A New Class of Generalized Power Lindley. Distribution with Applications to Lifetime Data

Transcription

1 Theoretical Mathematics & Applications, vol.5, no.1, 215, ISSN: (print, (online Scienpress Ltd, 215 A New Class of Generalized Power Lindley Distribution with Applications to Lifetime Data Mavis Pararai 1, Gayan Warahena-Liyanage 2 and Broderick O. Oluyede 3 Abstract A new class of distribution called the beta-exponentiated power Lindley (BEPL distribution is proposed. This class of distributions includes the Lindley (L, exponentiated Lindley (EL, power Lindley (PL, exponentiated power Lindley (EPL, beta-exponentiated Lindley (BEL, beta-lindley (BL, and beta-power Lindley distributions (BPL as special cases. Expansion of the density of BEPL distribution is obtained. Some mathematical properties of the new distribution including hazard function, reverse hazard function, moments, mean deviations, Lorenz and Bonferroni curves are presented. Entropy measures and the distribution of the order statistics are given. The maximum likelihood estimation technique is used to estimate the model parameters. Finally, real data examples are discussed to illustrate the usefulness and applicability of the proposed distribution. Mathematics Subject Classification: 6E5; 62E15 1 Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA, 1575, USA. pararaim@iup.edu 2 Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA, 1575, USA. g.j.warahenaliyanage@iup.edu 3 Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 346,USA. boluyede@georgiasouthern.edu Article Info: Received : August 7, 214. Revised : September 19, 214. Published online : January 1, 215.

2 54 A New Class of Generalized Power Lindley Distribution Keywords: Exponentiated power Lindley distribution; Power Lindley distribution; Lindley distribution; Beta distribution; Maximum likelihood estimation 1 Introduction Lindley [1] developed Lindley distribution in the context of fiducial and Bayesian statistics. Properties, extensions and applications of the Lindley distribution have been studied in the context of reliability analysis by Ghitany et al. [2], Zakerzadeh and Dolati [3], and Warahena-Liyanage and Pararai [4]. Several other authors including Sankaran [5], Asgharzadeh et al. [6] and Nadarajah et al. [7] proposed and developed the mathematical properties of various generalized Lindley distributions. (pdf of the Lindley distribution is given by f(y; β = β2 (1 + ye βy, y >, β >. The probability density function The power Lindley (PL distribution proposed by Ghitany et al. [8] is an extension of the Lindley (L distribution. Using the transformation X = Y 1 α, Ghitany et al. [8] derived and studied the power Lindley (PL distribution with the probability density function (pdf given by f(x; α, β = αβ2 (1 + xα x α 1 e βxα, x >, α >, β >. The cumulative distribution function (cdf of the power Lindley distribution is F (x = 1 S(x = 1 (1 + βxα e βxα for x >, α, β >. Warahena-Liyanage and Pararai [4] studied the properties of the exponentiated Power Lindley (EPL distribution. The EPL cdf and pdf are given by G EP L (x; α, β, ω = [ ] ω 1 (1 + βxα e βxα (1.1

3 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 55 and g EP L (x; α, β, ω = αβ2 ω (1 + xα x α 1 e βxα [ ] ω 1 1 (1 + βxα e βxα (1.2, respectively, for x >, α >, β >, ω >. The hazard rate function of the EPL distribution is given by h GEP L (x; α, β, ω = = g(x; α, β, ω G(x; α, β, ω ( ] ω 1 αβ 2 ω (1 + β+1 xα x α 1 e [1 βxα 1 + βxα e βxα β+1 [ ] ω. 1 1 (1 + e βxα βxα The r th moment of the EPL distribution is given by E(X r = i= j+1 i ( ( ( ω 1 i j + 1 ( 1 i β j k rα 1 +1 Γ(k + rα i j k ( i+1 (i + 1 (k+rα 1 +1 j= k= The purpose of this paper is to develop a five-parameter alternative to several lifetime distributions including the gamma, Weibull, exponentiated Weibull, exponentiated Lindley, lognormal, beta Weibull geometric (BWG [9], and beta Weibull Poisson (BWP [1] distributions. In this context, we propose and develop the statistical properties of the beta exponentiated power Lindley (BEPL distribution and show that it is a competitive model for reliability analysis. Our aim in this paper is to discuss some important statistical properties of the BEPL distribution. This discussion includes the shapes of the density, hazard rate and reverse hazard rate functions, moments, moment generating function and parameter estimation by using the method of maximum likelihood. Finally, applications of the model to real data sets in order to illustrate the applicability and usefulness of the BEPL distribution are presented. This paper is organized as follows. In section 2, the model, sub-models and some of its statistical properties including shapes and behavior of the hazard function are presented. Moments, conditional moments, reliability and related measures are given in section 3. Mean deviations, Bonferroni and Lorenz curves are presented in section 4. Section 5 contains distribution of order statistics and measures of uncertainty. In section 6, we present the maximum likelihood

