Scholars Journal of Physics, Mathematics and Statistics                                                                                                     
                  Abbreviated Key Title: Sch J Phys Math Stat 
                  ISSN 2393-8056 (Print) | ISSN 2393-8064 (Online)   
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                  Improved Estimation of Population Variance Utilizing Known Auxiliary 
                  Parameters 
                                          1*                        1
                  Shiv Shankar Soni , Himanshu Pandey  
                   
                  1Department of Mathematics and Statistics, DDU Gorakhpur University Gorakhpur, Civil Lines, Gorakhpur, Uttar Pradesh 273009, 
                  India 
                   
                  DOI: 10.36347/sjpms.2022.v09i06.001                                    | Received: 03.07.2022 | Accepted: 09.08.2022 | Published: 13.08.2022 
                   
                  *Corresponding author: Shiv Shankar Soni 
                  Department of Mathematics and Statistics, DDU Gorakhpur University Gorakhpur, Civil Lines, Gorakhpur, Uttar Pradesh 273009, 
                  India 
                   
                   Abstract                                                                                                            Original Research Article 
                   
                   Even  similar  things,  whether  created  artificially  or  naturally,  can  vary.  We  should  therefore  try  to  estimate  this 
                   variation. For improved population variance estimate, we propose a Searls ratio type estimator in the current research 
                   employing data on the tri-mean and the third quartile of the auxiliary variable. Up to the first-degree approximation, 
                   the suggested estimator's bias and mean squared error (MSE) are determined. The characterising scalar's ideal value is 
                   discovered,  and  given  this  ideal  value,  the  least  MSE  is  discovered.  The  mean  squared  errors  of  the  suggested 
                   estimator and the competing estimators are contrasted conceptually and experimentally. Given that it has the lowest 
                   MSE of the above competing estimators, the recommended estimator has been shown to be the most effective. 
                   Keywords: Population Variance, Estimator, Main and Auxiliary variables, Bias, MSE, PRE. 
                  Copyright © 2022 The Author(s): This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International 
                  License (CC BY-NC 4.0) which permits unrestricted use, distribution, and reproduction in any medium for non-commercial use provided the original 
                  author and source are credited. 
                   
                  1. INTRODUCTION                                                                    crosses through the origin, product type estimators are 
                             One  of  the  key  indicators  of  dispersion  is                       utilized to improve population variance estimation. In 
                  population  variance,  which  is  important  for  making                           either scenario, the known auxiliary variable is used in 
                  day-to-day business decisions. The variance is obvious                             conjunction with regression type estimators to improve 
                  and occurs naturally. The literature has a very strong                             population variance estimation of the primary variable. 
                  foundation  for  the  accurate  estimation  of  the                                             
                  parameters.  It  is  advantageous  for  big  populations  to                                   Using  the  auxiliary  data,  Singh  and  Singh 
                  reduce  errors  since  doing  so  will  ultimately  result  in                     (2001) proposed a ratio-type estimator for a enhanced 
                  time  and  planning  and  decision-making  cost  savings.                          estimation of the population variance. Later, Singh and 
                  Making  accurate  estimates  is  essential  for  timely                            Singh  (2003)  provided  an  improved  regression 
                  policymaking.  The  sample  variance,  which  has  the                             approach for estimating population variance in a two-
                  desirable characteristics of a good estimator, is mostly                           phase  sample  design.  A  useful  family  of  chain 
                  used to estimate variance. The sample variance of this                             estimators was also proposed by Jhajj et al., (2005) for 
                  approach could be quite considerable, which is one of                              the elevated estimation of the population variance under 
                  its  major  downsides.  Finding  an  estimator  with  a                            the  sub-sampling  method.  Furthermore,  Shabbir  and 
                  sample distribution that is tightly distributed around the                         Gupta (2007) focused on the development of auxiliary 
                  population variance is therefore necessary. As a result,                           parameter-based variance estimation. Then, Kadilar and 
                  the auxiliary data is necessary to achieve this goal.                              Cingi  (2007)  proposed  various  enhancements  to  the 
                                                                                                     simple random sampling scheme's variance estimation. 
                             The auxiliary variable, denoted by  X , which                           Using the understanding of the kurtosis of an auxiliary 
                  has a strong association with the study variable, denoted                          variable  in  sample  surveys,  Singh  et  al.,  (2008) 
                  by  Y ,  provides  additional  information.  When  Y and                           proposed  a  virtually  impartial  ratio  and  product  type 
                   X have a strong positive correlation and the regression                           estimator of the finite population variance. A correction 
                  line  of  one  passes  through  the  origin,  the  ratio                           remark  on  the  improved  estimation  of  population 
                  estimators  are  employed  to  estimate  the  enhanced                             variance  using  auxiliary  parameters  was  reported  by 
                  population  variance.  When  Y and  X have  a  strong                              Grover (2010). Additionally, Singh and Solanki (2012) 
                  negative  correlation  and  the  regression  line  of  one 
                    Citation:  Shiv  Shankar  Soni  &  Himanshu Pandey.  Improved  Estimation  of  Population Variance  Utilizing  Known                                
                    Auxiliary Parameters. Sch J Phys Math Stat, 2022 Aug 9(6): 92-101.                                                                               92 
                   
