POWER AKASH DISTRIBUTION AND ITS APPLICATION Rama SHANKER PhD, Uiversity Professor, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: shakerrama009@gmail.com Kamlesh Kumar SHUKLA,* PhD, Associate Professor, Departmet of Statistics, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: kkshukla@gmail.com * Correspodig author Abstract A two-parameter power Akash distributio (PAD), which icludes Akash distributio itroduced by Shaker (05) as a particular case, has bee proposed ad its importat statistical properties icludig shapes of the desity, hazard rate fuctio, momets, skewess ad kurtosis measures, ad stochastic orderig have bee discussed. The maimum likelihood estimatio has bee discussed for estimatig its parameters. Fially, the goodess of fit PAD has bee discussed with a real lifetime data set from egieerig ad the fit has bee foud better as compared with two-parameter power Lidley distributio (PLD) ad oe parameter Akash, Lidley ad epoetial distributios. Key words: Akash distributio; Hazard rate fuctio; Momets; Stochastic orderig;. Itroductio Maimum likelihood estimatio; Goodess of fit The probability desity fuctio (pdf) of Akash distributio itroduced by Shaker (05) is give by where y f y; y e ; y 0, 0 ; ; p f y p f y (.)
p y f y; e ; y 0 y ye f y; ; y 0 The pdf i (.) reveals that the Akash distributio is a two compoet miture of a epoetial distributio (with scale parameter ) ad a gamma distributio (with shape parameter ad scale parameter ), with miig proportio p. Shaker (05) has discussed some of its mathematical ad statistical properties icludig its shape, momets, skewess, kurtosis, hazard rate fuctio, mea residual life fuctio, stochastic orderig, mea deviatios, order statistics, Boferroi ad Lorez curves, etropy measure, stressstregth reliability, ad applicatios of Akash distributio for modelig lifetime data from egieerig ad biomedical scieces. However, there are some situatios where the Akash distributio may ot be suitable from either theoretical or applied poit of view. Recetly, Shaker et al (06) have doe a critical study o the applicatios of oe parameter Akash, Lidley ad epoetial distributios ad observed that each of these distributios has some advatage over the others ad there are some situatios where these distributios do ot provide satisfactory fit. Shaker (07) has also obtaied a Poisso miture of Akash distributio ad amed it Poisso-Akash distributio ad showed its superiority over Poisso- Lidley distributio ad Poisso distributio for modelig cout data. The correspodig cumulative distributio fuctio (cdf) of (.) is give by y y F y; y e ; y 0, 0 (.) I this paper, a power Akash distributio (PAD) has bee itroduced ad its various properties icludig shapes of desity fuctio for varyig values of parameters, survival fuctio, hazard rate fuctio, momets ad stochastic orderig has bee studied. The maimum likelihood estimatio for estimatig its parameters has bee discussed. Fially a applicatio of PAD has bee illustrated with a real lifetime data from egieerig ad the PAD shows satisfactory fit over PLD, Akash, Lidley ad epoetial distributios.. Power Akash distributio Assumig the power trasformatio i (.), the pdf of the radom variable ca be obtaied as where f e ;, ; 0, 0, 0 ;, ;, (.) p g p g (.)
p g ;, e ; 0, 0, 0 e g ;, ; 0, 0, 0 We would call the desity i (.) a power Akash distributio (PAD). It is obvious that the PAD is also a two- compoet miture of Weibull distributio (with shape parameter ad scale parameter ), ad a geeralized gamma distributio (with shape parameters, ad scale parameter ) with miig proportio (.) ca be obtaied as p F ;, e ; 0, 0, 0. The correspodig cdf of To study the ature ad behavior of the pdf of PAD, various graphs of the pdf of PAD for varyig values of parameters have bee draw ad preseted i figure. From the graphs of the pdf of PAD, it is clear that it takes differet shapes for varyig values of parameters. As the values of icreases, the shapes of PAD become ormal. Further, if the value of, the the shapes of PAD become leptokurtic. (.) Figure. Graphs of the pdf of PAD for varyig values of the parameters ad To study the ature ad behavior of the cdf of PAD, various graphs of cdf of PAD for varyig values of parameters have bee draw ad preseted i figure.