4 56 A New Class of Generalized Power Lindley Distribution method for estimating the parameters of the distribution. Applications are given in section 7 followed by concluding remarks. 2 The Model, Sub-models and Some Properties In this section, we present the BEPL distribution and derive some properties of this class of distributions including the cdf, pdf, expansion of the density, hazard and reverse hazard functions, shape and sub-models. Let G(x denote the cdf of a continuous random variable X and define a general class of distributions by F (x = B G(x(a, b, (2.1 B(a, b where B G(x (a, b = G(x t a 1 (1 t b 1 dt and 1/B(a, b = Γ(a + b/γ(aγ(b. The class of generalized distributions above was motivated by the work of Eugene et al. [11]. They proposed and studied the beta-normal distribution. Some beta-generalized distributions discussed in the literature include work by Jones [12], Bidram et al. [9]. Nadarajah and Kotz [13], Nadarajah and Gupta [14], Nadarajah and Kotz [15], Barreto-Souza et al. [16] proposed the beta-gumbel, beta-frechet, beta-exponential (BE, beta-exponentiated exponential (BEE distributions, respectively. Gusmao et al. [17] presented results on the generalized inverse Weibull distribution. Pescim et al. [18] proposed and studied the beta-generalized half-normal distribution which contains some important distributions such as the half-normal and generalized half normal (Cooray and Ananda [19] as special cases. Furthermore, Cordeiro et al. [2] presented the generalized Rayleigh distribution and Carrasco et al. [21] studied the generalized modified Weibull distribution with applications to lifetime data. More recently, Oluyede and Yang [22] studied the beta generalized Lindley distribution with applications. By considering G(x as the cdf of EPL distribution we obtain the betaexponentiated power Lindley (BEPL distribution with a broad class of distributions that may be applicable in a wide range of day to day situations

5 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 57 including applications in medicine, reliability and ecology. The cdf and pdf of the five-parameter BEPL distribution are given by F BEP L (x; α, β, ω, a, b = 1 B(a, b GEP L (x;α,β,ω t a 1 (1 t b 1 dt = B G(x(a, b, (2.2 B(a, b and f BEP L (x; α, β, ω, a, b = = 1 B(a, b [G EP L(x] a 1 [1 G EP L (x] b 1 g EP L (x, αβ 2 ω B(a, b( (1 + xα x α 1 e βxα [ ] ωa 1 1 (1 + βxα e βxα { [ ] ω } b (1 + βxα e βxα, (2.3 respectively, for x >, α >, β >, ω >, a >, b >. Plots of the pdf of BEPL distribution for several combinations of values of α, β, ω, a and b are given in Figure 2.1. The plots indicate that the BEPL pdf can be decreasing or right skewed. The BEPL distribution has a positive asymmetry. Figure 2.1: Plots of the PDF for different values of α, β, ω, a and b 2.1 Expansion of density The expansion of the pdf of BEPL distribution is presented in this section.

6 58 A New Class of Generalized Power Lindley Distribution For b > a real non-integer, we use the series representation ( b 1 (1 G EP L (x b 1 = ( 1 i [G EP L (x] i, (2.4 i i= where G EP L (x; α, β, ω = [ ] ω 1 (1 + βxα e βxα. If a is an integer, from Equation (2.3 and the above expansion (2.4, we can rewrite the density of the BEPL distribution as f BEP L (x; α, β, ω, a, b = g EP L(x ( b 1 B(a, b i i= = αβ2 ω (1 + xα x α 1 e βxα ( 1 i( b 1 i B(a, b i= i= ( 1 i [G EP L (x] a+i 1 (2.5 [ 1 (1 + βxα [ 1 (1 + βxα e βxα ] ω 1 e βxα ] ω(a+i 1 = αβ2 ω (1 + xα x α 1 e βxα ] ω(a+i 1 l i [1 (1 + βxα e βxα, (2.6 where the coefficients l i are l i = l i (a, b = ( 1i( b 1 i B(a, b and i= l i = 1, for x >, α >, β >, ω >, a >, b >. If a is real non-integer, we can expand [G EP L (x] a+i 1 as follows: [G EP L (x] a+i 1 = {1 [1 G EP L (x]} a+i 1 ( a + i 1 = ( 1 j [1 G EP L (x] j, j j= with [1 G EP L (x] j = j k= ( j ( 1 k [G EP L (x] k, k

7 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 59 so that [G EP L (x] a+i 1 = j= k= j ( a + i 1 j ( j k ( 1 j+k [G EP L (x] k. (2.7 From Equations (2.5 and (2.7, the BEPL density can be rearranged in the form f BEP L (x; α, β, ω, a, b = g EP L (x where the coefficients l i,j,k are i,j= k= j l i,j,k [G EP L (x] k (2.8 = αβ2 ω (1 + xα x α 1 e βxα j ] ω(k+1 1 l i,j,k [1 (1 + βxα e βxα, i,j= k= l i,j,k = l i,j,k (a, b = ( 1i+j+k( b 1 i B(a, b ( a+i 1 j ( j k and i,j= j k= l i,j,k = 1, for x >, α >, β >, ω >, a >, b >. Hence, for any real non-integer, the BEPL density is given by three (two infinite and one finite weighted power series sums of the baseline cdf G EP L (x. By changing j j= k= to k= j=k in Equation (2.8, we obtain f BEP L (x; α, β, ω, a, b = g EP L (x where the coefficient p i is i,k= p i [G EP L (x] k i,k= = αβ2 ω (1 + xα x α 1 e βxα ] ω(k+1 1 p i [1 (1 + βxα e βxα, with p i = p i (a, b = ( 1i( b 1 qk (a + i 1 i, B(a, b q k = q k (a + i 1 = ( ( a + i 1 j ( 1 j+k, j k j=k

