A Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data

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1 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) 235 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural A Method of Proposig New Distributio ad its Applicatio to Bladder Cacer Patiets Data Diesh Kumar 1,, Umesh Sigh 2 ad Sajay Kumar Sigh 2 1 Departmet of Statistics, The Uiversity of Burdwa, West Begal , Idia 2 Departmet of Statistics ad DST- CIMS, Baaras Hidu Uiversity, Uttar Pradesh , Idia Received: 21 Ju. 2015, Revised: 15 Jul. 2015, Accepted: 16 Jul Published olie: 1 Sep Abstract: A ew distributio is proposed by the use of some baselie distributio. As a applicatio part, it is derived for the baselie distributio as expoetial distributio. The ew distributio, thus obtaied have bee show to fit the bladder cacer patiets data. Further, maximum likelihood estimator (MLE) ad Bayes estimators uder geeral etropy loss fuctio (GELF) ad squared error loss fuctio (SELF) have bee derived. The estimators have bee compared through their Simulated risks. Keywords: Life Time Distributio, Reliability Aalysis, Maximum Likelihood Estimatio ad Bayesia Ifereces. 1 Itroductio I statistical literature, there are several method to propose ew distributio by the use of some 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(x) of baselie distributio as, where, α > 0 is the shape parameter of the proposed oe. G 1 (x)=(f(x)) α, Aother idea of geeralizig a baselie distributio is to trasmute it by usig quadratic rak trasmutatio map (QRTM) (see, Shaw ad Buckley [6]). If G 2 (x) be the cdf of trasmuted distributio correspodig to the baselie distributio havig cdf F(x), the where λ 1. G 2 (x)=(1+λ)f(x) λ{f(x)} 2, Recetly, various geeralizatios has bee itroduced based o QRTM. For example, trasmuted extreme value distributio (see, Aryal ad Tsokos [8]), trasmuted iverse Weibull distributio (see, Kha et al. [15]), trasmuted modified Weibull distributio (see, Kha ad Kig [14]), trasmuted log-logistic distributio (see, Aryal [12]) ad may more. I the preset study, we propose aother method to get ew distributio by the use of some baselie distributio. If f(x) ad F(x) be the probability desity fuctio (pdf) ad cdf of some baselie distributio, the the pdf g(x) of ew distrbutio is proposed by, g(x)= 1 e 1 f(x) ef(x) (1) We will call the trasformatio (1) as DUS trasformatio for frequetly used purpose i the preset article or elsewhere. It is clearly a trasformatio, ot a geeralizatio, hece it will produce a parsimoious distributio i terms of Correspodig author diesh.ra77@gmail.com

2 236 D. Kumar et al.: A Method of Proposig New Distributio computatio ad iterpretatio as it ever cotai ay ew parameter other tha the parameter(s) ivolved i the baselie distributio. The cdf ad hazard rate fuctio correspodig to the pdf g(x) are give by, ad respectively. G(x)= 1 e 1 h(x)= [ ] e F(x) 1 1 e e F(x) f(x) ef(x) (3) The rest of the paper is orgaized as follows: I sectio 2, we propose a ew distribtio, as obtai by DUS trasformatio (1) by usig Exp(θ)- distributio as the baselie distributio. Further, i sectio 3, we have show the applicability of the ew distributio obtaied i the sectio 2, 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 this ew distributio uder GELF ad SELF. Fially, compariso ad coclusio has bee show i sectios 5 ad 6 respectively. (2) 2 DUS trasformatio of Exp(θ)- distributio Let the baselie distributio is expoetial distributio with pdf, ad the correspodig cdf is give by, f(x)= θ e θx ; x>0 (4) F(x)=1 e θx (5) Here, θ > 0 is the rate parameter or iverse scale parameter of the expoetial distributio. Let g(x) be the pdf of the ew distributio; obtaied by DUS trasformatio (1), correspodig to the baselie pdf (4), the g(x)= 1 e 1 θ e θx e 1 e θx ; x>0 (6) For simplicity i terms of use, we ame/call the distributio havig pdf (6) as DUS trasformatio of Exp (θ)-distributio ad will write it as DUS E (θ)-distributio. The cdf ad hazard rate fuctio of DUS E (θ)-distributio are give by, ad respectively. G(x)= 1 e 1 [ ] e 1 e θx 1 h(x)=θe θx [ e e θx 1] 1 (8) The plots of pdf ad hazard rate fuctio of DUS E (θ)-distributio for differet values of θ are show the Figures 1 ad 2 respectively. (7)

