A New Method of Estimation of Size-Biased Generalized Logarithmic Series Distribution

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1 The Open Statistics and Pobability Jounal, 9,, - A New Method of Estimation of Size-Bied Genealized Logaithmic Seies Distibution Open Access Khushid Ahmad Mi * Depatment of Statistics, Govt Degee College (Boys Baamulla, Khmi, India Abstact: In this pape, a size-bied genealized logaithmic seies distibution (SBGLSD is intoduced and its moments ae obtained The estimates of the paametes of SBGLSD ae obtained by employing the method of moments and a poposed new method of estimation The new poposed method of estimation uses the non-zeo fequency of a vaiable only up to a finite value In this method, the estimation of only one paamete is needed and of the othe is obtained by the elationship among the paametes by counting the numbe of non-zeo fequency clses The method is found vey simple and quick to apply in pactice Extensive simulations ae pefomed to compae the pefomances of the poposed and the moment method of estimation mainly with espect to thei bies and mean squaed eos (MSE s, fo diffeent sample sizes and of diffeent paametic values Compaison h been made among diffeent estimation methods by means of Peon s Chi-squae, Akaike Infomation Citeion (AIC and Bayesian Infomation Citeion (BIC techniques Key Wods: Size-bied genealized logaithmic seies distibution, Non- zeo fequency clses, Chi-squae AIC, BIC INTRODUCTION The genealized logaithmic seies distibution (GLSD chaacteized by two paametes and w defined by Jain and Gupta [] The pobability function of the GLSD model is given by P[ X = x] = ( x x x x ( x! ( x x + x =,,, and - ( Whee = log( The model ( educes to the simple logaithmic seies distibution when = The GLSD model is a membe of Gupta s [] modified powe seies distibution and of Consul and Shenton s [] Lagangian pobability distibutions The model ( is also a limiting fom of zeo-tuncated fom of Jain and Consul s [] genealized negative binomial distibution Patel [5] defined GLSD and obtained the estimates of the paametes by the method of moments Famoye [6] showed that the GLSD is unimodal and the mode is at the point x = Some methods of sampling fom the model ( ae povided by Famoye [7] Famoye [8] obtained the moment estimatos, Jani and Shah [9] discussed the maximum likelihood and moment method of estimation fo two paamete GLSD model Misha and Tiway [] suggested an *Addess coespondence to this autho at the Depatment of Statistics, Govt Degee College (Boys Baamulla, Khmi, India; khshdmi@yahoocom ; altenative method of estimation bed on the fist thee moments and showed that the GLSD povides a vey close fits to the obseved data fom vaious fields such entomolgy, medicine, engineeing etc Famoye [] discussed the fitting of GLSD Tipathi and Gupta [] studied anothe genealization of GLSDA bief list of authos and thei woks can be seen in Johnson, Kotz and Kemp [] and Consul and Famoye [] The fist fou moments about oigin of GLSD ae given = è( â ( = ( ( ( 5 = ( ( ( + ( = ( 7 ( ( ( The ecuence elation among the cental moments is given ( d = + d + Which gives the fist fou cental moments ( ( ( = (7 (5 ( /9 9 Bentham Open

2 The Open Statistics and Pobability Jounal, 9, Volume Khushid Ahmad Mi = ( 5 ( ( + ( ( + ( (8 = ( ( ( ( + ( + ( ( ( ( ( ( In this pape, a size-bied genealized logaithmic seies distibution (SBGLSD taking the weights of the pobabilities the vaiate values, is defined The moments of the poposed model ae obtained The estimates of the paametes of SBGLSD ae obtained by employing the method of moments and a poposed new method of estimation It is vey difficult to compae the theoetical pefomances of diffeent estimatos poposed in this pape Theefoe, we pefom extensive simulations to compae the pefomances of the diffeent methods of estimation mainly with espect to thei bies and mean squaed eos (MSE s, fo diffeent sample sizes and of diffeent paametic values Goodness of fit test is done in ode to see that poposed new method of estimation gives bette esult in compaison to the method of moments SIZE-BIASED GENERALIZED LOGARITHMIC SERIES DISTRIBUTION (SBGLSD Size-bied distibutions ae a special ce of the moe geneal fom known weighted distibutions Fishe [5] intoduced these distibutions to model cetainment bi and wee late fomalized in a unifying theoy by Rao [6] These distibutions aise in pactice when obsevations fom a sample ae ecoded with unequal pobability and povide a unifying appoach fo the poblems whee the obsevations fall in the non-expeimental, non- eplicated, and nonandom categoies If the andom vaiable X h distibution f ( ; (9 x, with unknown paamete, then the coesponding size-bied distibution is of the fom * xf( x; f ( x; = ( E( x whee E( x = ( xf x; dx fo continuous ce and = p(x=x fo discete ce E( x x Using the citeia defined in equation ( and by using the equations ( and (, the pobability function of sizebied genealized logaithmic distibution (SBGLSD is obtained x P[X = x] =, whee = ( log( x ( ( xx x x ( = ( x! x-x + On simplification, the above equation is educed to x x- x xx ( ( = ( Since the above sum equals to, theefoe, it epesents a pobability distibution and we name it size-bied genealized logaithmic seies distibution (SBGLSD and is epesented x = = x x xx P[X x] ( ( and ;, = fo x t if t-t- ( < < When =, the SBGLSD educes to size- bied logaithmic seies distibution (SBLSD with pobability function P[X x] ( x = = ;, (5 Moments The th moment ( s of SBGLSD about oigin is obtained ( s = x P[X = x] ; =, x = x( ( x x xx Obviously ( s = and fo (= s x x ( x = x x = (= s x xx + P[ X + = x] (6 (7 whee + is the ( + th moments about oigin of GLSD ( The moments of SBGLSD can be obtained by using equations ( and ( in (7 ( s ( =Mean = ( ( ( ( + s = ( (8 (9