                   
                                                                                                                                                                                                                                                                                                                           
                                                                                                                                                                                                                                                                                                      
                                                                                                                                                     Shiv Shankar Soni & Himanshu Pandey., Sch J Phys Math Stat, Aug, 2022; 9(6): 92-101 
                                proposed  a  novel  method  utilising  auxiliary  data  for                                                                                            The              suggested                      estimators                     and             their              sample 
                                variance estimate in simple random sampling.                                                                                                           characteristics  up  to  the  first  order  approximation  are 
                                                                                                                                                                                       described  in  Section  3.  The  efficiency  comparison  of 
                                                     Yadav and Kadilar (2014), on the other hand,                                                                                      the  proposed estimator with the competing estimators 
                                suggested a two-parameter increased variance estimator                                                                                                 and  the  requirements  for  the  proposed  estimator's 
                                using  auxiliary  parameters.  An  improved  family  of                                                                                                superiority over competing estimators are explained in 
                                estimators  for  estimating  population  variance  using                                                                                               Section 4. The empirical research presented in Section 5 
                                auxiliary variable quartiles was proposed by Singh and                                                                                                 is the one in which the biases and MSEs for the actual 
                                Pal (2016). Yadav et al., (2017) suggested an improved                                                                                                 natural  population  were  computed.  The  conclusions 
                                variance estimator using the auxiliary variable's known                                                                                                drawn from the numerical study's findings are discussed 
                                tri-mean and interquartile range. Using the well-known                                                                                                 in Section 6. The conclusion of the results of the study 
                                tri-mean  and  third  quartile  of  the  auxiliary  variable,                                                                                          is  presented in Section 7 and the paper ends with the 
                                Yadav  et  al.,  (2019)  have  proposed  an  increased                                                                                                 references.  
                                estimator  of  the  population  variance.  When  outliers                                                                                               
                                were  present,  Naz  et  al.,  (2020)  offered  ratio-type                                                                                             2. LITERATURE REVIEW 
                                estimators  of  population  variance  and  employed                                                                                                                         Let the finite population U is made up of N 
                                unconventional  dispersion  measures  of  the  auxiliary                                                                                               different  and  recognizable  units  U ,U ,..........,U                                                                            
                                variable, which had a high correlation with the primary                                                                                                                                                                                       1         2                            N
                                variable  under  discussion.  Olayiwola  et  al.,  (2021)                                                                                              and             the           ‘Simple                  Random  Sampling  Without 
                                worked  on  a  new  exponential  ratio  estimator  of                                                                                                  Replacement’ (SRSWOR) method is used to collect a 
                                population variance and shown improvement over many                                                                                                    sample of size n units from this population, assuming 
                                existing estimators of population variance. Bhushan et                                                                                                 that Y and X has a strong correlation between them. Let 
                                al.,  (2022)  suggested  some  new  modified  classes  of                                                                                              (Y , X )  be  the  observation  on  the  ith  unit  of  the 
                                population  variance  utilizing  the  known  auxiliary                                                                                                       i          i
                                parameters.                                                                                                                                            population, i 1,2,..., N . 
                                                                                                                                                                                                             
