A New Distribution Using Sine Function- Its Application To Bladder Cancer Patients Data

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1 J. Stat. Appl. Pro. 4, No. 3, Joural of Statistics Applicatios & Probability A Iteratioal Joural A New Distributio Usig Sie Fuctio- Its Applicatio To Bladder Cacer Patiets Data Diesh Kumar 1,, Umesh Sigh ad Sajay Kumar Sigh 1 Departmet of Statistics, The Uiversity of Burdwa, West Begal , Idia Departmet of Statistics ad DST- CIMS, Baaras Hidu Uiversity, Uttar Pradesh- 1005, Idia Received: 31 Jul. 015, Revised: 16 Aug. 015, Accepted: 6 Aug. 015 Published olie: 1 Nov. 015 Abstract: A ew life time distributio is proposed by the use of Sie fuctio i terms of some life time distributio as baselie distributio. It is derived for the baselie distributio as expoetial distributio ad some statistical properties of the ew distributio, thus obtaied have bee studied. The ew distributio have bee show better fit to the bladder cacer patiets data as compared to some well kow distributios available i the statistical literature through Akaike iformatio criteria AIC, Bayesia iformatio criteria BIC, - log-likelihood ad the associated Kolmogorov-Smirov KS test values. Keywords: Life Time Distributio, Reliability Aalysis ad Bayesia Ifereces. 1 Itroductio I statistical literature, several methods are available to propose ew life time distributio by the use of some existig life time distributio as baselie distributio. For example, Gupta et al. [4] proposed the cumulative distributio fuctio cdf G 1 x of ew distributio correspodig to the cdf F 1 x of baselie distributio as, G 1 x=f 1 x α, where, α > 0 is the shape parameter of the proposed oe. For α = 1, the ew distributio ad the baselie distributio are the same. Several researchers geeralise some useful distributios by the idea of Gupta et al. [4]. For example, Nadarajah ad Kotz [6] itroduced four expoetiated type distributios that are the geeralizatios of the stadard gamma, stadard Weibull, stadard Gumbel ad the stadard Frechet distributios ad studied some mathematical properties for each distributio. Nadarajah [7] derived expoetiated stadard Gumbel distributio with a hope that it would attract wider applicability i climate modelig as the stadard Gumbel distributio do. May other geralizatios ca be foud i the statistical literatures. Aother idea of geeralizig a baselie distributio is to trasmute it by usig the quadratic rak trasmutatio map QRTM see, Shaw ad Buckley [8]. If G x be the cdf of trasmuted distributio correspodig to the baselie distributio havig cdf F x, the G x=1+λf x λ{f x}, where λ 1. For λ = 0, the ew distributio is same as the baselie distributio. Recetly, various geeralizatios has bee itroduced based o QRTM. For example, trasmuted extreme value distributio see, Aryal ad Tsokos [9], trasmuted log-logistic distributio see, Aryal [13], trasmuted modified Weibull distributio see, Kha ad Kig [15], trasmuted iverse Weibull distributio see, Kha, Kig ad Hudso Correspodig author diesh.ra77@gmail.com Natural Scieces Publishig Cor.

