |
0.02825 |
0.04539 |
65.6500 |
|
Singh
(2009) T4 |
0.00133 |
0.01002 |
289.5600 |
0.00058 |
0.01134 |
262.8300 |
|
Audu
et al. (2021b) T5 |
-0.02768 |
0.01163 |
249.4300 |
-0.04152 |
0.01544 |
192.9900 |
|
Musa
et al. (2023) |
|
|
|
|||
|
T6 (a=Cx,b=Sx) |
-1.08465 |
1.25058 |
2.3200 |
-1.09581 |
1.26833 |
2.3500 |
|
T6 (a=Cx,b=Skw) |
-1.18521 |
1.47997 |
1.9600 |
-1.19667 |
1.50113 |
1.9800 |
|
T6 (a=Cx,b=Kurt) |
-1.20351 |
1.52412 |
1.9000 |
-1.2151 |
1.5463 |
1.9300 |
|
T6 (a=Sx,b=Cx) |
-1.2069 |
1.5324 |
1.8900 |
-1.21852 |
1.55478 |
1.9200 |
|
T6 (a=Sx,b=Skw) |
-1.20694 |
1.53248 |
1.8900 |
-1.21855 |
1.55486 |
1.9200 |
|
T6 (a=Sx,b=Kurt) |
-1.20916 |
1.53792 |
1.8900 |
-1.22079 |
1.56043 |
1.9100 |
|
T6 (a=Skw,b=Sx) |
-1.08394 |
1.24905 |
2.3200 |
-1.09511 |
1.26679 |
2.3500 |
|
T6 (a=Skw,b=Cx) |
-1.1847 |
1.47874 |
1.9600 |
-1.19615 |
1.49987 |
1.9900 |
|
T6 (a=Skw,b=Kurt) |
-1.20344 |
1.52395 |
1.9000 |
-1.21502 |
1.54612 |
1.9300 |
|
T6 (a=Kurt,b=Cx) |
-1.13035 |
1.35207 |
2.1400 |
-1.14157 |
1.37098 |
2.1700 |
|
T6 (a=Kurt,b=Sx) |
-1.00982 |
1.09422 |
2.6500 |
-1.0211 |
1.11102 |
2.6800 |
|
T6 (a=Kurt,b=Skw) |
-1.13097 |
1.35346 |
2.1400 |
-1.14219 |
1.3724 |
2.1700 |
|
Estimators
of Proposed Scheme 1 |
|
|
|
|||
|
Tp11 (G,D) |
-0.00293 |
0.00959 |
302.3200 |
-0.00521 |
0.00993 |
300.0000 |
|
Tp12 (G,S) |
-0.00311 |
0.00965 |
300.6500 |
-0.005 |
0.01024 |
290.9600 |
|
Tp13 (D,G) |
-0.00323 |
0.00972 |
298.4500 |
-0.00479 |
0.01053 |
282.8800 |
|
Tp14 (D,S) |
-0.00328 |
0.00976 |
297.2700 |
-0.00468 |
0.01067 |
279.2400 |
|
Tp15 (S,G) |
-0.00303 |
0.00962 |
301.6000 |
-0.0051 |
0.01009 |
295.4400 |
|
Tp16 (S,D) |
-0.0029 |
0.00959 |
302.4800 |
-0.00524 |
0.00989 |
301.3200 |
|
Estimators
of Proposed Scheme 2 |
|
|
|
|||
|
Tp21 (G,D) |
0.01144 |
0.03097 |
93.6300 |
0.00188 |
0.03217 |
92.6300 |
|
Tp22 (G,S) |
0.01264 |
0.0324 |
89.5200 |
0.00312 |
0.0339 |
87.9000 |
|
Tp23 (D,G) |
0.01363 |
0.0336 |
86.3000 |
0.00416 |
0.03538 |
84.2300 |
|
Tp24 (D,S) |
0.01407 |
0.03413 |
84.9600 |
0.00461 |
0.03603 |
82.6900 |
|
Tp25 (S,G) |
0.01206 |
0.0317 |
91.4700 |
0.00252 |
0.03305 |
90.1500 |
|
Tp26 (S,D) |
0.01126 |
0.03075 |
94.2900 |
0.0017 |
0.0319 |
93.3900 |
|
Estimators
of Proposed Scheme 3 |
|
|
|
|||
|