                                                     Sharma et al., (2022) and Searls (1964) served                                                                                                         The  most  suitable  estimator  for  population 
                                as  inspiration  for  this  investigation.  To  improve  the                                                                                           variance  S2 is the sample variance  s2, given by, 
                                population  variance  estimation  of  the  key  variable  in                                                                                                                     y                                                            y
                                this study, we propose a Searls type estimator and use a                                                                                                               2             1          n                        2
                                                                                                                                                                                        t     s                                    (y  y)   
                                known population tri-mean and third quartile. Bias in                                                                                                     0            y         n1 i
                                sampling is examined up to an approximation of order                                                                                                                                          i1
                                one,  and  mean  squared  error  (MSE)  is  as  well.  The                                                                                              
                                remaining portions of the essay have been divided into                                                                                                 The variance of  t0 for an approximation of degree one 
                                sections. Review of population variance estimators for                                                                                                 is, 
                                the           research                  variable                 using              auxiliary                  variable                                V(t )  S4( 1)…………………….. (1) 
                                parameters that are known can be found in Section 2.                                                                                                             0                     y         40
                                 
                                Where,  
                                                                                                                    N                                                                         n                                                      N
                                            1           1                               2               1                                        2                                    1                                                      1                                                            rs
                                                           ,                     S                                    (y Y) ,                                         y                       y ,                        Y                           y ,                         rs                            , 
                                            n          N                                y          N1 i                                                                             n i                                                  N i                                                       r / 2     s / 2
                                                                                                                   i1                                                                      i1                                                     i1                                             20 02
                                                      1            N
                                                                      (Y Y)r(X X)s  
                                     rs          N1 i                                                  i
                                                                  i1
                                                      
                                                     Isaki (1983) utilized the known positively correlated auxiliary information and suggested the following usual 
                                ratio estimator of  S 2 as, 
                                                                              y
                                                           2
                                 t     s2Sx  
                                   r            y  s2 
                                                    x 
                                 
                                It is a biased estimator and its MSE up to the first order of approximation is, 
                                 MSE(t ) S4[( 1)( 1)2( 1)]………………. (2) 
                                                  r                    y          40                          04                            22
                                                      
                                Upadhyaya and Singh (1999) used the known coefficient of kurtosis of  X and introduced an estimator of S2 as, 
                                                                                                                                                                                                                                                                                           y
                                © 2022 Scholars Journal of Physics, Mathematics and Statistics | Published by SAS Publishers, India                                                                   93                        
                                 
                                 
                                                                                                                                                                    
                                                                                                                                                          
                                                                              Shiv Shankar Soni & Himanshu Pandey., Sch J Phys Math Stat, Aug, 2022; 9(6): 92-101 
                               2
                 t  s2Sx 2 
                  1      y  s2       
                            x        2 
                  
                 The MSE of t for an approximation of order one is, 
                                  1
                 MSE (t ) S4[(            1)R2( 1)2R ( 1)]……………… (3) 
                          1         y     40          1    04            1   22
                  
                 Where, 
                            S2                       1     N
                 R           x      and S2                  (x X)2  
                   1     S2                x    N1 i
                           x      2                        i1
                  
                 Kadilar and Cingi (2006) suggested three estimators of  S2 utilizing                 2,      and C as, 
                                                                                      y             Sx 2             x
                               2                          2                              2
                 t   s2Sx Cx, t  s2Sx2 Cx, t  s2SxCx 2 
                  2      y  s2 C          3      y  s2 C             4      y  s2C  
                            x        x               x 2          x              x x          2 
                  
                 The MSEs of ti (i  2,3,4) for an approximation of order one is, 
                 MSE(t ) S4[( 1)R2( 1)2R ( 1)] ……………….. (4) 
                           i        y     40          i    04            i   22
                  
                 Where, 
                             S2                   S2                      S2C                        S
                 R            x     ,  R          x   2     ,  R           x   x     and C          x  
                    2    S2 C           3     S2 C             4     S2C                   x     X
                           x       x             x   2      x             x   x      2
                  
                 Subramani & Kumarpandiyan (2012a) utilized the known median  Md  of  X and proposed the following estimator of 
                 S2as, 
                    y
                               2
                 t   s2Sx Md  
                  5      y  s2  M       
                            x         d 
                  
