Applied Mathematical Scieces, Vol. 3, 209, o., 33-44 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.209.879 Record Values from T-X Family of Pareto-Epoetial Distributio with Properties ad Simulatios Noor Waseem Departmet of Statistics Kiaird College for Wome, Lahore, Pakista Shakila Bashir Departmet of Statistics Forma Christia College, Lahore, Pakista Copyright 209 Noor Waseem ad Shakila Bashir. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, upper record values from Pareto-Epoetial distributio are studied with its statistical properties ad applicatios. Graphs are also give alog with Probability desity fuctio (PDF) ad cumulative desity fuctio (cdf). Reliability aalysis ad various properties which icluded Survival Fuctio, Hazard Fuctio, Cumulative Hazard rate, reversed Hazard rate, Geometric mea ad momets are discussed. Moreover, the epressios for Reyi etropy has also bee derived. Fially, Mote Carlo simulatio study is carried to geerate data of size 50 with a sample of 5 from Pareto-Epoetial distributio ad upper records has bee recorded. Keywords: Record values, Epoetial Pareto distributio, Momets, etropy. Itroductio Chadler [8] proposed record statistics as a model for ordered radom variable. Theory of record values defied i a sequece of idepedet ad idetically distributed radom variable. Ahsaullah [] studied some characteristics of upper
34 Noor Waseem ad Shakila Bashir record values from the Epoetial distributio. Ahsaullah, et. al. [2] established record values from the classical Pareto distributio. Bashir ad Ahmed [5] derived record values from size-biased Pareto distributio ad its characterizatios. Bashir ad Akhar [6] itroduced record values risig from studet s t Distributio. Paul ad Thomas [3] preseted a article o geeralized upper (k) record values from Weibull distributio. Chacko ad Muraleedhara [7] developed the lower k-record values arisig from a two parameter geeralized epoetial distributio ad applied the iferece statistics o it. Kumar [] gave some epressios of recurrece relatios of k-th lower record values from Dagum distributio derived by sigle ad product momets. Sulta [5] established some ew recurrece relatios of record values from the modified Weibull distributio betwee the sigle momets ad double momets. Sulta ad Moshref [6] derived the eact eplicit epressios for sigle, double, triple ad quadruple momets of the upper record values from a geeralized Pareto distributio. Balakrisha ad Ahsaullah [4] established recurrece relatio of upper record values from the geeralized Pareto distributio. Kha, et.al [0] obtaied the relatios for momets of k-th record values from epoetial-weibull lifetime distributio with its characterizatio. Dey, et. al. [9] itroduced the statistical iferece for the geeralized Rayleigh distributio based o upper record values. Miimol ad Thomas [2] preseted some properties of Makeham distributio usig geeralized record values ad its characterizatio. Sara ad Nai [4] developed the Relatioships for momets of k th record values from doubly trucated p th order epoetial ad geeralized Weibull distributios. The Pdf of upper record values f() is give by: The joit pdf of X U( j) ad X U(i) is R() f () f (), () i, j R() R(y) R() i ji i ji f (, y) r() f (y) (2) Where, R() l F(),0 F(), y j i 2. Record Values from Pareto-Epoetial Distributio Pareto-Epoetial distributio (PED) was derived by usig the CDF of the T-X family of distributio give by Alzaatreh, et al. [3], where radom variable T follows
Record values from T-X family of Pareto-epoetial distributio 35 Pareto distributio ad X follows the epoetial distributio. Followig are the PDF ad CDF of Pareto-Epoetial distributio f ( ) ( ), 0,, 0 (3) F ( ) (4) ( ) Where, is a scale parameter ad is a shape parameter. By substitutig eq (3) & (4) i eq (), the epressios of PDF ad CDF of Upper Record Values arisig from Pareto-Epoetial (UPE) Distributio are give as follows f( ) l( ) ( ),, 0,, 0. (5) F ( ), l (6) Lemma : The area uder the curve is uity f ( ) d Proof: f( ) l( ) ( ) d, 0,, 0 0 let, yl( ) the, dy d ( ) Whe 0 the y 0 Whe the y e y y y e dy 0 f ( ) 0 y y e dy Graphical represetatio of PDF of the UPE Distributio for various parameters value i figure.