Tp31 (G,D) |
-0.00219 |
0.00965 |
300.5000 |
-0.00558 |
0.00934 |
319.0300 |
|
Tp32 (G,S) |
-0.0019 |
0.00974 |
297.6400 |
-0.00564 |
0.00924 |
322.4900 |
|
Tp33 (D,G) |
-0.00165 |
0.00985 |
294.5000 |
-0.00566 |
0.0092 |
323.9800 |
|
Tp34 (D,S) |
-0.00154 |
0.0099 |
292.9400 |
-0.00566 |
0.00919 |
324.2300 |
|
Tp35 (S,G) |
-0.00204 |
0.00969 |
299.1500 |
-0.00561 |
0.00928 |
321.0400 |
|
Tp36 (S,D) |
-0.00224 |
0.00964 |
300.8400 |
-0.00557 |
0.00936 |
318.3400 |
Table 2: Biases, MSEs and PREs of
Existing Estimators and Proposed Estimators of the proposed schemes 1, 2 and 3
using N=2000, =1000, n=500,
r=300 (Case 1 & 2)
|
Estimators |
Case 1 |
Case 2 |
||||
|
Biases |
MSEs |
PREs |
Biases |
MSEs |
PREs |
|
|
Sample
mean T0 |
-0.00227 |
0.01853 |
100 |
0.00277 |
0.01875 |
100 |
|
Lee
et al. (1994) T1 |
-0.00414 |
0.00423 |
438.36 |
-0.00157 |
0.00645 |
290.86 |
|
Singh
& Horn (2000) T2 |
-0.00441 |
0.00407 |
454.81 |
-0.00221 |
0.00685 |
273.55 |
|
Kadilar
& Cingi (2008) T3 |
0.01484 |
0.01824 |
101.6 |
0.01666 |
0.02677 |
70.04 |
|
Singh
(2009) T4 |
-0.00115 |
0.00411 |
450.69 |
0.00135 |
0.00699 |
268.38 |
|
Audu
et al. (2021b) T5 |
-0.02433 |
0.00531 |
348.68 |
-0.0247 |
0.00846 |
221.71 |
|
Musa
et al. (2023) |
|
|
|
|||
|
T6 (a=Cx,b=Sx) |
-0.99812 |
1.02416 |
1.81 |
-0.97778 |
0.99538 |
1.88 |
|
T6 (a=Cx,b=Skw) |
-1.12083 |
1.28358 |
1.44 |
-1.1005 |
1.25366 |
1.5 |
|
T6 (a=Cx,b=Kurt) |
-1.14291 |
1.33356 |
1.39 |
-1.12257 |
1.30354 |
1.44 |
|
T6 (a=Sx,b=Cx) |
-1.14699 |
1.34293 |
1.38 |
-1.12666 |
1.31289 |
1.43 |
|
T6 (a=Sx,b=Skw) |
-1.14703 |
1.34303 |
1.38 |
-1.1267 |
1.31299 |
1.43 |
|
T6 (a=Sx,b=Kurt) |
-1.14971 |
1.34918 |
1.37 |
-1.12938 |
1.31913 |
1.42 |
|
T6 (a=Skw,b=Sx) |
-0.99725 |
1.02243 |
1.81 |
-0.97691 |
0.99367 |
1.89 |
|
T6 (a=Skw,b=Cx) |
-1.12021 |
1.28219 |
1.45 |
-1.09988 |
1.25227 |
1.5 |
|
T6 (a=Skw,b=Kurt) |
-1.14282 |
1.33337 |
1.39 |
-1.12249 |
1.30334 |
1.44 |
|
T6 (a=Kurt,b=Cx) |
-1.05418 |
1.13884 |
1.63 |
-1.03384 |
1.10943 |
1.69 |
|
T6 (a=Kurt,b=Sx) |
-0.90522 |
0.84804 |
2.18 |
-0.88493 |
0.82079 |
2.28 |
|
T6 (a=Kurt,b=Skw) |
-1.05493 |
1.14042 |
1.62 |
-1.03459 |
1.111 |
1.69 |
|
Estimators
of Proposed Scheme 1 |
|
|
|
|||
|
Tp11 (G,D) |
-0.00472 |
0.00392 |
472.18 |
-0.00221 |