2 418 D. Kumar et al.: A New Distributio Usig Sie Fuctio... [19] ad may more. I the preset study, we propose a method to get ew life distributio by the use of ay baselie life distributio. If fx ad Fx be the pdf ad cdf of some baselie life distributio, the the cdf Gx of ew life distrbutio is proposed by, π Gx=si Fx 1 Further, if gx be the pdf ad hx be the hazard rate fuctio correspodig to the cdf Gx, the, gx= π fx cos π Fx ad respectively. hx= π fx ta π 4 + π 4 Fx 3 We will call the trasformatio 1 ad as SS trasformatio for frequetly used purpose i the preset article or elsewhere. The rest of the paper is orgaized as follows: I sectio, we propose a ew distribtio, as obtai by SS trasformatio by cosiderig Expθ-distributio as the baselie distributio ad studied some of its statistical characteristics; like momet geeratig fuctio MGF, momets, media ad mode. Further, i sectio 3, we have show the applicability of the ew distributio obtaied i the sectio, to the bladder cacer patiets data i terms of assesig its fittig i compariso to some available distributios. I sectio 4, we have derived MLE ad Bayes estimators of the parameter θ of the distributio, thus obtaied uder GELF ad SELF. Fially, compariso ad coclusio has bee show i the sectios 5 ad 6 respectively. SS trasformatio of Expθ-distributio Let the baselie distributio is Expθ-distributio with pdf, ad the correspodig cdf is give by, Here, θ > 0 is the rate parameter or iverse scale parameter of Expθ-distributio. fx= θ e θx ; x>0 4 Fx=1 e θx 5 Let gx be the pdf of the ew distributio; obtaied by SS trasformatio, correspodig to the baselie pdf 4, the gx= π θ e θx si π e θx ; x>0 6 For simplicity i terms of use, we ame/call the distributio havig pdf 6 as SS trasformatio of Exp θ-distributio ad we will write it as SS E θ-distributio. The cdf ad hazard rate fuctio of SS E θ-distributio are give by, Gx=cos π e θx 7 ad hx= π θ e θx cot π 4 e θx 8 respectively. The plots of pdf ad hazard rate fuctio, for differet values of θ are show i Figures 1 ad respectively. Natural Scieces Publishig Cor.

3 J. Stat. Appl. Pro. 4, No. 3, / Momet Geeratig Fuctio of SS E θ-distributio The momet geeratig fuctio of SS E θ-distributio havig pdf 6 is obtaied as follows, M X t=θ k=0 1 k π k+ k+1! { 1 } k+θ t 9 provided t < θ.. Raw Momets of SS E θ-distributio The r th momet about origi i.e. raw momet of SS E θ-distributio is obtaied as follows, [ µ r ] M X t r = t r = r! θ r k=0 t=0 k+ 1 k π k+1!k+ r Media of SS E θ-distributio The media of SS E θ-distributio is the solutio of the followig equatio for M, GM= 1 ad the same is obtaied as follows, M = 1 θ l Mode of SS E θ-distributio Differetiatig 6 partially w. r. to x o both sides, we get g x= π { π θ e si θx e θx + π π cos e θx e θx} 1 Clearly g x<0 x, θ ad this shows that gx is a decreasig fuctio of x > 0 θ ad hece its mode is x = 0. 3 Estimatio of the parameter θ of SS E θ-distributio 3.1 Maximum Likelihood Estimator Let idetical items are put o a life testig experimet ad suppose X=X 1,X,...,X be their idepedet lives such that each X i [1] follow SS E θ-distributio havig pdf 6. The the likelihood fuctio for X is give by, LX θ= gx i 13 Natural Scieces Publishig Cor.