3 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) / Fig. 1: Plots of pdf of DUS E (θ)-distributio for differet values of θ Fig. 2: Plots of hazard rate fuctio of DUS E (θ)-distributio for differet values of θ

4 238 D. Kumar et al.: A Method of Proposig New Distributio MGF ad Raw Momets of DUS E (θ)- distributio: The momeg Geeratig Fuctio (MGF) of DUS E (θ)-distributio havig pdf (6) is obtaied as follows, provided t < θ. M X (t)=e[e tx ] = e e 1 k=0 ( 1) k (k+1)! { } t 1 1 (9) (k+1)θ The raw momets (i.e. r th momet about origi) of DUS E (θ)- distributio is obtaied as follows, [ µ r ] M X (t) r = t r = e e 1 r! θ r t=0 k=0 ( 1) k k!(k+1) r+1 (10) The ifiite series represetatio of µ r is coverget for every θ ad r. Thus, µ r exist for every r ad for all θ. Media of DUS E (θ)- distributio: The media of DUS E (θ)-distributio is the solutio of the followig, G(M)= 1 2 for M ad the same is obtaied as follows, M = 1 θ l (1 l ( )) e+1 2 (11) Mode of DUS E (θ)- distributio: Differetiatig (6) with respect to x, we get g (x)= e ( ) e 1 θ 2 e θ x e e θ x e θ x 1 (12) Clearly g (x) < 0 r, θ, this shows that g(x) is a decreasig fuctio of x ad hece x = 0 is the mode of DUS E (θ)- distributio. A compariso betwee mea, media ad mode of DUS E (θ)- distributio is show i Figure 3 ad it is clear that Mea > Media>Mode, i.e. DUS E (θ)- distributio is positively skewed. 3 Estimatio of the parameter θ of DUS E (θ)- distributio 3.1 Maximum Likelihood Estimator Let idetical items are put o life testig experimet ad suppose X=(X 1,X 2,...,X ) be their idepedet lives such that each X i ( [1]) follow DUS E (θ)-distributio havig pdf (6). The the likelihood fuctio for X is give by, L X (θ)= g(x i ) (13)

5 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) / Fig. 3: Comparative plots of mea, media ad mode of DUS E (θ)-distributio for differet values of θ Puttig the value of g at x i from (6) i (13), we get [ ] 1 L X (θ)= e 1 θ e θx i e 1 e θx i ( ) e = θ e θ x i e e θx i e 1 (14) The log- likelihood fuctio for X is obtaied as, where K = l ( e e 1) is a costat. l = ll X (θ) = K+ lθ θ Hece, the log- likelihood equatio for estimatig θ is give by, θ x i + x i l θ = 0 e θx i (15) x i e θx i = 0 (16) Above is a implicit equatio i θ, hece it ca ot be solved aalytically for θ. We propose Newto- Raphso method for its umerical solutio. 3.2 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. [10]. Aother, importat elemet is the choice of the appropriate prior distributio that covers all the prior kowledge

6 240 D. Kumar et al.: A Method of Proposig New Distributio regardig the parameter of iterest. For the criteria of choosig a approprriate prior distributio, see Sigh et al. [11]. With the above philosophical poit of view, we are motivated to take the prior for θ as G(α,β)-distributio with the pdf π(θ)= αβ Γ(β) 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. [11]). The mea ad variace of the prior distributio (17) are β α ad β respectively. Thus, we may take M = β α 2 α ad V = β, givig α = M2 α 2 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. For more applicatios regardig the use of gamma prior, readers may refer to Sigh et al. [16], Kudu [7] etc. The posterior pdf of θ give X correspodig to the cosidered prior pdf π(θ) of θ is give by, ψ(θ X)= L X(θ) π(θ) L X (θ) π(θ) θ 0 = e ( e 0 ( a+ x i a+ ) θ θ b+ 1 e e θx i (18) x i )θ θ b+ 1 e 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 DUS E (θ)-distributio for fixed values =15, θ = 2, M = 2, V = 0.25 (showig a higher cofidece i the guessed value) ad V = 1000 (showig a weak cofidece i the guessed value). The sample thus geerated is, X =( , , , , , , , , , , , , , , ) The graphs are show i Figures 4 ad 5 respectively. From the Figures 4 ad 5 of the prior ad correspodig posterior pdfs, it is quite clear that the posterior is bell shaped ad showig cocetratio aroud the true value of the parameter θ for whatever may be the ature of the prior (iformative/o-iformative). The loss fuctios 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 respectively. The Bayes estimators of θ uder GELF (19) ad SELF (20) are give by ad L S ( ˆθ, θ)=( ˆθ θ) 2 (20) [ { 1 ˆθ G = E θ X}] δ δ (21) ˆθ S = E[θ X] (22) respectively. It is easy to see that whe δ = 1, the Bayes estimator (21) uder GELF reduces to the Bayes estimator(22) uder SELF.

