Birnbaum Saunders Distribution with Application in Environmental Sciences

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1 Available online at.pelagiaresearhlibrary.om Advanes in Applied Siene Researh 5:-7 ISSN: CODEN SA: AASRFC Birnbam Sanders Distribtion ith Appliation in Environmental Sienes Kishore K. Das and Bhanita Das * Department of Statistis Gahati niversity Assam India _ ABSTRACT In this paper e have onsidered transformed Birnbam Sander s distribtion to obtain eighted and length biased distribtion. To obtain a valid eighted probability distribtion the transformation as made. The statistial distribtion of Zin onentration in drinking ater qality data follos BWBSD. Here e develop BD of eighted Birnbam Sanders Distribtion WBSD. Moment properties estimation of parameter and hazard fntion of BD of WBSD along ith its behavior have been onsidered. Finally e illstrate the appliability of BD of WBSD in environmental sienes sing drinking ater qality data based on Zin onentration. Keyords: WBSD length biased distribtion BD moments estimation of parameter hazard fntion drinking ater qality. _ INTRODCTION Althogh the effetiveness for analysis of nivariate data here the variability of measrement may be affeted by many fators sh as sampling analytial proedre and physio-hemial environment simple statistial appliations viz. mean mode SD skeness krtosis et. on environmental data provides only the pattern of distribtion symmetri and asymmetri of data Stephenson. Several distribtion theories and order statistis related to drinking ater qality data have been reported by varios athors e.g. Vanrolleghm et al. 996 and eiva et al. 8. Bt similar approahes sing ater qality data in Assam is laking. As sh an attempt has been made distribtion theory ith appliation in ater qality data. In environmental stdies observations fall in the non-erimental non-repliated and nonrandom ategories. Therefore making random seletion from the observed poplation is impossible. Ths problems of model speifiation and data interpretation aqire great importane. A ay of onfronting this problem is by onsidering observations hih are seleted ith probability proportional to their length. The reslting distribtion is alled the length-biased

2 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 distribtion Patil. On the other hand non-observation and damage of data set reslts redtion of vale or adoption of a sampling plan hih gives neqal probabilities to the varios nits and annot be onsidered as a random sample from the theoretial distribtion. For these types of data sets the eighted distribtion an be applied Rao 985. Maority of ater qality data are non niform data therefore some ork has been done in this paper sing the onept of eighted distribtion WD and length biased B distribtions. Patil reported that if X be a non negative random variable r.v. ith pdf f X then the X ith eight fntion hose density fntion is alled the eighted r.v. is denoted by eighted distribtion. The pdf of eighted distribtion is given by f X f X E[ ] ; [ ]< i.e. the first moment of eists. The length biased distribtion is a partilar ase of eighted distribtion hih an be denoted by a r.v t hih has pdf ressed as t f X t ft t ; t ; Assming that E. The length biased distribtion has been applied in varios fields sh as biometry eology environmental sienes and reliability and srvival analysis. A revie of this distribtion and their appliations are inlded in Gpta and Kirmani 99 Das et al. and Das and Roy. eiva et al. 8 reported that length biased Inverse Gassian distribtion and Birnbam Sander s distribtion have been developed by varios athors sing Where E[ ] environmental data and orked sessflly. As sh in this paper eighted Birnbam Sander s distribtion WBSD as ell as length biased distribtions BD has been derived and the behavior of these nely developed distribtions also have been stdied setting varios properties. The Birnbam Sanders BS distribtion is an important positively skeed probability model hih has attrative properties. De to the interesting theoretial argments the BS model is appropriate for desribing mlative degradation proesses. The Birnbam Sanders distribtion as derived from a model shoing that failre is de to the development and groth of a dominant rak Birnbam and Sanders 969. The to parameter Birnbam Sanders distribtion as originally proposed by Birnbam and Sanders 969. Althogh the BS distribtion as originally proposed as a failre time distribtion for fatige failre nder the assmption that the failre is de to the development and groth of a dominant rak a more general deviation as provided by Desmond 985 based on a biologial model. Some ork on the BS distribtion has been fond in Balakrishnan et. al. 7 Chang and Tang 99 and Chang and Tang 994 Dphis and Mills 998 Oen 6 and Xie and Wei 7 A onise revie of different developments of BS distribtion is fond in the book by Johnson et al Mann et al. 974] mentioned that hazard fntion of the BS distribtion is not an inreasing fntion. For details abot theoretial aspets and old and ne appliations of the BS