4 40 D. Kumar et al.: A New Distributio Usig Sie Fuctio... Puttig the value of g at x i from 6 i 13, we get LX θ= The log- likelihood fuctio for X is obtaied as, = l = llx θ = K+ lθ θ { π θ e θx i π si i} e θx π θ e θ x i x i + π si e θx i { π } l si e θx i where K = l π is a costat. Hece, the log- likelihood equatio for estimatig θ is give by, θ x i π { x i e θx i cot π e θx i l θ = 0 } = 0 16 Above equatio is ot solvable aalytically for θ. We propose Newto- Raphso method for its umerical solutio. 3. Bayes Estimators A importat elemet i Bayesia estimatio problem is the specificatio of the loss fuctio. The choice is basically depeds o the problem i had. For more discussio o the choice of a suitable loss fuctio, readers may refer to Sigh et al. [11]. Aother, importat elemet is the choice of the appropriate prior distributio that covers all the prior kowledge regardig the parameter of iterest. For the criteria of choosig a approprriate prior distributio, see Sigh et al. [1]. With the above philosophical poit of view, we are motivated to take the prior for θ as Gα,β-distributio with the pdf πθ= αβ Γβ e αθ θ β 1 ; θ > 0 17 where α > 0 ad β > 0 are the hyper- parameters. These ca be obtaied, if ay two idepedet iformatios o θ are available, say prior mea ad prior variace are kow see, Sigh et al. [1]. The mea ad variace of the prior distributio 17 are β α ad β respectively. Thus, we may take M = β α α ad V = β, givig α = M α V ad β = M V. For ay fiite value of M ad V to be sufficietly large, 17 behaves as like as o-iformative prior. The posterior pdf of θ give X correspodig to the cosidered prior pdf πθ of θ is give by, LX θ πθ ψθ X= LX θ πθ θ 0 = 0 e α+ x i θ θ β+ 1 e α+ x i θ θ β+ 1 si π e θx i si 18 π e θx i θ Now, to have a idea about the shapes of the prior ad correspodig posterior pdfs for differet cofidece levels i the guessed value of θ as its true value, we radomly geerate a sample from SS E θ-distributio for fixed values = 15, Natural Scieces Publishig Cor.

5 J. Stat. Appl. Pro. 4, No. 3, / 41 θ =, M =, V = 0.1 showig a higher cofidece i the guessed value ad V = 500 showig a weak cofidece i the guessed value. The sample thus geerated is, X = , , , , , , , , , , , , , , The graphs are show i Figures 3 ad 4 respectively. The loss fuctios we cosidered here are geeral etropy loss fuctio GELF ad squared error loss fuctio SELF, which are defied by, δ ˆθ ˆθ L G ˆθ, θ= δ l 1 19 θ θ ad L S ˆθ, θ=ˆθ θ 0 respectively. The Bayes estimators of θ uder GELF 19 ad SELF 0 are give by [ { ˆθ G = E θ X}] δ 1 δ 1 ad ˆθ S = E[θ X] respectively. It is easy to see that whe δ = 1, the Bayes estimator 1 uder GELF reduces to the Bayes estimator uder SELF. It is ame-worthy to ote here that GELF 19 was proposed by Calabria ad Pulcii [3] ad SELF 0 was proposed at first by Legedre [1] ad Gauss [] whe he was developig the least square theory. For more applicatios related to GELF, readers may refer to Sigh et al. [16,17,18]. Now, the Bayes estimator of the parameter θ of SS E θ-distributio havig pdf 6 uder GELF is obtaied as follows, [ { }] 1 ˆθ G = E θ δ δ X = 0 e α+ x i θ θ β δ+ 1 0 e α+ x i θ θ β+ 1 si π e θx i θ si π e i θx θ 1 δ 3 Further, if ˆθ S deotes the Bayes estimator of θ uder SELF, the it ca be obtaied by puttig δ = 1 i 3 ad therefore the same is give by, 0 ˆθ S = e 0 e α+ x i θ θ β+ si π e θx i θ α+ x i θ θ β+ 1 si 4 π e θx i θ The itegral ivolved i Bayes estimators do ot solved aalytically, therefore we propose Gauss - Lagurre s quadrature method for their umerical evaluatio. Natural Scieces Publishig Cor.

