Discrete Generalized Burr-Type XII Distribution

Transcription

1 Journal of Modern Applied Statistial Methods Volume 13 Issue 2 Artile Disrete Generalized Burr-Type XII Distribution B. A. Para University of Kashmir, Srinagar, India, parabilal@gmail.om T. R. Jan University of Kashmir, Srinagar, India, drtrjan@gmail.om Follow this and additional works at: Part of the Applied Statistis Commons, Soial and Behavioral Sienes Commons, and the Statistial Theory Commons Reommended Citation Para, B. A. and Jan, T. R. (2014) "Disrete Generalized Burr-Type XII Distribution," Journal of Modern Applied Statistial Methods: Vol. 13 : Iss. 2, Artile 13. DOI: /jmasm/ Available at: This Regular Artile is brought to you for free and open aess by the Open Aess Journals at DigitalCommons@WayneState. It has been aepted for inlusion in Journal of Modern Applied Statistial Methods by an authorized editor of DigitalCommons@WayneState.

2 Journal of Modern Applied Statistial Methods November 2014, Vol. 13, No. 2, Copyright 2014 JMASM, In. ISSN Disrete Generalized Burr-Type XII Distribution B. A. Para University of Kashmir Srinagar, India T. R. Jan University of Kashmir Srinagar, India A disrete analogue of generalized Burr-type XII distribution is introdued using a general approah of disretizing a ontinuous distribution. It may be worth eploring the possibility of developing a disrete version of the si parameter generalized Burr-type XII distribution for use in modeling a disrete data. This distribution is suggested as a suitable reliability model to fit a range of disrete lifetime data, as it is shown that hazard rate funtion an attain monotoni inreasing (deeasing) shape for ertain values of parameters. The equivalene of disrete generalized Burr-type XII (DGBD-XII) and ontinuous generalized Burr-type XII (GBD-XII) distributions has been established. The inreasing failure rate property in the disrete setup has been ensured. Various theorems relating this new model to other probability distributions have also been proved. Keywords: Disrete generalized Burr-type XII distribution, disrete lifetime models, reliability, failure rate Introdution In reliability theory a number of ontinuous life models is now available in the subjet to portray the survival behavior of a omponent or a system. Many ontinuous life distributions have been studied in details (see for eample Kapur & Lamberson, 1997; Lawless, 1982; Sinha, 1986). However, it is sometimes impossible or inonvenient in life testing eperiments to measure the life length of a devie on a ontinuous sale. For eample the lifetime of an on/off swithing devie is a disrete random variable, or life length of a devie reeiving a number of shoks it sustain before it fails is also a disrete random variable. Reently, the speial roles of disrete distributions have reeived reognition from survival analysts. Many ontinuous distributions have been disretized. For eample, the Geometri and Negative binomial distributions are the disrete B. A. Para is a Researh Sholar in the Department of Statistis. him parabilal@gmail.om. T. R. Jan is an Assistant Professor in the Post-Graduate Department of Statistis. him at drtrjan@gmail.om. 244

3 PARA & JAN versions of Eponential and Gamma distributions. Nakagawa (1975) disretized the Weibull distribution. The disrete versions of the normal and Rayleigh distributions were also proposed by Roy (2003, 2004). Disrete analogues of Mawell, two parameter Burr-type XII and Pareto distributions were also proposed by Krishna and Punder (2007, 2009). Reently the inverse Weibull distribution was also disretized by Jazi, Lai and Alamatsaz (2010). This artile addresses the problem of disretization of generalized Burr-type XII (GBD-XII) distribution, beause there is a need to find more plausible disrete lifetime distributions to fit to various life time data. The Disrete Generalized Burr XII Distribution: Roy (1993) pointed out that the univariate geometri distribution an be viewed as a disrete onentration of a orresponding eponential distribution in the following manner 1 p X s s when 0,1,2, where X is disrete random variable following geometri distribution with probability mass funtions as p 1, 0,1, 2, where s() represents the survival funtion of an eponential distribution of the form s() = ep( λ), learly θ = ep( λ), 0 < θ < 1. Thus, one to one orrespondenes between the geometri distribution and the eponential distribution an be established, the survival funtions being of the same form. The general approah of disretizing a ontinuous variable is to introdue a greatest integer funtion of X i.e., [X] (the greatest integer less than or equal to X till it reahes the integer), in order to introdue grouping on a time ais. If the underlying ontinuous failure time X has the survival funtion s() = p(x > ) and times are grouped into unit intervals, so that the disrete observed variable is dx = [X]. 245