36 Noor Waseem ad Shakila Bashir (a) (b) (c) Figure (d) Iterpretatio: From Fig. (a, b, c ad d) it ca be see that the pdf of the proposed model is positivley skewed for various values fo parameters. Figure 2 CDF plot for α=2, λ=.5, =2,3,4 Figure 2 displays the plot of CDF fuctio for various values of whe α ad λ are fied
Record values from T-X family of Pareto-epoetial distributio 37 3. Reliability Aalysis For the UPE distributio several measures of reliability are derived i this sectio. Reliability fuctio, hazard rate fuctio, reversed hazard rate fuctio ad cumulative hazard rate fuctio are give as respectively, R ( ), l (7) l( ) ( ) h ( ), l l( ) ( ) r ( ), l H ( ) l, l (8) (9) (0) Figure 3 Reliability plot for α=2, λ=.5, =2,3,4
0 2 3 4 5 6 7 8 9 0 25 75 25 200 h() 38 Noor Waseem ad Shakila Bashir 9,00E-0 8,00E-0 7,00E-0 6,00E-0 5,00E-0 4,00E-0 3,00E-0 2,00E-0,00E-0 0,00E+00 Hazard Fuctio =2 =3 =4 4. Statistical properties Figure 4 Hazard plot for α=2, λ=.5, =2,3,4 I this sectio, various statistical properties are computed for proposed distributio. The rth Momet about origi is as follows r r E( ( ) ) f( ) d r r E ( ( ) ) k0 k k rk r k () For r,2,3&4 we get first four raw momets of the upper record values from the Epoetial-Pareto Distributio are as follows respectively, / (2) / 2 2 2 2 / 3 3 3 3 3 2 (3) (4)
Record values from T-X family of Pareto-Epoetial distributio 39 / 4 6 4 4 4 4 3 2 (5) Mea ad Geometric mea is give by respectively Mea (6) k. GM i k0 k k Variace is as follows 2 2 2 2 2 k (7) 2 (8) Co-efficiet of Variatio, skewess ad kurtosis are give by respectively 2 2 2 2 (9) 2 3 2 3 3 3 3 2 2 (20) 3 2 3 2 3 2 2 4 6 3 2 3 4 4 3 2 2 4 2 2 2 2 2 4 (2)
40 Noor Waseem ad Shakila Bashir 5. Measure of iequality ad ucertaity I this sectio some Measure of iequality ad ucertaity are derived. Lorez ad Boferroi curves are as follows 2 2 L ( ),l( ) ( ),l( ) a 2 2,l( ) ( ),l( ) a B ( ), l Reyi Etropy is give by v v f ( ) l f ( ) d v ( ) l l( ) v f ( ) d v ( ) 0 (22) (23) (24) 6. Recurrece Relatios for Sigle ad Product Momet of UPE Distributio I this sectio recurrece relatios of the sigle ad product momets of the UPE distributio have bee derived. These relatios ca have used to fid momet of the model i recursive maer. The relatio betwee CDF ad PDF of EP distributio give i equatio (3) ad (4), is ( ) f F ( ) (25) Theorem : For > ad r = 0,,2,3,. r r r E( XU ( ) ) E( XU ( ) ) Proof: From equatio (2) ad, for > ad r = 0,,2,3,.. r r U ( ) 0 U ( ) E( X ) ( X ) f ( ) d (26)
Record values from T-X family of Pareto-epoetial distributio 4 r E( X ) ( ) l F( ) F( ) d U( ) r ( )! 0 r r E( XU( ) ) ( ) l F( ) F( ) d ( )! 0 After some simplificatio we get the result i equatio (26). Theorem 5.2: For i < m < 2 ad r, s = 0,,2,3. r s r s s E ( XU ( i) ) ( XU ( j ) ) E ( XU ( i) ) ( XU ( j) ) (27) for j = i + ad r, s = 0,,2,3. r s rs s r s E ( XU ( i) ) ( X ( ) ) U i E ( XU ( i) ) E ( XU ( i) ) ( XU ( i) ) (28) Proof: Let X U(i) ad X U(j) are from UEP distributio i equatio (5) ad, ad r, s = 0,,2,3. r s r i f( ) U ( i) U ( j) ( ) 0 U i E ( X ) ( X ) ( i )!( j i )! ( X ) l F( ) I( ) d F( ) (29) where, j i s I( ) ( y) l( F( y)) l( F( )) f ( y) dy s j i I( ) ( y) l( F( y)) l( F( )) F( y) dy ( y) j i s ( ) ( ) l( ( )) l( ( )) ( ) s ( y) j i s ( y) I y F y F F y dy I( ) l( F( y)) l( F( )) f ( y) dy s j i 2 ( j i ) l( F( y)) l( F( )) f ( y) dy Substitute the above equatio i equatio (29), we get the result i equatio (27) ad for j=i+, the result i (28). 7. Simulatio Mote Carlo simulatio study is carried out usig the R software to geerate data from the Pareto-Epoetial distributio. F()-u=0 equatio is used where F() is the CDF of the distributio ad u is a observatio from uiform distributio (0,). s
