International Journal of Statistical Distributions an Applications 2017; 3(3): 25-31 http://www.sciencepublishinggroup.com/j/ijsa oi: 10.11648/j.ijs.20170303.11 ISSN: 2472-3487 (Print); ISSN: 2472-3509 (Online) A New Discrete amily of Reuce Moifie Weibull Distribution Aemola Lateef Oloko, Osebekwin Ebenezer Asiribo, Ganiyu Abayomi Dawou, Mathew Omonigho Omeike, Nurueen Ayobami Ajai *, Abayomi Olumuyiwa Ajayi Department of Statistics, College of Physical Sciences, eeral University of Agriculture, Abeokuta, Ogun, Nigeria Email aress: ajainurueen2014@yahoo.com (N. A. Ajai) * Corresponing author To cite this article: Aemola Lateef Oloko, Osebekwin Ebenezer Asiribo, Ganiyu Abayomi Dawou, Mathew Omonigho Omeike, Nurueen Ayobami Ajai, Abayomi Olumuyiwa Ajayi. A New Discrete amily of Reuce Moifie Weibull Distribution. International Journal of Statistical Distributions an Applications. Vol. 3, No. 3, 2017, pp. 25-31. oi: 10.11648/j.ijs.20170303.11 Receive: August 16, 2017; Accepte: August 31, 2017; Publishe: October 26, 2017 Abstract: Discretization of continuous lifetime istribution is an interesting an intuitively appealing approach to erive a iscrete lifetime moel. This stuy erive a iscretize form of Reuce Moifie Weibull istribution known as the Marshall-Olkin Discrete Reuce Moifie Weibull (MDRMW) istribution. The mathematical an statistical properties of MDRMW istribution were erive an compare with existing istributions of Discrete Reuce Moifie Weibull istribution (DRMW), Exponentiate Discrete Weibull istribution (EDW) an Two Parameters Discrete Linley istribution (TDL). Maximum likelihoo metho was use to erive the statistics of MDRMW parameters. The Aarset Reliability ataset was fitte for the existing an erive istribution an AIC an olmogorov Smirrnoff (S) were compare. The shape of MDRMW istribution was unimoal an monotonic ecreasing. The plot of hazar rate function coul be ecreasing or bathtub. The AIC an S values of Aarset reliability ata analysis were 483.9 an 0.17579; 507.8 an 0.24435; 485.2 an 0.17897 for MDRMW, DRMW an TDL respectively. The AIC an S values of Leukemia survival ata analysis were 668.2 an 0.11053; 751.9 an 0.39285 respectively. The Aarset reliability ata analysis showe that MDRMW compare favorably with existing istributions. The MDRMW an DRMW hanle Leukemia survival ata set as against EDW an TDL. The values of AIC an S for MDRMW were lower than DRMW, EDW an TDL. This showe that MDRMW was better than the existing istributions. eywors: Weibull, TDL, DRMW, EDW, MDRMW 1. Introuction The Weibull istribution is a lifetime istribution, this make it to be important an esirable. Weibull istribution can be use in ifferent fiels with many applications. Its survival an hazar rate functions have simple expression an its flexibility make it useful to fit ifferent lifetime ata sets in ifferent fiels. The cumulative istribution function (CD) of the twoparameter Weibull istribution is given by the ( ) ( x θ ) x = 1 exp α, x > 0, (1) Where α > 0, an θ > 0 are the scale an shape parameters respectively. The probability ensity function (PD) is given by ( ) xθ ( xθ ) f x An the hazar rate function is = αθ 1 exp α, x > 0. (2) ( ) = x θ 1, x > 0, h x αθ (3) This can either increase; ecrease or constant epening on > 1, < 1 or = 1. or many years, using ifferent techniques, many researchers have evelope various moifie forms of the Weibull istribution to achieve nonmonotonic shapes. Bebbington et al (2007) propose that the hazar rate function of the two-parameter flexible Weibull