0.00697 |
268.78 |
|
Tp12 (G,S) |
-0.00462 |
0.00402 |
460.55 |
-0.0021 |
0.00726 |
258.38 |
|
Tp13 (D,G) |
-0.00452 |
0.00414 |
447.8 |
-0.00198 |
0.00752 |
249.15 |
|
Tp14 (D,S) |
-0.00446 |
0.0042 |
441.54 |
-0.00193 |
0.00765 |
245.03 |
|
Tp15 (S,G) |
-0.00467 |
0.00397 |
466.73 |
-0.00216 |
0.00711 |
263.53 |
|
Tp16 (S,D) |
-0.00473 |
0.00391 |
473.57 |
-0.00223 |
0.00694 |
270.31 |
|
Estimators
of Proposed Scheme 2 |
|
|
|
|||
|
Tp21 (G,D) |
-0.00096 |
0.0203 |
91.26 |
0.00419 |
0.02055 |
91.22 |
|
Tp22 (G,S) |
-0.00002 |
0.0216 |
85.78 |
0.00524 |
0.02188 |
85.67 |
|
Tp23 (D,G) |
0.00079 |
0.02271 |
81.58 |
0.00614 |
0.02303 |
81.4 |
|
Tp24 (D,S) |
0.00114 |
0.0232 |
79.85 |
0.00654 |
0.02354 |
79.63 |
|
Tp25 (S,G) |
-0.00048 |
0.02097 |
88.37 |
0.00473 |
0.02123 |
88.3 |
|
Tp26 (S,D) |
-0.00111 |
0.02011 |
92.15 |
0.00404 |
0.02035 |
92.12 |
|
Estimators
of Proposed Scheme 3 |
|
|
|
|||
|
Tp31 (G,D) |
-0.00481 |
0.00389 |
476.51 |
-0.00237 |
0.00645 |
290.58 |
|
Tp32 (G,S) |
-0.00477 |
0.00396 |
467.51 |
-0.00233 |
0.00637 |
294.16 |
|
Tp33 (D,G) |
-0.00471 |
0.00406 |
456.56 |
-0.00226 |
0.00635 |
295.35 |
|
Tp34 (D,S) |
-0.00468 |
0.00411 |
450.95 |
-0.00222 |
0.00635 |
295.36 |
|
Tp35 (S,G) |
-0.00479 |
0.00392 |
472.48 |
-0.00235 |
0.0064 |
292.72 |
|
Tp36 (S,D) |
-0.00481 |
0.00388 |
477.45 |
-0.00237 |
0.00647 |
289.82 |
4. 2 Interpretation of the Results
The interpretation of the simulation
results is carried out based on the key aims of the proposed modifications,
namely the reduction of bias, minimization of Mean Square Error (MSE), and
improvement in efficiency in estimating the population mean within Survey Sampling. To evaluate the
performance of the proposed estimators with respect to these objectives,
simulation data were generated and analyzed under different sampling
configurations. The results obtained from the simulation study are summarized
in Tables 1–2 for Cases 1 and 2.
These tables present the computed
values of Biases, Mean Square Errors (MSEs), and Percentage Relative
Efficiencies (PREs) of the proposed class of estimators and compare them with
those of several existing estimators developed by Lee
et al. (1994)
, Singh and Horn (2000)
, Kadilar and Cingi (2008)
, Singh (2009)
, Audu et al.