8 6 A New Class of Generalized Power Lindley Distribution for x >, α >, β >, ω >, a >, b >, respectively. Note that the BEPL density is given by three infinite weighted power series sums of the baseline distribution function G EP L (x. When b > is an integer, the index i in the previous series representation stops at b Some sub-models of the BEPL distribution Some sub-models of the BEPL distribution for selected values of the parameters α, β, ω, a and b are presented in this section. (1 a = b = 1 When a = b = 1, we obtain the exponentiated power Lindley (EPL distribution whose cdf and pdf are given in (1.1 and (1.2, (Warahena- Liyanage and Pararai [4]. (2 ω = 1 When ω = 1, we obtain the beta-power Lindley (BPL distribution. The BPL cdf is given by F BP L (x; α, β, a, b = 1 B(a, b GP L (x;α,β t a 1 (1 t b 1 dt for x >, α >, β >, a >, b >. The corresponding pdf is given by f BP L (x; α, β, a, b = αβ 2 B(a, b( (1 + xα x α 1 e βxα [ ] a 1 ] b 1 1 (1 + βxα e [(1 βxα + βxα e βxα for x >, α >, β >, a >, b >. (3 α = 1 When α = 1, we obtain beta-exponentiated Lindley (BEL distribution (Oluyede and Yang [22]. The BEL cdf is given by F BEL (x; β, ω, a, b = 1 B(a, b GEL (x;β,ω t a 1 (1 t b 1 dt

9 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 61 for x >, β >, ω >, a >, b >. The corresponding pdf is given by f BEL (x; β, ω, a, b = β 2 ω (1 + xe βx B(a, b( [ ( βx ] ωa 1 e βx { [ 1 1 ( 1 + βx e βx ] ω } b 1 for x >, β >, ω >, a >, b >. (4 ω = α = 1 When ω = α = 1, we obtain beta-lindley (BL distribution (Oluyede and Yang [22]. The BL cdf and pdf are given by and F BL (x; β, a, b = 1 B(a, b GL (x;β,ω t a 1 (1 t b 1 dt f BL (x; β, a, b = β 2 (1 + xe βx B(a, b( [ ( βx ] a 1 [( e βx 1 + βx ] b 1 e βx, respectively, for x >, β >, ω >, a >, b >. (5 ω = a = b = 1 When ω = a = b = 1, we obtain the power Lindley (PL distribution (Ghitany et al. [8]. The PL cdf and pdf are respectively given by F P L (x; α, β = 1 (1 + βxα e βxα and f P L (x; α, β = β 2 ( (1 + xα x α 1 e βxα for x >, α >, β >.

10 62 A New Class of Generalized Power Lindley Distribution (6 α = a = b = 1 When α = a = b = 1, we obtain exponentiated-lindley (EL distribution. The EL cdf is given by F EL (x; β, ω = [ ( βx ] ω e βx for x >, β >, ω >. The corresponding pdf is given by [ ( β 2 ω f EL (x; β, ω = (1 + xe βx βx ] ω 1 e βx ( for x >, β >, ω >. (7 α = ω = a = b = 1 When α = ω = a = b = 1, we obtain Lindley distribution. The Lindley cdf and pdf are respectively given by ( F L (x; β = βx e βx and for x >, β >. f L (x; β = β 2 (1 + xe βx ( (8 ω = a = 1 When ω = a = 1, the cdf of BEPL distribution reduces to ] b F BP L (x; α, β, b = 1 [(1 + e βxα βxα for x >, α >, β >, b >. The corresponding pdf is ] b 1 f BP L (x; α, β, b = bαβ2 ( (1 + xα x α 1 e [(1 βxα + βxα e βxα for x >, α >, β >, b >. (9 α = a = 1 When α = a = 1, the cdf of BEPL distribution reduces to { [ ( F BEL (x; α, β, b = βx ] ω } b e βx

11 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 63 for x >, β >, ω >, b >. The corresponding pdf is given by f BEL (x; β, ω, b = bωβ2 (1 + xe βx ( [ ( βx ] ω 1 e βx { [ ( βx ] ω } b 1 e βx, for x >, β >, ω >, b >. distribution with parameters β, ω and b. (1 α = ω = a = 1 This is the Kumaraswamy Lindley When α = ω = a = 1, the cdf of BEPL distribution reduces to F BL (x; β, b = 1 [( 1 + βx ] b e βx for x >, β >, b >. The corresponding pdf is given by f BL (x; β, b = for x >, β >, b >. [( bβ 2 (1 + xe βx 1 + βx ] b 1 e βx ( 2.3 Hazard and Reverse Hazard Functions The hazard and reverse hazard functions of the BEPL distribution are presented in this section. Graphs of these functions for selected values of parameters α, β, ω, a and b are also presented. The hazard and reverse hazard functions of the BEPL distribution are given respectively by and h BEP L (x; α, β, ω, a, b = f BEP L(x; α, β, ω, a, b F BEP L (x; α, β, ω, a, b = g EP L(x [G EP L (x] a 1 [1 G EP L (x] b 1 B(a, b B GEP L (x(a, b

12 64 A New Class of Generalized Power Lindley Distribution τ BEP L (x; α, β, ω, a, b = f BEP L(x; α, β, ω, a, b F BEP L (x; α, β, ω, a, b = g EP L(x [G EP L (x] a 1 [1 G EP L (x] b 1, B GEP L (x(a, b for x >, α >, β >, ω >, a >, b >, where G EP L (x and g EP L (x are the cdf and pdf of the EPL distribution given by Equations (1.1 and (1.2, respectively. Plots of the hazard function for selected values of parameters α, β, ω, a and b are given in Figures 2.2 and 2.3. The graphs of the hazard function for several combinations of the parameters represent various shapes including monotonically increasing, monotonically decreasing, bathtub and upside down bathtub shapes. This attractive flexibility makes BEPL hazard rate function useful and suitable for non-monotone empirical hazard behaviors which are more likely to be encountered or observed in real life situations. Figure 2.2: Plots of the hazard function for different values of α, β, ω, a and b 2.4 Monotonicity Properties The monotonicity properties of the BEPL distribution are discussed in this section. Let V (x = G P L (x; α, β = 1 (1 + βxα e βxα.