26 Aemola Lateef Oloko et al.: A New Discrete amily of Reuce Moifie Weibull Distribution extension can be increasing, ecreasing or bathtub shape. Also Muholkar an Srivastave (1993) propose a twoparameter moel, calle the exponentiate Weibull istribution. The authors are intereste in using the Marshal-Olkin iscretize family to iscretize the moifie Weibull istribution (MW) ue to the wie ability to iscretize any continuous istribution unlike the general metho being use. There is no limit at which the metho can iscretize continuous istributions in as much as the continuous istribution has a survival function. Also, a shape parameter from the Marshal-Olkin family is an ae avantage in the analysis of ata. The propose istribution is referre to as Marshall-Olkin Discrete Reuce Moifie Weibull istribution (MDRMW). The statistical properties of the propose istribution were erive. The propose istribution is compare with existing competing istributions an applie to meical an reliability. The new istribution has a esire characteristic, it has a close form an it can be highly monotonically ecreasing in shape, its hazar function can be bathtub an it is useful in moeling meical an reliability ata. Marshall-Olkin Discrete amily A methoology to a a parameter to obtain a new family of istributions was introuce by Marshall an Olkin (1997). The family of Marshall Olkin istributions with survival function given as ;=, (4) where >0, =1 an is a survival function of the continuous istribution. ere, the authors consier the class of continuous scale family of istributions with scale parameter >0 an enote probability ensity function an the cumulative istribution function of the same by. an. respectively. The survival function of the new scale family of continuous istributions on the positive real line using equation 4 is given by ;,=, (5) where =1. The survival function 5 can be consiere as a generalization of the scale family of istributions an the corresponing family of istributions. Let be a iscrete ranom variable associate to a continuous ranom variable belonging to RMW. The probability mass function is given by! =;=" " = ; +1; " = where >0, =1 $%& '('%(,=0,1,2, (6) 2. Methoology A two-parameter iscrete istribution will be introuce. It will be a new iscrete istribution allowing for bathtub shape hazar rate functions. Its mathematical properties will be iscusse, an its applications to real ata sets will be stuie. The propose istribution will be shown to outperform the existing moels incluing the ones allowing for bathtub shape hazar rate functions. The propose iscrete istribution is base on a twoparameter moification of the Reuce Moifie Weibull (RMW) istribution propose by Almaki, (2014). The twoparameter istribution woul be shown to be flexible, have a bathtub shape hazar rate function. The authors refer to it as the Marshall-Olkin Discrete Reuce Moifie Weibull istribution an enote by MDRMW. 3. Propose Distribution: Marshall-Olkin Discrete Reuce Moifie Weibull Distribution (MDRMW) The cumulative istribution function an survival function of RMW istribution are respectively as follows an respectively. " = =1 +,-. 0 -. 1 23, (7) =+,-. 0 -. 1 23, (8) 45 3-. 6-789 3 : 5 37- -. 6-789 37- : ; '5 3-. -789 3 ('5 37- -. 6-789 37- :( (9) Equation 9 is referre to as Marshall-Olkin Discrete Reuce Moifie Weibull istribution (MDRMW) which is the propose moel. 3.1. Properties of Propose Distribution (MDRMW) The cumulative istribution function an survival function of the iscrete ranom variable having the probability mass function 9 is given by an ;,=,>,?= 537--. 6-789 37- : 5 37- -. ;,=,>,?= 53-. 6-789 3 : 5 3-. -789 3 6-789 37- :,=0,1,2, (10),=0,1,2, (11) 3.2. Shape of the Propose Distribution (MDRMW) In igure 1, it can be seen that the MDRMW istribution is flexible. Its PM can take one of the following shapes: i) a unimoal shape; ii) a monotonic ecreasing shape; iii) a