3 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 model inlding for eample the environmental area see Johnson et al Balakrishnan et. al. 7 and Sanheza et al. 8 Birnbam and Sander s distribtion 969a have proposed the distribtion ith pdf ; ; π p to represent the distribtion of lifetime of omponents nder speified onditions of ear a fatige life distribtion. In the rest of the paper in setion. WBSD and BWBSD have been derived. Varios properties of BWBSD have been stdied in setion.. Setion.4 and setion.5 attempts estimation of parameters of BWBSD and fitting of BWBSD. Finally onlsion has been given in setion.6.. Weighted and ength Biased Version of Birnbam Sander s Distribtion In this setion Weighted and ength Biased Version of transformed Birnbam Sander s distribtion has been derived. sing transformations to eqation e have π f Again sing eight fntion e have the eighted Birnbam- Sander s WBS and length biased Birnbam Sander s distribtion respetively as f 4 f. Properties of BD of WBSD The rth order moment of BD is given by

4 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 5 r r r moment generating fntion mgf of BD is given by 6 t t M The mlant generating fntion gf is given by 7 log t t K From the ression given in 6 and 7 e an find ot moments mean variane Skeness and krtosis of the BD of WBS distribtion as follos The mlative distribtion fntion for BD of WBSD is given by v F The orresponding srvival fntion hazard fntion and reverse hazard fntion are given by v S λ

5 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 4 τ. Behavior of BWBSD The probability density fntion f is tie differentiable fntion for a r.v.. Considering f f η e see that the first derivative of η eist i.e. η eist and η Again < η for therefore aording to Glaser s 98 lemma the hazard fntion of BWBSD is monotonially inreasing fntion IFR in. Moreover onsidering derivative of the hazard fntion e have λ here By observing λ it is seen that a. The BWBSD has a dereasing failre rate DFR if b. The BWBSD has a inreasing failre rate IFR if <. The BWBSD has a onstant failre rate CFR if

6 Bhanita Das et al Adv. Appl. Si. Res. 5:-7.4 Estimation sing the method of moments the estimation of parameter for BD is given as m here m is the sample moment..5 Appliation Sampling as done sing a stratified simple random proedre from st Marh to st Otober 9 in three administrative sb-divisions of Nagaon distrit of Assam India. A total of 78 ater samples 66 Tbe Well 8 Deep Tbe Well Ring Well and Spply Water ere olleted randomly from 9 villages from three sb-divisions viz. Nagaon Kaliabor and Hoai. Three villages from Hoai sb-division viz. Nilbagan dali and Dimor five stations from Nagaon sb-division Kampr Ton Soth Haibargaon Haloabhakatgaon Barabli and Barghat and greater Kaliabor ton area ere seleted randomly for ater sampling hih is in 5 km minimm distane apart. Ot of the total bore ell or ring ell ater sores 74% sores are only sore of ater and 6% sores are omplementary sore of ater. Water samples ere olleted after mintes of initial pmping in pre leaned polythene ontainers of one liter apaity ere rinsed ot - 4 times and transported to laboratory at C. The ontainers ere filled p to the moths and then tightly stopper to avoid ontat ith air or to prevent agitation dring transport. The storage and preservation of samples ere done ith standard proedre APHA 995. Water temperatre and ph are measred at the time of olletion of the sample. In the laboratory ater samples ere vam filtered sing.45 m pre-ashed and pre-eighed membranes and the filtrates ontaining dissolved metals ere pored in to lean plasti ontainers then aidified ith onentrated nitri aid to ph to. Sine the government of Assam has delared 4 distrits of Assam as droght/droght like onditionn the year 9 inlding Nagaon distrit also the rainfall reeived % departre from the normal in this distrit dring the above period as -7%. Maimm and minimm temperatre reorded 6 C and C respetively dring the stdy period. The ater qality parameter estimation and alibration of eqipments ere done sing standard methods and tehniqes APHA 995. Zin is an essential element and is neessary for the effetive fntioning of varios enzyme systems defiieny of hih leads to groth retardation immatrity and anemia. Symptoms of zin toiity in hman inlde vomiting dehydration eletrolyte imbalane abdominal pain nasea lethargies dizziness and lak of mslar oordination. Table : The Observed Freqenies and Epeted Freqenies of Zin Conentration. Class Interval Observed FreqenyO i Epeted FreqenyE i Belo The hi-sqare test of goodness of fit is applied to drinking ater qality data. The no. of lass 5