3 A New Method of Estimation of SBGLSD The Open Statistics and Pobability Jounal, 9, Volume ( ( ( s = Vaiance = ( ( The highe moments of SBGLSD about oigin can be obtained similaly using equation (7 if so desied ESTIMATION OF SIZE-BIASED GENERALIZED LOGARITHMIC SERIES DISTRIBUTION In this section, we study the estimation of the paametes of SBGLSD by the method of moments and a new poposed method Also compaison is made between these two estimatos Method of Moments Replacing sample moments with population moments, we get ( x = ( ( ( ( S = ( Fom above two equations, we get S ( /x ( = x ( ( ( Solving above equation fo, we get the estimate fo and substituting that value in equation (, we get the estimate of Poposed New Estimato fo SBGLSD In this method, only one paamete is estimated with the help of the fist moment of the SBGLSD and the othe paamete is estimated bed on non-zeo fequency clses Thus, this method may be much ey and quick in pactice The condition in the SBGLSD that P[ X = x] = fo x t if t-t- ( gives a elationship between the paamete and the numbe of the clses of non-zeo fequencies of the GLSD Hence in those ces when the numbe of the clses of nonzeo fequency is finite, may be eadily estimated using the equation ( Let us suppose that in a sample of size n, the fist (t- clses have non-zeo fequencies and the est of the clses have zeo fequencies, then P[ X = x] if x < t = if x t (5 Fom the equation (, we have estimate of, say, t + t = (6 Thus the value of, is obtained diectly fom the nonzeo fequency clses and may be teated pedetemined n in the ce of binomial distibution Now substituting the estimate of in the expession ( fo the mean of the SBGLSD and eplacing by the sample mean x, we get ( x = (7 ( Solving this fo we get the estimate of Efficiency of Poposed Estimato In ode to check the usefulness of new poposed method, the efficiency of the paamete is studied Fo this pupose, an extensive compute simulation is done by taking n=5,,, 5,, =, 5,, and =,, 6, 5, 5 Fo each combination of n and we geneate a sample of size n fom SBGLSD and estimate by diffeent methods We epot the aveage values of ˆ and the coesponding aveage MSE s All the epoted esults ae bed on, eplications The esults ae pesented in Table Hee we epot the aveage values of ˆ fo each method and the coesponding MSE s ae epoted within backets Fom the table it is immediate that the aveage bies and the aveage MSE s decee sample size incees It indicates that all the methods povide ymptotically unbied and the consistent estimatos It is also obseved that the aveage bies and the aveage MSE s of ˆ depend on On compaing the pefomances of all the methods it is clea that fa the minimum bi is concened, the poposed estimato woks the best in almost all the ces Goodness of Fit An attempt is made to fit the SBGLSD to obseved data estimating the paametes and by suggested altenative method To know how much good o bad the fits ae due to this method in compaison to those due to method of moments, we have used the data sets of Guie et al [7] and Student [8] The expected fequencies accoding to both the methods along with the estimates of both the paametes and the values of chi-squae, AIC and BIC ae given in Tables and CONCLUSIONS It is encouaging to obseve fom the above tables that the poposed estimato is giving best esults in compaison to moment estimatos Futhemoe, the suggested method h