International Journal of Statistical Distributions an Applications 2017; 3(3): 25-31 27 ecreasing shape followe by an increasing shape followe by a ecreasing. Shape of MDRMW probability mass function 0.00 0.02 0.04 0.06 0.08 0.10 0.12 beta=2 beta=10 beta=0.15 0 5 10 15 20 x value igure 1. The shape of pmf of the propose istribution. 3.3. Series Expansion of the Propose Distribution ere, a series expansion for the survival function is erive, an hence for the probability mass function. Using the series expansion for the exponential, it can be written as = -. %@A 3 CD E 5 E. IJ 6 : > ' L CD L A! IJ MIJ ( M! (12) Now using the partial exponential Bell polynomial, N O (Comtet, 1974), efine by IJ PQ O R O S! T OI IJ V U! =QN O R O S! OI =Q WX = Y Q4 Z Z! U ; > C U!QN O ln?,ln?, W! CI ]^_ `aa,`aa, = CDE 5 E. CI C%Ẹ. (13) C! The representation in 13 can be use to erive similar expansions for the probability mass function as the following cqqq WX = Y N O ln?,ln?, CI C% Y QQQ WX = Y N O ln?,ln?, +1 C% Y CI
28 Aemola Lateef Oloko et al.: A New Discrete amily of Reuce Moifie Weibull Distribution c1 QQQ WX = Y N O ln?,ln?, IJIJ CI using the binomial expansion for non-integer powers: where?",g= hhhy hi%, i!! j=cqqq WX = Y N O ln?,ln?, IJIJCI k=c1 QQQ WX = Y N O ln?,ln?, Then CI C% Y c1 QQQ WX = Y N O ln?,ln?, +1 C% Y IJIJCI e+n f =Q?",ge hi N h, C% Y C% Y iij QQQQ?'Z 2 +W,"(WX = Y N O ln?,ln?, IJIJ CI hij c1 QQQQ?'Z 2 +W,"(WX = Y N O ln?,ln?, CI hij Y %Ch Y %Ch no c E p3 E. q^_ rs9,rs9, n7ẹ 94 Ẹ 7n,x;no E E p3 E. u u q^_ rs9,rs9, E u E Evw _vw nv_ u. 7ntx Et_n! Evw _vw nv_ u xvw Et_n! l = m E no y u E p3 E. q^_ rs9,rs9, n7ẹ 94 Ẹ 7n,x;no u E p3 E. q^_ rs9,rs9, E E _ nv_ zc u E u u. 7ntx Et_n! E _ nv_ xvw Et_n! (14) 3.4. azar Rate unction of the Propose Distribution The azar rate function is ",. h= =- %@A 3 = %-. 6%@A 37- : = -. %@A 3 61 = %-. %@A 37- : Shape of hazar function of MDRMW azar rate function 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 x value igure 2. The ecreasing hazar shape of the propose istribution.
International Journal of Statistical Distributions an Applications 2017; 3(3): 25-31 29 Shape of hazar function of MDRMW azar rate function 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 x value igure 3. The ecreasing then increasing hazar shape. 3.5. The Probability Generating unction of the Propose Distribution The probability generating function (pgf) of iscrete ranom variable having the MDRMW =,>,?, is given by " R=1+R 1QR 1, I =1+R 1QR I = -. %@A 3 1 = -. %@A 3 mean an variance of the ranom variable having MDRMW =,>,?, }=,>,?,=~ =Q =-. = I2 1 I %@A 3 1 =, -. %@A 3 6-789 3 Y : I ; 5 3-. -789 3 5 3-. 6-789 3 : 4 5 3-. 5 3-. -789 3 3.6. Recurrence Relation of Probability Generating unction of the Propose Distribution (15) The recurrence relation for generating probabilities given by " % = +1 +2 +1 where " J =. then, " = 537--. 6-789 37-: 5 37.-. 6-789 37. : 1. 1 +2 ",=0,1,2 5 3-. 6-789 3 :. 5 3-. -789 3 5 37--. 6-789 37- : 5 37.-. 6-789 37. : ",= 0,1,2 (16) where " J = 5-789 5-789. 3.7. Maximum Likelihoo Estimation of the Propose Distribution (MDRMW) The maximum likelihoo estimation (MLE) is use to estimate unknown parameters. Consier a ranom sample, Y,, D from the Marshall-Olkin Discrete Reuce Moifie Weibull (MDRMW) istribution. The likelihoo function is given by D logƒ=xlog+ log'= L -. MI %@A 3L = L% -. %@A 3 L 7- ( D log'1 = L -. MI %@A 3 L 7- ( %@A 3 L ( D log'1 = L% -. MI (17) Two applications using well-known ata sets were use to emonstrate the robustness an applicability of the propose moel. These ata present ifferent egrees of skewness an kurtosis. The new istribution was compare with the existing istributions which are Discrete Reuce Moifie Weibull istribution (DRMW) (Amalki, 2014), Two parameter Discrete Linley istribution (TDL) (Tassaaq et al., 2016). 4. Numerical Illustration The gooness of fit statistic an the gooness of fit plot were provie in orer to check the moel that best fit the ata among the moels for each ataset use for this research. Two real ata sets were consiere in this section. One of them has bathtub shape hazar rate functions while the secon one has an increasing hazar rate function.