13 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 65 Figure 2.3: Plots of the hazard function for different values of α, β, ω, a and b From Equation (2.3 we can rewrite the BEPL pdf as f BEP L (x; α, β, ω, a, b = αβ 2 ω B(a, b( (1 + xα x α 1 e βxα [V (x] ωa 1 [1 V ω (x] b 1 for x >, α >, β >, ω >, a >, b >. It follows that ( αβ 2 ω log f BEP L (x = log + log(1 + x α + (α 1 log(x βx α B(a, b( + (ωa 1 log V (x + (b 1 log [1 V α (x], (2.9 and d log f BEP L (x dx = αxα x + α 1 αβx α 1 α x + (ωa 1(1 V ω (x ω(b 1V ω (x V (x. V (x [1 V ω (x] (2.1 Substituting V (x = dv (x/dx = (αβ 2 /((1 + x α x α 1 e βxα into Equation (2.1, we have d log f BEP L (x dx = αxα x + α 1 αβx α 1 + αβ2 α x (1 + xα x α 1 e βxα { } (ωa 1(1 V ω (x ω(b 1V ω (x. V (x [1 V ω (x]

14 66 A New Class of Generalized Power Lindley Distribution Since α >, β >, ω >, a > and b >, we have V (x = If x, then If x, then dv (x dx V (x = 1 V (x = 1 = αβ2 (1 + xα x α 1 e βxα >, x >. (2.11 (1 + βxα e βxα. (1 + βxα e βxα 1. Thus V (x is monotonically increasing from to 1. Note that, since < V (x < 1, < V ω (x < 1, ω >, < 1 V ω (x < 1, ω > and V (x >, we have V (x/v (x[1 V ω (x] >. If α 1/2, ωa < 1 and b > 1. we obtain d log f BEP L (x dx = αxα x + α 1 αβx α 1 α x + (ωa 1(1 V ω (x ω(b 1V ω (x V (x < V (x [1 V ω (x] (2.12 since [αx α 1 /(1 + x α ] + [(α 1/x] = [(2α 1x α + (α 1]/x(1 + x α <, (ωa 1(1 V ω (x ω(b 1V ω (x < and V (x/v (x[1 V ω (x] >. In this case, f BEP L (x; α, β, ω, a, b is monotonically decreasing for all x. If α > 1/2, f BEP L (x; α, β, ω, a, b could attain a maximum, a minimum or a point of inflection according to whether d 2 log f BEP L (x dx 2 <, d 2 log f BEP L (x dx 2 > or d 2 log f BEP L (x dx 2 =. 3 Moments, Conditional Moments and Reliability In this section, moments, conditional moments and reliability and related measures including coefficients of variation, skewness and kurtosis of the BEPL

15 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 67 distribution are presented. A table of values for mean, variance, coefficient of skewness (CS and coefficient of kurtosis (CK is also presented. 3.1 Moments The r th moment of the BEPL distribution, denoted by µ r is given by µ r = E(X r = x r f BEP L (xdx for r =, 1, 2,.... In order to find the moments of the BEPL distribution, consider the following lemma. Lemma 3.1. Let L 1 (α, β, m, r = then (1 + x α x α+r 1 [1 ] m 1 (1 + βxα e βxα e βxα dx, L 1 (α, β, m, r = j k+1 ( m 1 j j= k= l= ( j k ( k + 1 l ( 1 j β k Γ(l + rα α( j [β(j + 1] (l+rα Proof. Using the series expansion (1 z a 1 = where z < 1 and b > is a real non-integer, we have ( a 1 ( 1 i z i, (3.1 i i= L 1 (α, β, m, r = ( m 1 [ ] 1 + β(1 + x ( 1 j α j e jβxα (1 + x α x α+r 1 e βxα dx j j= = = ( m 1 ( 1 j j= j ( j ( j m 1 j= j k= ( j k j k= k+1 l= ( j β k (1 + x α k+1 x α+r 1 e ( jβxα βx α dx k ( k + 1 ( 1 j β k l ( j x α+αl+r 1 e ( jβxα βx α dx.

16 68 A New Class of Generalized Power Lindley Distribution Now consider, x α+αl+r 1 e ( jβxα βx α dx. (3.2 Let u = β(j + 1x α, then du dx = αβ(j + 1xα 1 and x = Consequently, [ ] 1/α u. β(j + 1 L 1 (α, β, m, r = j k+1 ( m 1 j j= k= l= ( j k ( k + 1 l ( 1 j β k Γ(l + rα α( j [β(j + 1] (l+rα Therefore, the r th moment of the BEPL distribution from equation (2.6 is given by µ r = αβ 2 ω ( b 1 ( 1 i B(a, b( i i= ] ω(a+i 1 x r (1 + x α x [1 α 1 (1 + βxα e βxα e βxα dx. Now, using Lemma 3.1 with m = ω(a + i, we have µ αβ 2 ω ( b 1 r = ( 1 i L 1 (α, β, ω(a + i, r. (3.3 B(a, b( i i= The mean, variance, coefficient of variation (CV, coefficient of skewness (CS and coefficient of kurtosis (CK are given by µ = µ αβ 2 ω ( b 1 1 = ( 1 i L 1 (α, β, ω(a + i, 1, (3.4 B(a, b( i i= and σ 2 = µ 2 µ 2, (3.5 CV = σ µ µ = 2 µ 2 µ 2 = 1, (3.6 µ µ 2 CS = E [(X µ3 ] [E(X µ 2 ] 3/2 = µ 3 3µµ 2 + 2µ 3 (µ 2 µ 2 3/2, (3.7 CK = E [(X µ4 ] [E(X µ 2 ] 2 = µ 4 4µµ 3 + 6µ 2 µ 2 3µ 4 (µ 2 µ 2 2, (3.8