7 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) / Fig. 4: Prior ad Posterior pdfs of θ for a radomly geerated sample from DUS E (θ)-distributio for fixed =15, θ = 2, M = 2 ad V=0.25 It is ame-worthy to ote here that GELF (19) was proposed by Calabria ad Pulcii [3] ad SELF (20) was proposed at first by Legedre [1] ad Gauss [2] whe he was developig the least square theory. Now, the Bayes estimator of the parameter θ of DUS E (θ)-distributio havig pdf (6) uder GELF is obtaied as follows [ { }] ˆθ G = E θ δ 1 δ X = 0 ( e a+ x i )θ θ b δ+ 1 e 0 e θx i θ ( e a+ ) x i θ θ b+ 1 e e θx i θ Further, if ˆθ S deotes the Bayes estimator of θ uder SELF, the it ca be obtaied by puttig δ = 1 i (23) ad therefore the same is give by, 0 ˆθ S = ( e 0 ( ) e a+ x i θ θ b+ e e θx i θ a+ 1 δ (23) (24) x i )θ θ b+ 1 e 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. 4 Bladder Cacer Data Applicatio To assess the applicability of DUS E (θ)-distributio, as obtaied by DUS trasformatio (1), by usig Exp(θ)- distributio as baselie distributio; i the real situatios, we have cosidered a real data of the remissio times of 128 bladder cacer patiets. The data is extracted from Lee ad Wag [5].

8 242 D. Kumar et al.: A Method of Proposig New Distributio Fig. 5: Prior ad Posterior pdfs of θ for a radomly geerated sample from DUS E (θ)-distributio for fixed =15, θ = 2, M = 2 ad V=1000 Kha et al. [15] showed the applicability of trasmuted iverse Weibull (TIW) distributio 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 value. They compared some life time distributios amely trasmuted iverse Rayleigh (TIR) distributio, trasmuted iverted expoetial (TIE) distributio ad iverse Weibull (IW) distributio i terms of their AIC, BIC, MSE ad KS test value ad foud that the TIW distributio has the lowest AIC, BIC, MSE ad KS test value, idicatig that TIW distributio provides a better fit tha the other three lifetime distributios. We compute MLE of the parameter θ of DUS E (θ)- distributio havig pdf (6) for the above data set ad foud it as The AIC, BIC ad KS test value for DUS E (θ)- distributio are calculated ad we get their values as i Table 1. We have extracted the values of AIC, BIC ad KS test value from Kha et al. [15] ad preset their values i the followig comparative Table 1. Table 1: AIC, BIC ad KS test value for DUS E (θ), TIW, TIR, TIE ad IW distributios Distributios AIC BIC KS test value DUS E (θ) TIW TIR TIE IW The plots of empirical cdf F ad fitted cdf G(x) of DUS E (θ)- distributio for the data of remissio times of 128 bladder cacer patiets are show i Figure 6.