4 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION The probability mass funtion of dx an be written as 1 1 s s 1, 0,1,2, p p dx p X being the umulative distribution funtion of random variable X. In reliability theory, many lassifiation properties and measures are diretly related to the funtional form of the survival funtion. The inreasing failure rate (IFR), dereasing failure rate (DFR), Inreasing failure rate average (IFRA), dereasing failure rate average (DFRA), new better than used (NBU), new worse than used (NWU), new better (worse) than used in epetation NBUE (NWUE) and inreasing (dereasing) mean residual lifetime IMRL (DMRL) et. are eamples of suh lass properties as may be seen from Barlow and Proshan (1975). If disretization of a ontinuous life distribution an retain the same funtional form of the survival funtion then many reliability measures and lass properties will remain unhanged. In this sense, the disrete onentration onept is onsidered herein as a simple approah that an generate a disrete life distribution model. Thus, given any ontinuous life variable with survival funtion s() a disrete lifetime variable X with probability mass funtion p() is defined by p s s 1, 0,1,2, Using this onept for the purpose of disretizing generalized Burr-type XII distribution, suppose that Y 1 and Y 2 are independently distributed ontinuous random variables. If Y 1 has an eponential density funtion y f y, e, y 0; 0 and Y 2 has a gamma distribution with pdf k y2 k1 f y2;, k e y2, 0; k 0; y2 0 X Y Y k, then the random variable 1 has a si parameter generalized Burr-type XII distribution with 2 parameters (μ = 0, σ = 1, α, θ,, k) with a density funtion 246

5 PARA & JAN k 1 k f ; 0, 0; 0; 0; k 0 k1 (1) The pdf plot for DGB-XII variate X for different values of parameters is shown in Figure 1. It is evident that the distribution of the random variable X is right skewed. Figure 1. PDF plot for GBD-XII (α, θ,, k) Introduing loation parameter μ and sale parameter σ in (1) results in 1 k 1 k f 1, 0, 0, 0, 0, 0, 0, k 0 k

6 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION Reliability measures of GBD-XII random variable X Various reliability measures of GBD-XII random variable X are given by Survival Funtion: r th Moment: r / 0 1 s p X f d a1 1 0 k k 1 k d k 1 r r E k B 1, k r /, 0; k r / ; 0; 0; r 0; 0 where B a, b Failure Rate: 1 Seond Rate of Failure: ab d 1 f k r, 0; k 0; 0; 0; 0 s SRF S 1 log S k k log 0; k 0; 0; 0; 0 1 A disrete generalized Burr XII variable, dx an be viewed as the disrete onentration of the ontinuous generalized Burr-type XII variate X distribution, where the orresponding probability mass funtion of dx an be written as 0 248

7 PARA & JAN 1 p dx p s S k 1 k k log log log 1 The pmf of DGBD-XII takes the form k p log 1 0 log log log 1 1, 2,3, (2) k where e ; 0 1 k 0; 0; 0; 0 Figures 2 through 5 give the pmf plot of (2) for (α = 0.3; θ = 0.1; = 4; β = 0.1), (α = 0.3; θ = 0.1; = 1; β = 0.1), (α = 0.3; θ = 0.01; = 0.5; β = 0.1), and (α = 1; θ = 1; = 2; β = 0.5), respetively. Figure 2. PMF plot for DGBD-XII Figure 3. PMF plot for DGBD-XII (α = 0.3; θ = 0.1; = 4; β = 0.1) (α = 0.3; θ = 0.1; = 1; β = 0.1) 249

8 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION Figure 4. PMF plot for DGBD-XII Figure 5. PMF plot for DGBD-XII (α = 0.3; θ = 0.01; = 0.5; β = 0.1) (α = 1; θ = 1; = 2; β = 0.5) The pmf at = 0 is independent of the shape parameter. It is monotoni dereasing if,, e log log log 2 where,, log 2 log log where e k ;0 1; k 0; 0; 0; 0 otherwise it is no longer monotoni dereasing but is unimodal, having a mode at = 1 i.e., it takes a jump at = 1 and then dereases for all 1.For α = θ = = 1 pmf of disrete generalized Burr-type XII distribution oinides with disrete pareto distribution and for α = θ = 1 DGBD-XII redues to disrete Burr-type XII distribution. To introdue loation parameter μ and sale parameter σ then the disretized version of f( 1 ) is given as 250