17 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 69 respectively. Table 3.1 lists the first six moments of the BEPL distribution for selected values of the parameters by fixing α = 1.5, β = 1. and ω = 1.5. These values can be determined numerically using R and MATLAB. Algorithms to calculate the pdf moments, reliability, mean deviations, Rényi entropy, maximum likelihood estimators and variance-covariance matrix of the BEPL distribution are provided in the appendix. Table 3.1: Moments of the BEPL distribution for some parameter values; α = 1.5, β = 1. and ω = 1.5 µ s a =.5, b = 1.5 a = 1.5, b = 1.5 a = 1.5, b = 2.5 a = 2.5, b = 1.5 µ µ µ µ µ µ Variance Skewness Kurtosis Conditional Moments For lifetime models, it is useful to know the conditional moments defined as E(X r X > x. In order to calculate the conditional moments, we consider the following lemma: Lemma 3.2. Let L 2 (α, β, m, r, t = then L 2 (α, β, m, r, t = t (1 + x α x α+r 1 [1 j k+1 ( m 1 j j= k= l= ( j k ] m 1 (1 + βxα e βxα e βxα dx. ( k + 1 ( 1 j β k Γ(l + rα 1 + 1, β(j + 1t α, l α( j [β(j + 1] (l+rα 1 +1

18 7 A New Class of Generalized Power Lindley Distribution where Γ(a, t = x a 1 s x dx is the upper incomplete gamma function. t Proof. Using the same procedure that was used in Lemma 3.1, this can be simplified into the following form. L 2 (α, β, m, r, t = t j k+1 ( m 1 j j= k= l= t ( j k ( k + 1 l ( 1 j β k ( j (3.9 x α+αl+r 1 e ( jβxα βx α dx. (3.1 Now consider, x α+αl+r 1 e ( jβxα βx α dx, and let u = β(j + 1x α, then [ ] 1/α du = αβ(j + u dx 1xα 1 and x =. The above integral can be rewritten by using the complementary incomplete gamma function Γ(a, t = β(j + 1 x a 1 e x dx. t Consequently, L 2 (α, β, m, r, t = j k+1 ( m 1 j j= k= l= ( j k ( k + 1 ( 1 j β k Γ(l + rα 1 + 1, β(j + 1t α. l α( j [β(j + 1] (l+rα 1 +1 Now using Lemma 3.2, the r th conditional moment of the BEPL distribution is given by E(X r αβ 2 ω ( b 1 X > x = B(a, b( i i= = αβ2 ω ( b 1 ( i i= ( 1 i L 2 (α, β, ω(a + i, r, x 1 F BEP L (x; α, β, ω, a, b ( 1 i L 2 (α, β, ω(a + i, r, x B(a, b B GELP (x(a, b. The mean residual lifetime function is given by E(X X > x x. The moment generating function (MGF of the BEPL distribution is given by M X (t = αβ 2 ω B(a, b( i= n= ( b 1 i ( 1 i tn n! L 1(α, β, ω(a + i, n. ( Reliability We derive the reliability R when X and Y have independent BEPL(α 1, β 1, ω 1, a 1, b 1 and BEPL(α 2, β 2, ω 2, a 2, b 2 distributions, respectively. Note from Equation

19 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 71 (2.2 that the BEPL cdf can be written as: F BEP L (x; α, β, ω, a, b = = 1 B(a, b 1 B(a, b GEP L (x;α,β,ω ( b 1 ( 1 j j a + j j= t a 1 (1 t b 1 dt, [ ] ω(a+j 1 (1 + βxα e βxα. (3.12 Now, from Equations (3.12 and (2.3, we obtain R = P (X > Y = = [ 1 f X (x; α 1, β 1, ω 1, a 1, b 1 F Y (x; α 2, β 2, ω 2, a 2, b 2 dx α 1 β1ω 2 1 B(a 1, b 1 (β (1 + xα 1 x α1 1 e β 1xα1 ( 1 + β ] 1x α 1 ω1 a 1 1 e β 1x α 1 β { [ B(a 2, b 2 ( 1 + β 1x α 1 β ( b2 1 ( 1 j j a 2 + j j= ] ω1 } b1 1 e β 1x α 1 We apply the following series representations: [ ( β ] 1x α 1 ω1 a 1 1 e β 1x α 1 β = = [ ( β ] 2x α 2 ω2 (a 2 +j e β 2x α 2 dx. β (3.13 ( ( ω1 a 1 1 ( 1 k 1 + β 1x α 1 k e β 1kx α 1 k β k ( ( ω1 a 1 1 k ( 1 k β1 m x mα 1 k m (β m e β 1kxα1, k= k= m= (3.14 { [ ( β ] 1x α 1 ω1 } b1 1 e β 1x α 1 β = l= p= n= p ( b1 1 l ( ω1 l p ( 1l+p β n 1 x nα 1 (β n e β 1pxα1 ( p n (3.15