6 4 D. Kumar et al.: A New Distributio Usig Sie Fuctio... 4 Bladder Cacer Patiets Data I this sectio, we aalyze a real data set to illustrate that SS E θ-distributio ca be a good lifetime model, comparig with may kow distributios available i statistical literature. For the purpose, we have cosidered a real data of the remissio times i moths of a radom sample of 18 bladder cacer patiets. The data is extracted from Lee ad Wag [5] ad is as show below: X =0.08,.09, 3.48, 4.87, 6.94, 8.66, 13.11, 3.63, 0.0,.3, 3.5, 4.98, 6.97, 9.0, 13.9, 0.40,.6, 3.57, 5.06, 7.09, 9., 13.80, 5.74, 0.50,.46, 3.64, 5.09, 7.6, 9.47, 14.4, 5.8, 0.51,.54, 3.70, 5.17, 7.8, 9.74, 14.76, 6.31, 0.81,.6, 3.8, 5.3, 7.3, 10.06, 14.77, 3.15,.64, 3.88, 5.3, 7.39, 10.34, 14.83, 34.6, 0.90,.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05,.69, 4.3, 5.41, 7.6, 10.75, 16.6, 43.01, 1.19,.75, 4.6, 5.41, 7.63, 17.1, 46.1, 1.6,.83, 4.33, 5.49, 7.66, 11.5, 17.14, 79.05, 1.35,.87, 5.6, 7.87, 11.64, 17.36, 1.40, 3.0, 4.34, 5.71, 7.93, 1.46, 18.10, 11.79, 4.40, 5.85, 8.6, 11.98, 19.13, 1.76, 3.5, 4.50, 6.5, 8.37, 1.0,.0, 13.31, 4.51, 6.54, 8.53, 1.03, 0.8,.0, 3.36, 1.07, 6.76, 1.73,.07, 3.36, 6.93, 8.65, 1.63,.69 Kha et al. [19] showed the applicability of trasmuted iverse Weibull distributio TIWD o this data by the fittig criteria i terms of Akaike iformatio criteria AIC, Bayesia iformatio criteria BIC, mea square error MSE ad the associated Kolmogorov-Smirov KS test values. They compared some life time distributios amely trasmuted iverse Rayleigh distributio TIRD, trasmuted iverted expoetial distributio TIED ad iverse Weibull distributio IWD i terms of their AIC, BIC, MSE ad KS test values ad foud that the TIWD has the lowest AIC, BIC, MSE ad KS test value, idicatig that TIWD provides a better fit tha the other three lifetime distributios to the bladder cacer patiets data. We have computed MLE of the parameter θ of SS E θ-distributio havig pdf 6 for the above data set ad foud it as The AIC, BIC ad KS test value for SS E θ-distributio are calculated ad we get their values as i Table 1. We have extracted the values of AIC, BIC, -log likelihood -LL ad KS test values for TIWD, TIED, IWD ad TIRD for the above cosidered data from Kha et al. [19] ad preset their values i the followig comparative Table 1. Table 1: AIC, BIC, -LL ad KS test values for SS E θ-distributio, TIWD, TIED, IWD ad TIRD Distributios AIC BIC -LL KS test value SS E θ-distributio TIWD TIED IWD TIRD The plots of empirical cdf F ad fitted cdf Gx of SS E θ-distributio havig pdf 6 for above data of the remissio times of a radom sample of 18 bladder cacer patiets are show i Figure 5. From Table 1, it is observed that SS E θ-distributio havig pdf 6 has the lowest AIC, BIC, -LL ad KS test value i comparisio to those of TIWD, TIED, IWD ad TIRD; idicatig that SS E θ-distributio provides a better fit tha the other four lifetime distributios amely TIWD, TIED, IWD ad TIRD. 5 Compariso of the estimators I this sectio, we compared the cosidered estimators i.e. ˆθ M, ˆθ S, ˆθ G of the parameter θ of pdf 6 i terms of simulated risks average loss over sample space uder GELF. It is clear that the expressios for the risks caot be obtaied i ice closed form. So, the risks of the estimators are estimated o the basis of Mote Carlo simulatio study of 5000 samples from pdf 6. It may be oted that the risks of the estimators will be a fuctio of umber of items put o test, parameter θ of the model, the hyper parameters α ad β of the prior distributio ad the GELF parameter δ. I order to cosider the variatio of these values, we obtaied the simulated risks for = 15, θ =, M = 1,,3, V = 0.1,0.5,1,,5,10,100,500 ad δ =±3. Natural Scieces Publishig Cor.