7 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 intervals over hih the omptations ere fond as 5 for Zin onentration. Fitting of BD of WBSD as performed by Simpson s Rle of nmerial integration ith the help of C. CONCSION Table. shos that esimated vale of the parameter BWBSD is ˆ. 8and χ. 65 for BWBSD. In this artile the BD of eighted Birnbam-Sander s WBSD as developed. This ne distribtion trns ot to be qite fleible for modeling ater qality. An appliation to the real data shoed that this to ne model is fleible alternative to other ell knon models. Ths e have developed a ne probability model hih might be of se for pratitioners in the environmental sienes. Here e postlated that BD of WBSD is appropriate for modeling ater qality. Aknoledgments The athors ish to thank Bhpen Kmer Barah and Abani K. Mishra Department of Chemistry Gahati niversity Gahati for providing valable data in this artile. REFERENCES [] APHA. Standard methods for the eamination of ater and asteater. Ne York APHA AWWA and WEF th edition 995. [] Balakrishnan N eiva V opez J Comm Stat Siml Comp [] Birnbam ZW SC Sander Jornal of Applied Probability [4] Chang DS C Tang IEEE Transations on Reliability [5] Chang DS C Tang Commniations in Statistis Theory and Methods [6] Das KK Das B Barah BK Mishra AK Advanes in Applied Siene Researh [7] Das KK Roy TD. Advanes in Applied Siene Researh 4-7. [8] Desmond AF Canadia Jornal of Statistis [9] Dphis DJ Mills JE IEEE Transations on Reliability [] Glaser RE J. Amer. Statistial Asso [] Gpta RC S Kirmani Comm. Stat. Theor. Meth [] Johnson N Kotz S and Balakrishnan N Continos nivariate distribtions 995 Wiley Ne York. [] Khadar Bab SK Karthikeyen K Ramanaiah MV and Ramanah D Advanes in Applied Siene Researh 8-. [4] Mann NR Shafer RE Singpralla ND Methods for Statistial Analysis of Reliability and ife Data. John Wiley & Sons Ne york 974. [5] Oen WJ A ne three-parameter etension to the Birnbam Sander s distribtion. IEEE Transations on Reliability [6] Patil GP Weighted distribtions. Enylopedia of environmetris Wiley [7] Rao CR Celebration of Statistis 985. [8] Sanheza A eiva V and Balakrishnan N Comm Stat Theor Meth [9] Singh S and Shah RR Advanes in Applied Siene Researh

8 Bhanita Das et al Adv. Appl. Si. Res. 5:-7 [] Stephenson DB Statistial onept in environmental siene. Dept. of metrology niversity of Reading. [] Vanrolleghm P Frontea C Baens W Evalation of design and operation of the seage transport and treatment system by an eqo/eqs based analysis of the reeiving ater immission harateristis. In proeedings WEF speiality onferene on rban et eather polltion: ontrolling seer overflos and stormater rnoff [] Xie FC Wei BC Comptational Statistis & Data Analysis