30 Aemola Lateef Oloko et al.: A New Discrete amily of Reuce Moifie Weibull Distribution 4.1. Discrete Aarset Data The ata are integer parts of the lifetimes of fifty evices. The ata are liste in Table 1. Nooghabi et al. (2011) ha shown that the DMW istribution provies a goo fit for this ata. Table 1. Aarset ata ( in weeks). Time of failure 0 1 2 3 6 7 11 12 18 21 No of failure 2 5 1 1 1 1 1 1 5 1 Time of failure 32 36 40 45 46 47 50 55 60 63 No of failure 1 1 1 1 1 1 1 1 1 2 Time of failure 67 72 75 79 82 83 84 85 86 No of failure 4 1 1 1 2 1 3 5 2 Table 2 shows the Maximum Likelihoo Estimation (MLEs) of the parameters an their stanar errors. Table 3 shows the AIC an BIC values for the fitte MDRMW, DRMW, TDL an EDW istributions. Table 2. The Maximum Likelihoo Estimation of the MDRMW istribution for the Aarset Data an Stanar Error. MODEL 0.6183 18.682 2.5905 0.1776 MDRMW 0.0443 10.7933 4.1552 0.2084 0.8475 7.4787 0.0817 DRMW 0.0198 22.5427 0.2335 0.9687 0.0275 TDL 0.0051 0.0234 Table 3. The AIC an BIC for the fitte istributions. MODEL -2Log-Likelihoo AIC BIC MDRMW 475.9 483.9 491.5 DRMW 501.8 507.8 513.5 TDL 481.2 485.2 489.0 4.2. Leukemia Data The ata set for this example was collecte from the Ministry of ealth ospital in Saui Arabia. The ata are lifetimes in ays of forty three bloo patients who ha leukemia. The ata set exhibits an increasing hazar rate. Table 4. The Leukemia ata. 115 181 255 418 441 461 516 739 743 789 807 865 1062 924 983 1025 1063 1165 1191 1222 1222 1251 1277 1290 1578 1357 1369 1408 1478 1549 1455 1578 1599 1603 1605 1696 1735 1799 1815 1852 1899 1925 1965 Table 5. The Maximum Likelihoo Estimation of the MDRMW istribution for the Leukemia Data an Stanar Errors. MODEL MDRM 2 0.8091(2.37 10 ) 3 1.494.8(1.615 10 ) 3 2214.9(6.78 10 ) 2 0.9274(2.676 10 ) DRMW 0.9706 1.0197 0.8921 0.0044 0.0548 Table 6. The AIC an BIC for the fitte istributions. MODEL -2Log-Likelihoo AIC BIC MDRMW 660.2 668.2 675.3 DRMW 745.9 751.9 757.2 5. Conclusion A new istribution calle Marshall-Olkin Discrete Reuce Moifie Weibull istribution (MDRMW) was propose an its properties stuie. In this work, the new iscrete RMW was use to analyze all the atasets use by Amalki, (2014) in orer to properly compare the new istribution with DRMW of Amalki. In aition, two most recent istributions which are Exponentiate Discrete Weibull istribution an Two Parameter iscrete Linley istribution (TDL) propose by Tassaaq et al., (2016) were also use to analyze the atasets. The MDRMW istribution is flexible to moel iscrete ata such as survival ata an over-isperse ata. Its hazar function is monotonically ecreasing, followe by an increasing shape an upsie own bathtub. The close form expressions for the moments, istribution of orer statistics were obtaine. Maximum likelihoo estimation technique was use to estimate the moel parameters. References [1] Almalki, S. J. (2014). Statistical analysis of lifetime ata using new moifie Weibull istributions, a thesis submitte to the University of Manchester for the egree of Doctor of Philosophy in the aculty of Engineering an Physical Sciences. [2] Bebbington, M., Lai, C. D. an Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering an System Safety, 92, 719-726. [3] Comtet, L. (1974). Avance Combinatorics: The art of finite an infinite expansions. Springer. [4] Marshall, A. W., Olkin, I. (1997). A new metho for aing a parameter to a family of istributions with application to the exponential an Weibull families, Biometrika 84(3), 641-652.
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