Figure. Graphs of the cdf of PAD for varyig values of the parameters ad Recall that Ghitay et al (0) obtaied the power Lidley distributio (PLD) havig pdf ad cdf give by f e ;, ; 0, 0, 0 F ;, e ; 0, 0, 0 Ghitay et al (0) have discussed its mathematical ad statistical properties ad established its goodess of fit over other distributios.. Survival ad hazard rate fuctios (.4) (.5) h, of PAD ca be obtaied a The survival fuctio, S, ad hazard rate fuctio, 4
S F ;, e ; 0, 0, 0 h The behavior of f ; 0, 0, 0 S h at 0 if 0 h0 if 0 if ad, respectively, are give by 0 if h if if, ad The shapes of hazard rate fuctio of PAD for varyig values of the parameters are show i the figure. The graphs of hazard rate fuctio of PAD shows that it takes differet shapes icludig mootoically icreasig ad decreasig for varyig values of parameters. (.) Figure. Graphs of h of PAD for varyig values of the parameters ad 4. Momets ad related measures Usig the miture represetatio (.), the r th momet about origi of PAD ca be obtaied as r r ;, r ;, r E p g d p g d 0 0 5
It should be oted that at r ; r,,,... r r r r origi of Akash distributio ad is give by, the above epressio will reduce to the r r r r r! ; r,,,... Therefore, the mea ad the variace of PAD, respectively, are obtaied as r (4.) th momet about 6 The skewess ad kurtosis measures of PAD, upo substitutig for the raw momets, ca be obtaied usig the epressios Skewess = 5. Stochastic orderig ad Kurtosis 4 4 4 6 4. Stochastic orderig of positive cotiuous radom variables is a importat tool for is said to be smaller tha a ra- judgig their comparative behavior. A radom variable dom variable i the (i) stochastic order F F (ii) hazard rate order (iii) mea residual life order (iv) likelihood ratio order if st if h h hr lr for all for all mrl if m m f if f for all decreases i. The followig importat iterrelatioships due to Shaked ad Shathikumar (994) are well kow for establishig stochastic orderig of distributios lr hr mrl st The PAD is ordered with respect to the strogest likelihood ratio orderig as show i the followig theorem: Theorem: Let PAD ad PAD,. If, ad ( or 6
ad ) the Proof: We have Now f f lr ad hece hr, e mrl ad st ; 0 f log log log log f This gives meas that d 4 4 f log f d Thus for ad (or ad lr ad hece hr, mrl ad st.. ), d f log 0 f d. This 6. Maimum likelihood estimatio of parameters Let,,,..., be a radom sample of size from PAD,. The loglikelihood fuctio is give by l L l f i i l l l l i l i i. i i i The maimum likelihood estimate (MLE) ˆ, ˆ of, of PAD (.) are the solutios of the followig equatios l L 0 i i l L l l l 0 i i i i i i i i These two likelihood equatios do ot seem to be solved directly because these caot be epressed i closed form. However, Fisher s scorig method ca be applied to solve these equatios iteratively. For, we have l L 7
The MLE where l i i i i i l L 4 i l log L i ˆ, ˆ i l i of, of PAD (.) are the solutio of the followig equatios l L l L l L l L ˆ 0 l L l L ˆ 0 ˆ 0 ˆ 0 ad 0 0. These equatios are solved iteratively till sufficietly close estimates of estimate the parameters 7. Data aalysis are iitial values of ˆ ad ˆ ad ˆ 0 ˆ 0 i are obtaied. I this paper, R-software has bee used to ad for the give data sets. I this sectio, we preset the goodess of fit of PAD usig maimum likelihood estimates of parameters to a real data set from egieerig ad compare its fit with the oe parameter epoetial, Lidley ad Akash distributios ad two-parameter PLD. The followig real lifetime data have bee cosidered for the goodess of fit of the proposed distributio. Data Set: The followig data represet the tesile stregth, measured i GPa, of 69 carbo fibers tested uder tesio at gauge legths of 0mm, Bader ad Priest (98)..4.479.55.700.80.86.865.944.958.966.997.006.0.07.055.06.098.40.79.4.40.5.70.7.74.0.0.59.8.8.46.44.45.478.490.5.54.55.554.566.570.586.69.6.64.648.684.697.76.770.77.800.809.88.8.848.880.954.0.067.084.090.096.8..4.585.585 I order to compare these distributios, l L, AIC (Akaike Iformatio Criterio), K-S Statistic ( Kolmogorov-Smirov Statistic) for the real data set have bee computed usig maimum likelihood estimates ad preseted i table. The formulae for computig AIC ad K-S Statistics are as follows: ad F F AIC l L k parameters, = the sample size, fuctio ad F0 K-S Sup 0, where F k = the umber of is the empirical (sample) cumulative distributio is the theoretical cumulative distributio fuctio. 8