9 PARA & JAN P 1 P dx p p X P X where ( 1 ) represents umulative distribution funtion of random variable X. The survival funtion of disrete generalized Burr-type XII random variable dx is given by 1 1 pdx 0 pdx 1 p dx 1 s p dx p dx k log log log a 0,1, 2, a 0; 0; k 0; 0 Thus, survival funtion of disrete generalized DGBD-XII is same as ontinuous GBD-XII for integer points of. The failure rate of disrete generalized Burr-type XII random variable is given by p log 1 r 1 s 0,1, 2, ; a 0; 0; k 0; 0 (3) i.e., the onditional probability that failure ours at a time given that the system has not failed by 1. log 2 Note that r(0) = r(1) gives a (for eample). If α = θ = 1, the log 2 hazard funtion of disrete generalized Burr-type XII distribution oinides with two parameter disrete Burr-type XII distribution. Note that r() is dereasing in if 0 < < a and for = a; r(0) = r(1) and for > a, r(0) > r(1) i.e., monotoni inreasing, and r() < r( 1) > 2 i.e., r() dereases for all > 1, uniformly in β. 251

10 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION For different values of α and θ, an take different values as: a) θ =.5; α = 100 gives = b) θ =.99; α =.5 gives = ) θ = 9999; α =.9 gives = d) θ = 1; α = 1 gives = Taking part a) it ould be seen that for θ =.5; α = 100 gives = (for eample as above) r(0) > r(1), for 0 < < a (for eample when = 1) and for = a; r(0) = r(1) and for > a, r(0) < r(1) implies a monotoni inreasing and r() < r( 1) 2 i.e., r() takes a jump at r = 1 and dereases for all 1, uniformly in β. Similarly, for all other ases where an take different values for different values of α and θ, r() will show its monotoniity aordingly as in the above ase. For disrete generalized Burr-type XII distribution i.e., DGBD-XII(α, θ,, β) SRF log log s log log s log 1 log 1 log log, 0; 0; k 0; 0 1 and to see whether SRF() shows the same monotoniity as r() SRF log log 1 For SRF(0) = SRF(1) log 2 log2 2 (for eample) log 2 252

11 PARA & JAN Using the same proedure as in (3) it is lear that SRF() shows the same monotoniity as that of r(). Figure 6. seond rate of failure Figure 7. seond rate of failure plot of DGBD-XII plot of DGBD-XII (α = 1; θ = 1; = 2; β = 0.1) (α = 1; θ = 1; = 1.585; β = 0.1) Figure 8. seond rate of failure Figure 9. seond rate of failure plot of DGBD-XII plot of DGBD-XII (α = 1; θ = 1; = 0.5; β = 0.1) (α = 1; θ = 1; = 0.1; β = 0.1) Figures 6 through 9 illustrate the alternative hazard rate plot for DGBD-XII for different values of parameters. Note that SRF() is dereasing in if 0 < < a 253

12 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION and for = a; SRF(0) = SRF(1) and for > a, SRF(0) > SRF(1) i.e., monotoni inreasing, and SRF() < SRF( 1) > 2 i.e., SRF() dereases for all > 1. The r th moment of disrete generalized Burr-type XII distribution, i.e. DGBD (α, θ,, β), is as follows r r r E X 0 p 0 r r 1 1 S log log log (4) r log 1 1 k r1 r r 1 k E X r s r log r r Now, E r 1 k 1, k r1 and from (4), E s 1 log log 1 1 and s E 2 log log 2 1 E s E 1 E will take a finite value if k > r, E whih are infinite series and annot be written as losed form. The parameter β of DGBD-XII (α, θ,, β) and the parameter k of GBD-XII (α, θ,, k), are mathed via β = e k. It is therefore observed from the survival funtions of DGBD-XII and GBD-XII distributions log log k k log log d In other words, μ d 1 < μ < μ d where μ and μ d are the means of the ontinuous and disrete generalized Burr XII distributions, respetively. 254