20 72 A New Class of Generalized Power Lindley Distribution and [ ( β ] 2x α 2 ω2 (a 2 +j e β 2x α 2 β = q= q ( ω2 (a 2 + j t= q ( q t ( 1 q β t 2x tα 2 (β 2 t e β 2qxα2. (3.16 By substituting Equations (3.14, (3.15 and (3.16 into Equation (3.13, we obtain R = = α 1 β1ω 2 1 B(a 1, b 1 (β (1 + xα 1 x α1 1 e β 1xα1 k ( ( ω1 a 1 1 k ( 1 k β1 m x mα 1 k m (β k= m= m e β 1kxα1 p ( ( ( b1 1 ω1 l p ( 1 l+p β1 n x nα 1 e β 1pxα1 l p n (β l= p= n= n 1 ( b2 1 ( 1 j B(a 2, b 2 j a j= 2 + j q ( ( ω2 (a 2 + j q ( 1 q β2x t tα 2 e β 2qxα2 dx q t (β 2 t q= t= α 1 ω 1 k p B(a 1, b 1 B(a 2, b 2 k,l,p,j,q= m= n= ( ( ( ( p b2 1 ω2 (a 2 + j q n j q t q ( ( ( ( ω1 a 1 1 k b1 1 ω1 l k m l p t= ( 1 k+l+p+j+q β m+n+2 1 β t 2 (β m+n+1 (β t (a 2 + j (1 + x α 1 x (m+n+1α 1+tα 2 1 exp( [β 1 (1 + p + kx α 1 + β 2 qx α 2 ]dx. (3.17 Note that, (1 + x α 1 x (m+n+1α 1+tα 2 1 e β 1(1+p+kx α 1 β 2 qx α 2 dx = s= r= ( α1 r ( 1 s β s 1(1 + p + k s x (m+n+s+1α 1+tα 2 +r 1 e β 2qx α 2 dx. (3.18

21 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 73 Using the definition of gamma function, we have x (m+n+s+1α 1+tα 2 +r 1 e β 2qx α 2 dx = Γ((m + n + s + 1α 1α rα 1 α 2 (β 2 q (m+n+s+1α 1α 1 2 +rα 1 Substituting Equation (3.19 into Equation (3.18, we obtain (1 + x α 1 x (m+n+1α 1+tα 2 1 e β 1(1+p+kx α 1 β 2 qx α 2 dx = s= r= ( α1 r 2 + t 2 +t. ( 1 s β s 1(1 + p + k s Γ((m + n + s + 1α 1α2 1 + rα2 1 + t α 2 (β 2 q (m+n+s+1α 1α 1 2 +rα 1 Finally, substituting Equation (3.2 into (3.17, we obtain α 1 ω 1 k p q ( ω1 a 1 1 R = B(a 1, b 1 B(a 2, b 2 k k,l,p,j,q= m= n= t= ( ( ( ( p b2 1 ω2 (a 2 + j q n j q t ( α1 ( 1 s β s r 1(1 + p + k s s= r= Γ((m + n + s + 1α 1α2 1 + rα2 1 + t α 2 (β 2 q (m+n+s+1α 1α 1 2 +rα 1 α 1 ω 1 k p = B(a 1, b 1 B(a 2, b 2 ( ( ( p b2 1 ω2 (a 2 + j n j q ( k m ( b1 1 ( t. ( 1 k+l+p+j+q β m+n+2 1 β t 2 (β m+n+1 (β t (a 2 + j 2 +t. k,l,p,j,q,s,r= m= n= t= ( α1 ( q t Γ((m + n + s + 1α 1α2 1 + rα2 1 + t α 2 (β 2 q (m+n+s+1α 1α 1 2 +rα 1 r 2 +t. q ( ( ( ω1 a 1 1 k b1 1 k m l ( 1 k+l+p+j+q+s β m+n+s+2 l (3.2 ( ω1 l p ( ω1 l p 1 β t 2(1 + p + k s (β m+n+1 (β t (a 2 + j 4 Mean Deviations, Bonferroni and Lorenz Curves In this section, we present the mean deviation about the mean, the mean deviation about the median, Bonferroni and Lorenz curves. Bonferroni and

22 74 A New Class of Generalized Power Lindley Distribution Lorenz curves are income inequality measures that are also useful and applicable in other areas including reliability, demography, medicine and insurance. The mean deviation about the mean and mean deviation about the median are defined by D(µ = x µ f(xdx and D(M = x M f(xdx, respectively, where µ = E(X and M = Median(X = F 1 (1/2 is the median of F. These measures D(µ and D(M can be calculated using the relationships: D(µ = 2µF (µ 2µ + 2 µ xf(xdx = 2µF (µ 2 µ and M D(M = µ + 2 xf(xdx = µ 2 xf(xdx. M Now using Lemma 3.2, we have D(µ = 2µF BEP L (µ 2µ + and D(M = µ + 2αβ 2 ω B(a, b( 2αβ 2 ω B(a, b( xf(xdx, ( b 1 ( 1 i L 2 (α, β, ω(a + i, 1, µ i i= ( b 1 ( 1 i L 2 (α, β, ω(a + i, 1, M. i i= Lorenz and Bonferroni curves are given by L(F BEP L (x = x xf BEP L (xdx E(X, and B(F BEP L (x = L(F (x BEP L, F BEP L (x or L(p = 1 q xf µ BEP L (xdx, and B(p = 1 q xf pµ BEP L (xdx, respectively, where q = F 1 (p. Using Lemma 3.2, we can re-write Lorenz BEP L and Bonferroni curves as B(p = 1 pµ = 1 pµ = 1 pµ q xf BEP L (xdx [ ] xf BEP L (xdx xf BEP L (xdx q [ αβ 2 ω ( ] b 1 µ ( 1 i L 2 (α, β, ω(a + i, 1, q, B(a, b( i i=