(2021b)
and Musa et al. (2023)
. Through
these comparisons, the effectiveness of the modified regression-cum-dual mean
imputation estimators is assessed in terms of their ability to produce
estimates with smaller bias, lower MSE, and higher efficiency relative to the
existing estimators. The subsequent interpretation therefore examines the
results in relation to each modification aim in order to demonstrate the
statistical advantages of the proposed estimators within the framework of Statistics.
5.0
Conclusion
This study examined the statistical properties of the proposed modified
estimators for estimating the population mean in the presence of non-response
within the framework of Survey Sampling. The analytical properties of the
estimators, particularly their Biases and Mean Square Errors (MSEs), were
derived up to the third-order approximation using the Taylor Series
Expansion. These derivations provided a theoretical basis for
evaluating the performance of the modified imputation estimators and for
establishing conditions under which they outperform some existing estimators
considered in the study.
Furthermore, the efficiency conditions of the modified estimators were
obtained by comparing the minimum MSE expressions of the proposed estimators
with the MSE (or minimum MSE) expressions of the competing estimators. The bias
and MSE expressions of the proposed imputation schemes were derived using
binomial expansion techniques up to the third order, which allowed for a more
accurate approximation of the estimators’ sampling properties. This analytical
framework enabled the determination of efficiency conditions and provided
insight into the circumstances under which the proposed estimators yield
improved estimation performance.
In addition to the theoretical derivations, an empirical study based on
simulated data was conducted to further evaluate the performance of the
modified estimators. The simulation results confirmed the theoretical findings,
demonstrating that the proposed estimators consistently exhibit smaller biases,
lower MSEs, and higher Percentage Relative Efficiencies (PREs) when compared
with the existing estimators. Overall, the results of both the theoretical
analysis and the simulation study indicate that the proposed imputation
estimators provide more accurate and efficient estimates of the population
mean, thereby offering a useful methodological contribution to Statistics
in handling missing data under two-phase sampling designs.
References
Adejumobi,
A., Audu, A., Yunusa, M. A. and Singh, R. V. K. (2022). Efficiency of Modified
Generalized Imputation Scheme for Estimating Population Mean with Known
Auxiliary Information. Bayero Journal of Pure and Applied Sciences,
15(1): 105-112
Ahmed,
M.S., Al-Titi, O., Al-Rawi and Abu-Dayyeh, W. (2006). Estimation of a
population mean using different imputation methods. Statistics in
Transition, 7(6): 1247 -1264.
Al-Omari,
A. I., Bouza, C. N. and Herrera, C. (2013). Imputation methods of missing data
for estimating the population mean using simple random sampling with known
correlation coefficient. Quality and
Quantity, 47, 353-365.
Audu, A., Ishaq,
O. O., Singh, R. V. K., Danbaba, A. and Manu, F. (2023a). On the study of
efficiency of exponential-type estimator of population mean using robust
regression methods. Quality and Reliability Engineering International,
39,190-205.
Audu, A., Singh, R. and
Khare, S. (2023b): New Regression-Type Compromised Imputation Class of Estimators with known
Parameters of Auxiliary Variable. Communication in Statistics-Simulation and Computation, 52:10, 4789-4801.
Audu, A., Ishaq,
O.O., Isah, U., Muhammed, S., Akintola, K. A., A. Rashida, A., and
Abubakar. A. (2020a) On the Class of
Exponential-Type Imputation Estimators of Population Mean with Known Population
Mean of Auxiliary Variable. NIPES Journal of Science and Technology
Research 2(4) 2020 pp.1 - 11
Audu, A., Ishaq O. O., Zakari, Y., Wisdom, D. D., Muili, J. and Ndatsu.
A. M. (2020b): Regression-cum-exponential ratio imputation class of estimators
of population mean in the presence of non-response. Science Forum Journal of Pure and Applied Science, 20, 58-63
Audu, A. Danbaba, A.,
Ahmad, S. K., Musa, N., Shehu, A., Ndatsu, A. M. and Joseph, A. O. (2021b). On the Efficient of Almost Unbiased
Mean Imputation When the Population Mean
of Auxiliary Variable is Unknown. Asian Journal of Probability and
Statistics, 15(4): 235-250.