                 The MSE of t5 for an approximation of order one is, 
                 MSE(t ) S4[( 1)R2( 1)2R ( 1)] ……………… (5) 
                           5        y     40          5    04            5   22
                  
                 Where, 
                             S2
                 R            x       
                   1     S2 M
                           x        d
                             
                 Subramani & Kumarpandiyan (2012b) utilized the known quartiles of  X and their functions and suggested the following 
                 five estimators of  S 2 as, 
                                         y
                               2                          2                          2                          2                           2
                         2 S    Q                2 S    Q                2 S Q                   2 S    Q                 2 S Q 
                 t   s        x      1   ,  t  s        x      3   ,  t s         x      r   ,  t s         x      d   ,  t   s        x      a    
                  6      y  s2 Q          7      y  s2 Q          8      y  s2 Q          9      y  s2 Q          10      y  s2 Q 
                            x        1               x        3               x        r               x        d                x        a 
                  
                 The MSEs of ti (i  6,7,...,10) for an approximation of order one is, 
                 MSE(t ) S4[( 1)R2( 1)2R ( 1)] …………………. (6) 
                           i        y     40          i    04            i   22
                  
                  
                  
                 © 2022 Scholars Journal of Physics, Mathematics and Statistics | Published by SAS Publishers, India                                                                   94                        
                  
                  
                                                                                                                                                        
                                                                                                                                              
                                                                        Shiv Shankar Soni & Himanshu Pandey., Sch J Phys Math Stat, Aug, 2022; 9(6): 92-101 
               Where, 
                          S2                  S2                  S2                  S2                   S2
                R          x     ,  R         x     ,  R         x     ,  R         x     ,  R          x      and Q Q Q ,  
                  6    S2 Q          7    S2 Q          8    S2 Q          9    S2 Q          10    S2 Q              r      3     1
                         x      1            x      3            x      r            x     d              x     a
                       Q Q                 Q Q
                Q  3           1 , Q        3     1 . 
                  d         2          a        2
               Subramani & Kumarpandiyan (2013) suggested a new estimator of  S2 using known S2, M and C as, 
                                                                                             y                  x      d        x
                             2
                t   s2SxCx Md  
                 11     y  s2C M 
                          x x          d 
                
               The MSE of t5 for an approximation of order one is, 
                MSE(t ) S4[( 1)R2( 1)2R ( 1)] ……………… (7) 
                        11        y    40         11   04           11   22
                
               Where, 
                            S2C
                R            x   x      
                  11    S2C M
                          x   x      d
                          
               Khan & Shabbir (2013) utilized the known third quartile  Q3of  X and correlation coefficient between  Y and  X and 
               suggested an estimator of  S2 as,  
                                               y
                              2
                        2 S  Q 
                t   s       x        3    
                 12     y  s2 Q 
                           x         3 
                
               The MSE of t for an approximation of order one is, 
                               12
                MSE(t ) S4[( 1)R2 ( 1)2R ( 1)] …………………. (8) 
                        12        y    40          12  04           12   22
                
               Where, 
                           S2
                R           x
                  12    S2Q
                          x        3  
                          
               Maqbool and Javaid (2017) utilized known  S2, TM and Q of  X and suggested the following estimator of S2 as,  
                                                                   x               a                                                        y
                             2
                        2 S   (TM Q )
                t   s       x               a     
                 13     y  s2  (TM Q )
                           x                a  
                
               The MSE of t13for an approximation of order one is, 
                MSE(t ) S4[( 1)R2( 1)2R ( 1)] ……………… (9) 
                        13        y    40         13   04           13   22
                
               Where, 
                                 S2                        Q 2Q Q
                R                x           and TM  1              2     3  
                  13    S2 (TM Q )                                4
                          x               a
                          
               Khalil et al., (2018) suggested the following three estimators of  S2 using the known auxiliary parameters as, 
                                                                                         y
                             2                             2                           2
                t   s2Sx CxSx, t s2Sx CxX , t s2Sx CxMd  
                 14     y  s2 C S          15     y  s2 C X          16     y  s2 C M 
                           x       x  x               x       x                 x        x   d 
               © 2022 Scholars Journal of Physics, Mathematics and Statistics | Published by SAS Publishers, India                                                                   95