23 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 75 and q L(p = 1 xf BEP L (xdx µ = 1 [ xf µ BEP L (xdx q [ = 1 αβ 2 ω ( b 1 µ µ B(a, b( i i= ] xf BEP L (xdx ( 1 i L 2 (α, β, ω(a + i, 1, q ]. 5 Order Statistics and Measures of Uncertainty In this section, we present distribution of order statistics, Shannon entropy [23], [24], as well as Rényi entropy [25] for the BEPL distribution. The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and is a good measure of randomness or uncertainty. 5.1 Distribution of Order Statistics Order Statistics play an important role in probability and statistics. In this section, we present the distribution of the order statistics for the BEPL distribution. Suppose that X 1, X 2,..., X n is a random sample of size n from a continuous pdf, f(x. Let X 1:n < X 2:n <... < X n:n denote the corresponding order statistics. If X 1, X 2,..., X n is a random sample from BEPL distribution, it follows from Equations (2.2 and (2.3 that the pdf of the k th order statistic, say Y k = X k:n is given by f k (y k = n! (k 1!(n k! f (y n k BEP L k l= ( n k l ( 1 l [ BGEP L (y k ;α,β,ω(a, b B(a, b αβ 2 ω B(a, b( (1 + yα k y α 1 k exp( βyk α [V (y k ] ωa 1 [1 V ω (y k ] b 1 = αβ2 ωn!(1 + yk αyα 1 k exp( βyk α n k b 1 ( ( n k b 1 ((k 1!(n k! l m l= m= ( 1 l+m (B (B(a, b k+l 1 GEP L(yk;α,β,ω(a, b k+l 1 [V (y k ] ω(a+m 1, ] k+l 1

24 76 A New Class of Generalized Power Lindley Distribution ( where V (y k = G P L (y k ; α, β, ω = βyα k exp( βyk α and G EP L (y k ; α, β, ω = V ω (y k. The corresponding cdf of Y k is F k (y k = = = n j n ( n j j=k j=k l= n j n ( n j j=k l= l= ( n j l ( n j l ( 1 l [F BEP L (y k ] j+l [ ] ( 1 l BGEP L (y k ;α,β,ω(a, b j+l B(a, b n j n ( ( n n j ( 1 l [ j l [B(a, b] j+l BGEP L (y k ;α,β,ω(a, b ] j+l. 5.2 Shannon Entropy Shannon entropy [23],[24] is defined by H [f BEP L ] = E [ log(f BEP L (X; α, β, ω, a, b]. Thus, we have H [f BEP L ] = [ ] B(a, b( log E [log(1 + X α ] αβ 2 ω (α 1E [log(x] + βe [X α ] [ { }] (ωa 1E log 1 (1 + βxα e βxα 1 + β [ { [ ] ω }] (b 1E log 1 1 (1 + βxα e βxα. ( β Note that, for x < 1, using the series representation log(1+x = ( 1 q+1 x q q=1, q we obtain E [log(1 + X α ( 1 q ] = E [X qα ], (5.2 q E [log(x] = p p=1 s= q=1 ( p ( 1 s E [X s ], (5.3 s p E [ { }] log 1 (1 + βxα e βxα 1 + β t ( t β u = u t( u t=1 u= E [ ] X uα e βtxα (5.4

25 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 77 and E [ { [ ] ω }] log 1 1 (1 + βxα e βxα 1 + β = d c=1 d= e= ( ωc d ( d e ( 1 d+1 β e c( e E [ X eα e βdxα ]. (5.5 By using the results in Lemma 3.1, we can calculate Equations (5.2, (5.3, (5.4 and (5.5. Now, we obtain Shannon entropy for the BEPL distribution as follows: H [f BEP L ] = [ ] B(a, b( log + αβ 2 ω [ ( 1 q L 1 (α, β, ω(a + i, qα q q=1 + (α 1 p=1 s= αβ 2 ω B(a, b( ( b 1 ( 1 i i i=1 ( p ( 1 s L 1 (α, β, ω(a + i, s s p + βl 1 (α, β, ω(a + i, α ( t β u+v t v 1 ( 1 v + (ωa 1 u ( u+1 v! t=1 v= u= L 1 (α, β, ω(a + i, α(u + v + d ( ( ωc d β e+f d f ( 1 d+f (b 1 d e c( e+1 f! c=1 d= e= f= ] L 1 (α, β, ω(a + i, α(e + f. ( Rényi Entropy Rényi entropy [25] is an extension of Shannon entropy. Rényi entropy is defined to be I R (v = 1 1 v log ( fbep v L(x; α, β, ω, a, bdx, v 1, v >. (5.7 Rényi entropy tends to Shannon entropy as v 1. Note that by using the