7 J. Stat. Appl. Pro. 4, No. 3, / 43 Table shows the risks of the estimators of θ whe guessed value of θ M = 1 is less tha its true value θ = ad we observed that whe over estimatio is more serious tha uder estimatio, the estimator ˆθ G performs better i the sese of havig smallest risk i compariso to ˆθ S ad ˆθ M for lower cofidece i the guessed value ad for high cofidece i guessed value, ˆθ S performs better tha ˆθ G ad ˆθ M. But i the reverse situatio, the oly chage is oted that for high cofidece i the guessed value, ˆθ M performs better tha ˆθ G ad ˆθ S. Further, Table 3 shows the risks of the estimators of θ whe guessed value of θ M = is same as its true value θ = ad it is observed that the estimator ˆθ G performs better tha the other estimators for moderate ad lower cofidece i the guessed value, while for higher cofidece i the guessed value, ˆθ S performs better for whatever may be the situatio is serious. Fially, Table 4 shows the risks of the estimators of θ whe guessed value of θ M = 3 is greater tha its true value θ = ad it is observed that whe over estimatio is more serious tha uder estimatio, the estimator ˆθ G performs better i compariso to ˆθ S ad ˆθ M for lower cofidece i the guessed value ad for high cofidece i guessed value, ˆθ S performs better. But i the reverse situatio, the estimator ˆθ M performs well for higher cofidece, ˆθ S ;performs batter for moderate cofidece ad for lower cofidece, the estimator ˆθ G performs better. Table : Risks of the estimators of θ uder GELF for fixed =15, θ =, M = 1 ad δ =±3 V R G ˆθ M δ = 3 δ =+3 R G ˆθ S R G ˆθ G R G ˆθ M R G ˆθ S R G ˆθ G Table 3: Risks of the estimators of θ uder GELF for fixed =15, θ =, M = ad δ =±3 V R G ˆθ M δ = 3 δ =+3 R G ˆθ S R G ˆθ G R G ˆθ M R G ˆθ S R G ˆθ G Natural Scieces Publishig Cor.

8 44 D. Kumar et al.: A New Distributio Usig Sie Fuctio... Table 4: Risks of the estimators of θ uder GELF for fixed =15, θ =, M = 3 ad δ =±3 V R G ˆθ M δ = 3 δ =+3 R G ˆθ S R G ˆθ G R G ˆθ M R G ˆθ S R G ˆθ G Fig. 1: Plots of probability desity fuctio of SS E θ-distributio for differet values of θ Fig. : Plots of hazard rate fuctio of SS E θ-distributio for differet values of θ Natural Scieces Publishig Cor.

9 J. Stat. Appl. Pro. 4, No. 3, / 45 Fig. 3: Prior ad Posterior pdfs of θ for a radomly geerated sample X from SS E θ-distributio for fixed =15, θ =, M = ad V=0.1 Fig. 4: Prior ad Posterior pdfs of θ for a radomly geerated sample X from SS E θ-distributio for fixed =15, θ =, M = ad V=500 Natural Scieces Publishig Cor.