42 Noor Waseem ad Shakila Bashir We simulate 5 samples each of size 50 from the Pareto-Epoetial distributio with the specified values of parameters takig α=3 ad λ=2. For upper record values from Pareto-Epoetial distributio we cosider the upper record from each sample of size 50. Mea Geometric Mea Harmoic Mea Media Descriptive Statistics Variace Stadard Deviatio Skewess Kurtosis Miimum Maimum.935.807.678.807 0.55 0.77 0.29 -.3 0.89 3.2 8. Coclusio I this article we cosidered the Pareto-Epoetial distributio derived by T-X family of distributios ad itroduced the upper record values from Pareto- Epoetial (UPE) distributio. The UPE distributio is a two parameters positively skewed cotiuous distributio. Various properties of the UPE distributio have bee derived. From the Fig. 4, it ca be see that the hazard fuctio of the UPE distributio is showig firstly icreasig (IHR) ad the decreasig (DHR) tred. Measure of iequality ad ucertaity amed Reyi Etropy, Lorez ad Boferroi curves for UPE distributio have bee derived. Recurrece relatio for the sigle ad product momets of UPE distributio have bee derived, the recurrece relatios ca be used to fid momets i a recursive maer. Fially, a Mote Carlo simulatio has bee doe by geeratig data of size 50 with sample size 5 ad upper record has bee oted. Descriptive measures of the UPE distributio have calculated. Refereces [] M. Ahsaullah, Some characterizatios of epoetial distributio by upper record values, Pakista Joural of Statistics, 26 (200), 69-75. [2] M. Ahsaullah, O the record values of the classical Pareto distributio, Pakista Joural of Statististics, 3 (997), 9-5. [3] A. Alzaatreh, F. Famoye ad C. Lee, Gamma-Pareto distributio ad its applicatios, Joural of Moder Applied Statistical Methods, (202), 78-94. https://doi.org/0.22237/jmasm/33584560 [4] N. Balakrisha ad M. Ahsaullah, Recurrece relatios for sigle ad product momets of record values from geeralized Pareto distributio, Commuicatios i Statistics-Theory ad Methods, 23 (994), 284-2852. https://doi.org/0.080/03609294088349
Record values from T-X family of Pareto-epoetial distributio 43 [5] S. Bashir ad M. Ahmad, Record Values from Size-Biased Pareto Distributio ad a Characterizatio, Iteratioal Joural of Egieerig Research ad Geeral Sciece, 2 (204), 209-2730. [6] S. Bashir ad K. Akhtar, Record Values o The Size-Biased Studet s t Distributio, Iteratioal Joural of Iovative Sciece, Egieerig & Techology, (204), 2348-7968. [7] M. Chacko ad L. Muraleedhara, Iferece Based o k-record Values from Geeralized Epoetial Distributio, Statistica, 78 (208), 37-56. [8] K.N. Chadler, The distributio ad frequecy of record values, Joural of Royal Society, Series B (Methodological), 4 (952), 220-228. https://doi.org/0./j.257-66.952.tb005. [9] S. Dey, T. Dey ad D. Luckett, Statistical iferece for the geeralized Rayleigh distributio based o upper record values, Model Assisted Statistics ad Applicatios, 2 (207), o., 5-29. https://doi.org/0.3233/mas-60380 [0] R.U. Kha, A. Kulshrestha ad M.A. Kha, Relatios for momets of k-th record values from epoetial-weibull lifetime distributio ad a characterizatio, Joural of the Egyptia Mathematical Society, 23 (205), 558-562. https://doi.org/0.06/j.joems.204..003 [] D. Kumar, K-th record values from Dagum distributio ad characterizatio, Discussioes Mathematicae Probability ad Statistics, 36 (206), 25-4. https://doi.org/0.75/dmps.83 [2] S. Miimol ad P.Y. Thomas, O some properties of Makeham distributio usig geeralized record values ad its characterizatio, Brazilia Joural of Probability ad Statistics, 27 (203), o. 4, 487-50. https://doi.org/0.24/-bjps78 [3] J. Paul ad P.Y. Thomas, O geeralized upper (k) record values from Weibull distributio, Statistica, 75 (205), 33-330. [4] J. Sara ad K. Nai, Relatioships for momets of kth record values from doubly trucated pth order epoetial ad geeralized Weibull distributios, ProbStat Forum, 5 (202), 42-49. [5] K.S. Sulta, Record values from the modified Weibull distributio ad applicatios, Iteratioal Mathematical Forum, 2 (2007), 2045-2054. https://doi.org/0.2988/imf.2007.0784
44 Noor Waseem ad Shakila Bashir [6] K.S. Sulta ad M.E. Moshref, Record values from geeralized Pareto distributio ad associated iferece, Metrika, 5 (2000), 05-6. https://doi.org/0.007/s00840000025 Received: December 4, 208; Published: Jauary, 209