13 PARA & JAN Referenes Barlow, R. E., & Proshan, F. (1975). The statistial theory of reliability and life testing. New York: Holt, Rinehart & Winston. Jazi, M. A., Lai, C. D., & Alamatsaz, M. H. (2010). A disrete inverse Weibull distribution and estimation of its parameters. Statistial Methodology, 7, Kapur, K. C., & Lamberson, L. R. (1997). Reliability in engineering design. New York: John Wiley & Sons. Krishna, H., & Pundir, P. S. (2007). Disrete Mawell Distribution. Interstat. Retrieved from Krishna, H., & Pundir, P. S. (2009). Disrete Burr and disrete Pareto distributions. Statistial Methodology, 6, Lawless, J. F. (1982). Statistial models and methods for lifetime data. New York: John Wiley & Sons. Nakagawa, T., & Osaki, S. (1975). The disrete Weibull distribution. IEEE Transations on Reliability, R-24(5), Olapade, A.K. (2008). On a si parameter generalized Burr XII distribution. Retrieved from Roy, D. (1993). Reliability measures in the disrete bivariate set up and related haraterization results for a bivariate geometri distribution. Journal of Multivariate Analysis, 46, Roy, D. (2003). The disrete normal distribution. Communiations in Statistis Theory and Methods, 32(10), Roy, D. (2004). Disrete Rayleigh distribution. IEEE Transations on Reliability, 53(2), Sinha, S. K. (1986). Reliability and life testing. New Delhi: Wiley Eastern Ltd. Srinath, L. S. (1985). Reliability engineering (4th ed.). New Delhi: East- West Press. Xie, M., Gaudoin, M., & Braquemond, C. (2002). Redefining failure rate funtion for disrete distributions. International Journal of Reliability, Quality and Safety Engineering, 9(3),

14 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION Appendi: Some theorems related to disrete generalized Burr XII distribution Lemma 1 If X is a ontinuous random variable with inreasing (dereasing) failure rate IFR (DFR) distribution, then dx = [X] has a disrete inreasing (dereasing) failure rate difr (ddfr). Proof: (See Roy and Dasgupta, 2001) Lemma 2 If X is a non-negative ontinuous random variable and Y is a non-negative integer valued disrete random variable, then X Y X Y Proof: Note that X Y X Y X Y X Y where the last equality holds beause Y is integer valued. Therefore X Y X Y Theorem 1 If X ~ DGBD-XII (μ = 0; σ = 1; α, θ,, β) then k Y log ~ Geo where e ; 0; 0; k 0; 0 256

15 PARA & JAN Proof: PY y Plog P Y y Plog y y e PX 1 1/ ylog 1 1/ y log e log log log 1 ylog by Lemma 2 y ~ Geo y = 0, 1, 2, As β y is the survival funtion of geometri random variable. Theorem 2 If X ~ GBD-XII (μ = 0; σ = 1; α, θ,, k) then Y = [X] ~ DGBD-XII (μ = 0; σ = 1; α, θ,, β); where β = e k ; α > 0; > 0; k > 0; θ > 0 Proof: P Y y P X y PX y by Lemma 2 k y k log logy Thus, Y = [X]~DGBD-XII (α, θ,, β) y = 0, 1, 2, β = e k ; 0 < β < 1 257

16 DISCRETE GENERALIZED BURR-TYPE XII DISTRIBUTION Theorem 3 If X ~ ep(k), the eponential distribution with failure rate k. Then e Y θ > 0. X log 1/ ~ DGBD-XII (α, θ,, β), where β = e k ; α > 0, > 0, k > 0, Proof: P Y y P Y y X log 1/ e P y X log 1/ e P y P X log y log klog y log e log k k y e log logy by Lemma 2 y = 0, 1, 2, whih is the survival funtion of DGBD-XII (α, θ,, β). Thus, Y ~ DGBD-XII (α, θ,, β). Theorem 4 Let X be a random variable following ontinuous generalized Burr XII distribution with E(X r ) < r = 1, 2, Then E(Y r ) < where Y = [X] ~ DGB (α, θ,, k) Proof: Proof is straightforward, beause 0 [X] X, so learly if E(X r ) < r = 1, 2,, then E([X] 2 ) <. 258