The best distributio is the distributio correspodig to lower values of AIC, ad K-S statistics ad higher p-value. l L, Table. MLE s, -l L, AIC, K-S Statistic ad p -value of the fitted distributios of the data set PAD PLD Akash Lidley Model Epoetial ML Estimates ˆ 0.69 ˆ.06 ˆ 0.050 ˆ.868 ˆ 0.9647 ˆ 0.65900 ˆ 0.40794 -l L AIC K-S p-value Statistic 98.0 0.0 0.08 0.999 98. 0. 0.044 0.998 4.8 6.8 0.48 0.00 8.8 40.8 0.90 0.000 6.74 6.74 0.44 0.000 From the goodess of fit of two-parameter PAD ad PLD ad oe parameter epoetial, Lidley ad Akash distributios i table, it is obvious that PAD is well competig with PLD ad gives better fit ad thus it ca be used for modelig lifetime data from egieerig over PLD ad other oe parameter lifetime distributios. The variace-covariace matri ad 95% cofidece itervals (CI s) for the parameters ˆ ad ˆ of PAD for data set has bee preseted i table. Table. Variace-covariace matri ad 95% cofidece itervals (CI s) for the parameters ˆ ad of PAD ˆ Parameters Variace-Covariace Matri 95% CI ˆ ˆ ˆ ˆ 0.00799-0.0009-0.0009 0.069 Lower Upper 0.0096 0.687.6007.570 ˆ The profile of likelihood estimatio for parameters of PAD for parameters for data has bee preseted i figure 4. ˆ ad Figure 4. Likelihood estimate for parameters of PAD ad Fitted distributios plots for the give data set 9
8. Cocludig remarks I this paper a two-parameter power Akash distributio (PAD), of which oe parameter Akash distributio itroduced by Shaker (05) is a special case, has bee proposed. Its importat statistical properties icludig shapes of the desity for varyig values of parameters, hazard rate fuctio, momets, skewess ad kurtosis measures have bee discussed. The maimum likelihood estimatio has bee discussed for estimatig its parameters. The goodess of fit of the proposed distributio for a real lifetime data set from egieerig has bee discussed ad it shows quite satisfactory over two-parameter PLD ad oe parameter Akash, Lidley ad epoetial distributios. Therefore, PAD ca be cosidered a importat lifetime distributio for modelig lifetime data from egieerig. 9. Refereces. Bader, M.G. ad Priest, A. M. Statistical aspects of fiber ad budle stregth i hybrid composites, i Hayashi, T., Kawata, K. ad Umekawa, S. (Eds), Progress i Sciece i Egieerig Composites, ICCM-IV, Tokyo, 98, pp. 9 6.. Ghitay, M.E., Al-Mutairi, D.K., Balakrisha, N., ad Al-Eezi, L.J. Power Lidley distributio ad Associated Iferece, Computatioal Statistics ad Data Aalysis, Vol. 64, 0, pp. 0.. Lidley, D.V. Fiducial distributios ad Bayes Theorem, Joural of the Royal Statistical Society, Series B, Vol. 0, 958, pp. 0 07. 4. Shaker, R. Akash Distributio ad Its Applicatios, Iteratioal Joural of Probability ad Statistics, Vol. 4, No., 05, pp. 65 75. 5. Shaker, R. The Discrete Poisso-Akash Distributio, Iteratioal Joural of Probability ad Statistics, Vol. 6, No., 07, pp. -0. 6. Shaker, R., Hagos, F. ad Sujatha, S. O Modelig of Lifetime Data Usig Oe Parameter Akash, Lidley ad Epoetial Distributios, Biometrics & Biostatistics Iteratioal Joural, Vol., No., 06, pp. 0. Prof. Rama Shaker has completed his Bachelor, Master ad Ph.D i Statistics from Departmet of Statistics, Pata Uiversity, Pata, Idia. He has more tha twety years of teachig ad research eperiece. Presetly, he is workig as Professor ad Head, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea. He is the foudig head of Departmet of Statistics uder College of Sciece at Eritrea Istitute of techology, Asmara, Eritrea. He was also the foudig Editor-i-Chief of Eritrea Joural of Sciece ad Egieerig (EJSE), a biaual Sciece ad Egieerig Joural, published from Eritrea Istitute of Techology, Eritrea. His research iterests iclude Distributio Theory, Modelig of Lifetime Data, Statistical Iferece, Mathematical Demography, Trasportatio ad Assigmet Problems. He has more tha 90 research papers published i atioal ad iteratioal jourals of Statistics, Mathematics ad Operatios research. He is the iteratioal advisory Board members of research jourals published from Idia, Sigapore, Turkey ad USA ad reviewers of research papers i distributio theory of may iteratioal jourals of Statistics. He has bee workig as iteratioal advisory committee members of cofereces/semiars i Statistics. He has delivered ivited/pleary talks i iteratioal/atioal cofereces/semiars i Statistics ad Statistics related disciplies. He is the eteral eamier of Ph.D thesis i Statistics for differet uiversities. Dr. Kamlesh Kumar Shukla is presetly workig as Associate Professor at Departmet of Statistics, Eritrea Istitute of Techology, Eritrea sice February, 06. He has bee awarded Ph.D. i Statistics from Baaras Hidu Uiversity, Varaasi, Idia, ad Masters Degree i Statistics (Gold Medalist) i 997. Dr. Shukla has worked as Assistat Professor i Adama Sciece ad Techology Uiversity, Ethiopia ad havig more tha 4 years of teachig eperiece of colleges/ Uiversities icludig Iteratioal eperiece. Author has worked o si projects i Iteratioal ad Natioal orgaizatio Viz: IIPS, WHO, DST, New Delhi ad preseted may papers i Iteratioal ad Natioal cofereces. He has published more tha 4 research papers i Iteratioal ad Natioal Joural icludig oe book etitled Elemets of Statistics for BCA Studets. 0