9 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) / Fig. 6: Plots of empirical cdf F ad fitted cdf G(x) of DUS E (θ)- distributio for remissio times of 128 bladder cacer patiets data DUS E (θ)- distributio havig pdf (6) has the lowest AIC, BIC ad KS test value i compariso to those of TIW, TIR, TIE ad IW distributios (see, Table 1), idicatig that DUS E (θ)- distributio havig pdf (6) provides a better fit tha the other four lifetime distributios amely TIW, TIR, TIE ad IW distributios. 5 Compariso of the estimators I this sectio, we compare the cosidered estimators i.e. ˆθ M, ˆθ S, ˆθ G of the parameter θ of DUS E (θ)- distributio havig 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 DUS E (θ)- distributio. 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 = 2, δ =±3, M = 1,2,3 ad V = 0.25,0.5,1,2,5,10,50,100,500,1000. From Tables 2-4, 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 whatever cofidece i the guessed value of θ as its true value. But if guessed value of θ is either more/less tha its true value, the estimator ˆθ G performs well for lower cofidece i such guessed value of θ, otherwise ˆθ M performs better (whe guessed value of θ is less tha its true value) or ˆθ S performs better (whe guessed value of θ is more tha its true value). Further, whe uder estimatio is more serious tha over estimatio, for whatever be the cofidece i the guessed value of θ as less tha its true value, the estimator ˆθ G performs better tha the estimators ˆθ S ad ˆθ M. But whe guessed value of θ is same as its true value, the estimator ˆθ G performs better for lower cofidece; otherwise ˆθ S performs better ad whe guessed value of θ is more tha its true value, the estimator ˆθ M performs well for higher cofidece; ˆθ S, performs batter for moderate cofidece ad for lower cofidece value, the estimator ˆθ G performs better. 6 Coclusio From the simulatio study, it is clear that the estimators of the parametr θ of DUS 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, DUS trasformatio (1) is full proof ad by its use, the distributio, thus foud may be appropriate for real life applicatios.

10 244 D. Kumar et al.: A Method of Proposig New Distributio Table 2: Risks of the estimators of θ uder GELF for fixed =15, θ = 2, M = 2 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 = 2, 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 ) Table 4: Risks of the estimators of θ uder GELF for fixed =15, θ = 2, 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 ) 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 paper. Refereces [1] Legedre, A. (1805): New Methods for the Determiatio of Orbits of Comets. Courcier, Paris. [2] 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,

11 J. Stat. Appl. Pro. Lett. 2, No. 3, (2015) / [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, 27 (4), 887?-904. [5] Lee, E. T. ad Wag, J. W. (2003): Statistical Methods for Survival Data Aalysis. Wiley, New York, DOI: / [6] Shaw, W. T. ad Buckley, I. R. (2007): The alchemy of probability distributios: beyod Gram-Charlier expasios ad a skewkurtotic-ormal distributio from a rak trasmutatio map. Research report. [7] Kudu, D. (2008): Bayesia Iferece ad Life Testig Pla for the Weibull Distributio i Presece of Progressive Cesorig. TECHNOMETRICS, 50 (2), [8] Aryal, G. R. ad Tsokos, C. P. (2009): O the trasmuted extreme value distributio with applicatio. Noliear Aalysis: Theory, Methods ad Applicatios, 71 (12), , [9] Sigh, S. K., Sigh, U. ad Kumar, D. (2011): Bayesia estimatio of the expoetiated gamma parameter ad reliability fuctio uder asymmetric loss fuctio. REVSTAT Statistical Joural, 9 (3), [10] Sigh, S. K., Sigh, U. ad Kumar, D. (2011): 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 (2), [11] Sigh, S. K., Sigh, U. ad Kumar, D. (2012): 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 29 May, 2012, DOI: / , [12] Aryal, G. R. (2013): Trasmuted log-logistic distributio. Joural of Statistics Applicatios ad Probability, 2 (1), 11?-20. [13] Sigh, S. K., Sigh, U. ad Kumar, D. (2013): Bayesia estimatio of parameters of iverse Weibull distributio. Joural of Applied Statistics, 40 (7), [14] Kha, M. S. ad Kig, R. (2013): Trasmuted modified Weibull distributio: A geeralizatio of the modified Weibull probability distributio. Europe. J. of Pure Appl. Math., 6 (1), [15] Kha, M. S., Kig, R. ad Hudso, I. L. (2014): Characterisatios of the trasmuted iverse Weibull distributio. ANZIAM J. 55 (EMAC-2013), C197 C217. [16] Sigh, S. K., Sigh, U., Yadav, A. S. ad Vishwkarma, P. K. (2015): O the estimatio of stress stregth reliability parameter of iverted expoetial distributio. Iteratioal Joural of Scietific World, 3 (1),

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