4 The Open Statistics and Pobability Jounal, 9, Volume Khushid Ahmad Mi Table- Aveage Relative Estimates and Aveage Relative Mean Squaed Eos of n b Method = =5 = = 5 Poposed Estimato Moment Estimato 6(6 (758 ( (57 5(5 5(5 (78 66( Poposed Estimato Moment Estimato ( 6(6 (9 (99 (8 ( (55 97(5 6 Poposed Estimato Moment Estimato (5 68(5 5( 56(5 9(5 (6 87( (5 5 5 Poposed Estimato Moment Estimato (6 (99 ( 55(7 5(5 5(9 (5 65(5 5 Poposed Estimato Moment Estimato 7(7 5(5 ( ( ( ( ( 5( Table Zeo-Tuncated Data on P nubilalis (Euopean Con Boe of Guie et al [7] No of Boes Pe Plant Obseved Fequency Method of Moments Expected Fequency Poposed Method Total AIC 85 BIC 5 Estimates ˆ ˆ Table Zeo-Tuncated Data of Haemocytomete Yet Cell Counts Pe Squae Obseved by STUDENT [8] No of Cells Pe Squae Obseved Fequency Method of Moments Expected Fequency Poposed Method Total AIC BIC Estimates ˆ ˆ

5 A New Method of Estimation of SBGLSD The Open Statistics and Pobability Jounal, 9, Volume an advantage ove the method of moments in cetain situations It can be applied wheneve, it is elatively vey quick to be obtained and so it may be pefeed to othes when vey quick esults ae equied ACKNOWLEDGEMENT The autho is highly thankful to the edito and the efeee fo thei valuable comments REFERENCES [] GC Jain, and RC Gupta, A logaithmic type distibution, Tabjos Estadist, vol, pp 99-5, 97 [] RCGupta, Modified powe seies distibution and some of its applications, Sankhya Se B, vol 5, pp 88-98, 97 [] PC Consul, and LR Shenton, Use of Lagangian expansion fo geneating genealized pobability distibutions, SIAM J Appl Math, vol, pp9-8, 97 [] GCJain, and PCConsul, A genealized negative binomial distibution, SIAM J Appl Math, vol, pp 5-5, 97 [5] ID Patel, A genealization of logaithmic seies distibution, J Indian Soc Agic Stat, vol 9, pp 9 -, 98 [6] F Famoye, A shot note on the genealized logaithmic seies distibution, Stat Pobab Lett, vol 5, pp 5-6, 987 [7] F Famoye, Sampling fom the genealized logaithmic seies distibution, Computing, vol 58, pp 65-75, 997 [8] F Famoye, On cetain methods of estimation fo the genealized logaithmic seies distibution, J Appl Stat Sci, vol pp - 7, 995 [9] PN Jani, and SM Shah, On fitting of the genealized logaithmic seies distibution, J Indian Soc Agicul Stat, vol (, pp -, 979 [] A Misha, and D Tiway, On genealized logaithmic seies distibution, J Indian Soc Agic Stat, vol 7(, pp 9-, 985 [] F Famoye, Goodness of fit tests fo genealized logaithmic seies distibution, J Comput Stat Data Anal, vol, pp 59-67, [] RC Tipathi, and RC Gupta, Anothe genealization of the logaithmic seies and the Geometic distibutions, Commun Stat, vol 7, pp 5-57, 988 [] NLJohnson, S Kotz and AWKemp, Univaiate discete distibutions, d ed, Wiley, 5 [] PC Consul and F Famoye, On the Lagangian pobability distibutions, Bikause publications, 6 [5] RAFishe, The effects of methods of cetainment upon the estimation of fequencies, Ann Eugen, vol 6, pp -5, 9 [6] CR Rao, On discete distibutions aising out of methods of cetainment in Clsic Contag Discete Distibutions GP Patil, Ed Statistical Publishing Society, Calcutta, pp -,965 [7] M Guie, U Judson, TA Bindley, and TA Bancoft, The distibution of Euopean con boe lavae pyausta nubilalis in field con, Biometics, vol, pp 65-78, 957 [8] Student, On the eo of counting with haemocytomete, Biometika, vol 5, pp 5-6, 97 Received: Febuay 5, 9 Revised: Mach, 9 Accepted: Mach, 9 Khushid Ahmad Mi; Licensee Bentham Open This is an open access aticle licensed unde the tems of the Ceative Commons Attibution Non-Commecial License ( by-nc// which pemits unesticted, non-commecial use, distibution and epoduction in any medium, povided the wok is popely cited

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