10 46 D. Kumar et al.: A New Distributio Usig Sie Fuctio... Fig. 5: Plots of empirical cdf F ad fitted cdf Gx of SS E θ-distributio for remissio times of 18 bladder cacer patiets data 6 Coclusio From the above simulatio study, it is clear that the Bayes estimators of the parameter θ of SS E θ-distributio havig pdf 6 may be recommeded for their use as per cofidece level i the guessed value of θ as discussed i the previous sectio. Further from real data aalysis, it is clear that SS trasformatio 1 is full proof ad by its use, the distributio, thus foud may be appropriate for real life applicatios. Ackowledgemet The authors are grateful to the Editor ad the aoymous referees for a careful checkig of the details ad for helpful commets that led to improvemet of the article. We devote the preset article to the parets Mr. Sukkhu Ram ad Mrs. Sushila Devi of the first author Diesh Kumar. Refereces [1] Legedre, A. 1805: New Methods for the Determiatio of Orbits of Comets. Courcier, Paris. [] Gauss, C.F. 1810: Least Squares method for the Combiatios of Observatios. Traslated by J. Bertrad 1955, Mallet-Bachelier, Paris. [3] Calabria, R. ad Pulcii, G. 1994: A egieerig approach to Bayes estimatio for the Weibull distributio. Micro Electro Reliab., 34, [4] Gupta, R.C., Gupta, R.D. ad Gupta, P.L. 1998: Modelig failure time data by Lehma alteratives. Commuicatio i Statistics- Theory ad Methods, 7 4, [5] Lee, E. T. ad Wag, J. W. 003: Statistical Methods for Survival Data Aalysis. Wiley, New York, DOI: / [6] Nadarajah, S. ad Kotz, S. 006: The Expoetiated Type Distributios. Acta Appl Math 9, , DOI /s [7] Nadarajah, S. 006: The expoetiated Gumbel distributio with climate applicatio. Evirometrics, 17, [8] Shaw, W. T. ad Buckley, I. R. 007: The alchemy of probability distributios: beyod Gram-Charlier expasios ad a skewkurtotic-ormal distributio from a rak trasmutatio map. Research report. [9] Aryal, G. R. ad Tsokos, C. P. 009: O the trasmuted extreme value distributio with applicatio. Noliear Aalysis: Theory, Methods ad Applicatios, 71 1, , [10] Sigh, S. K., Sigh, U. ad Kumar, D. 011: Bayesia estimatio of the expoetiated gamma parameter ad reliability fuctio uder asymmetric loss fuctio. REVSTAT Statistical Joural, 9 3, Natural Scieces Publishig Cor.

11 J. Stat. Appl. Pro. 4, No. 3, / 47 [11] Sigh, S. K., Sigh, U. ad Kumar, D. 011: Estimatio of Parameters ad Reliability Fuctio of Expoetiated Expoetial Distributio: Bayesia approach Uder Geeral Etropy Loss Fuctio. Pakista joural of statistics ad operatio research, VII, [1] Sigh, S. K., Sigh, U. ad Kumar, D. 01: Bayes estimators of the reliability fuctio ad parameter of iverted expoetial distributio usig iformative ad o-iformative priors. Joural of Statistical Computatio ad Simulatio, olie dated 9 May, 01, DOI: / , 1-1. [13] Aryal, G. R. 013: Trasmuted log-logistic distributio. Joural of Statistics Applicatios ad Probability, 1, [14] Sigh, S. K., Sigh, U. ad Kumar, D. 013: Bayesia estimatio of parameters of iverse Weibull distributio. Joural of Applied Statistics, 40 7, [15] Kha, M. S. ad Kig, R. 013: Trasmuted modified Weibull distributio: A geeralizatio of the modified Weibull probability distributio. Europe. J. of Pure Appl. Math., 6 1, [16] Sigh, S. K., Sigh, U. ad Kumar, M. 013: Estimatio of Parameters of Geeralized Iverted Expoetial Distributio for Progressive Type-II Cesored Sample with Biomial Removals. Joural of Probability ad Statistics, Volume 013, Article ID 18365, 1 1, [17] Sigh, S. K., Sigh, U. ad Kumar, M. 014: Bayesia estimatio for Poissio- expoetial model uder Progressive type-ii cesorig data with Biomial removal ad its applicatio to ovaria cacer data. Commuicatios i Statistics - Simulatio ad Computatio, DOI: / [18] Sigh, S. K., Sigh, U. ad Kumar, M. 014: Bayesia iferece for expoetiated Pareto model with applicatio to bladder cacer remissio time. Statistics i Trasitio ew series, Summer 014, 15 3, [19] Kha, M. S., Kig, R. ad Hudso, I. L. 014: Characterisatios of the trasmuted iverse Weibull distributio. ANZIAM J. 55 EMAC013, C197 C17. Natural Scieces Publishig Cor.