26 78 A New Class of Generalized Power Lindley Distribution series expansion in Equation (3.1, and Equation (2.3, we have f v BEP L(x; α, β, ω, a, bdx = Now let u = β(v + j + qx α, then ( αβ 2 v ω q ( ( v ωav v B(a, b( i j i,j,p,q= k= r= ( ( ( ( j bv v ωp q ( 1 j+p+q β k+r k p q r ( k+r x α(i+k+r+v v e β(v+j+qxα dx. x α(i+k+r+v v e β(v+j+qxα dx = (v 1 Γ(i + k + r + v α (v 1 i+k+r+v α [β(v + j + q] α. Consequently, Rényi entropy is given by I R (v = for v 1, v >. [ ( 1 1 v log αβ 2 v ω B(a, b( ( ( ( bv v ωp q ( 1 j+p+q β k+r p q r ( k+r Γ(i + k + r + v (v 1 α (v 1 i+k+r+v α [β(v + j + q] α q i,j,p,q= k= r= ( v i ( ωav v j ( j k ], ( s-entropy The s-entropy for the BEPL distribution is defined by H s [f BEP L (X; α, β, ω, a, b] = 1 [ ] 1 fbep s s 1 L(x; α, β, ω, a, bdx if s 1, s >, E [ log f(x] if s = 1. Now, using the same procedure that was used to derive Equation (5.8, we

27 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 79 have f s BEP L(x; α, β, ω, a, bdx = Consequently, s-entropy is given by H s [f BEP L (X; α, β, ω, a, b] = for s 1, s >. ( αβ 2 ω B(a, b( s ( ( ( ωas s j bs s j k p i,j,p,q= k= r= ( ωp q q ( s i ( q r ( 1j+p+q β k+r Γ(i + k + r + s (s 1 ( k+r α [β(s + j + q] α (s 1 i+k+r+s α 1 s 1 1 ( αβ 2 s ω s 1 B(a, b( q ( ( ( s ωas s j i j k i,j,p,q= k= r= ( ( ( bs s ωp q ( 1 j+p+q β k+r p q r ( k+r Γ(i + k + r + s (s 1 α (s 1 i+k+r+s α [β(s + j + q] α. 6 Maximum Likelihood Estimation In this section, the maximum likelihood estimates of the BEPL parameters α, β, ω, a and b are presented. If x 1, x 2,..., x n is a random sample from BEPL distribution, the log-likelihood function is given by log L(α, β, ω, a, b = ( αβ 2 ω n n log + log(1 + x α i B(a, b( i=1 + n n n (α 1 log(x i β x α i + (ωa 1 log V (x i + (b 1 i=1 i=1 n log [1 V ω (x i ], i=1 ( where V (x i = G P L (x i ; α, β = βxα i exp ( βx α i. The partial derivatives of log L(α, β, ω, a, b with respect to the parameters a, b, α, β and i=1

28 8 A New Class of Generalized Power Lindley Distribution ω are: log L(α, β, ω, a, b a log L(α, β, ω, a, b b log L(α, β, ω, a, b α = n [ψ(a + b ψ(a] + ω = n [ψ(a + b ψ(b] + = n α + n i=1 (ωa 1 + ω(b 1 n log V (x i, i=1 n log [1 V ω (x i ], i=1 [ x α log(x i i 1 + x α i n i=1 n i=1 V (x i / α V (x i ] βx α i + 1 [V (x i ] ω 1 V (x i / α, 1 V ω (x i log L(α, β, ω, a, b β = n(β + 2 n β( x α i (ωa 1 + ω(b 1 n i=1 i=1 n i=1 [V (x i ] ω 1 V (x i / β 1 V ω (x i V (x i / β V (x i and log L(α, β, ω, a, b ω = n ω (b 1 n i=1 V ω (x i log V (x i 1 V ω (x i + a n log V (x i, i=1 respectively, where V (x i α = β 2 log(x i(1 + x α i x α i exp( βx α i and V (x i β = [( ] 1 + βxα i 1 x α ( 2 i exp( βx α i. When all the parameters are unknown, numerical methods must be applied to determine the estimates of the model parameters since the system of equations is not in closed form. Therefore, the maximum likelihood estimates, ˆΘ of Θ = (α, β, ω, a, b can be determined using an iterative method such as the Newton-Raphson procedure.

29 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede Fisher Information Matrix In this section, we present a measure for the amount of information. This information can be used to obtain bounds on the variance of estimators and as well as approximate the sampling distribution of an estimator obtained from a large sample. Moreover, it can be used to obtain an approximate confidence interval in the case of a large sample. Let X be a random variable with the BEPL pdf f BEP L ( ; Θ, where Θ = (θ 1, θ 2, θ 3, θ 4, θ 5 T = (α, β, ω, a, b T. Then, Fisher information matrix (FIM is the 5 5 symmetric matrix with elements: I ij (Θ = E Θ [ log(fbep L (X; Θ θ i log(f BEP L (X; Θ θ j If the density f BEP L ( ; Θ has a second derivative for all i and j, then an alternative expression for I ij (Θ is [ ] 2 log(f BEP L (X; Θ I ij (Θ = E Θ. θ i θ j For the BEPL distribution, all second derivatives exist; therefore, the formula above is appropriate and most importantly significantly simplifies the computations. Elements of the FIM can be numerically obtained by MATLAB or MAPLE software. The total FIM I n (Θ can be approximated by [ ] J n ( ˆΘ 2 log L θ i θ j Θ= ˆΘ. 55 ]. (6.1 For real data, the matrix given in Equation (6.1 is obtained after the convergence of the Newton-Raphson procedure in MATLAB or R software. 6.2 Asymptotic Confidence Intervals In this section, we present the asymptotic confidence intervals for the parameters of the BEPL distribution. The expectations in the Fisher Information Matrix (FIM can be obtained numerically. Let ˆΘ = (ˆα, ˆβ, ˆω, â, ˆb be the maximum likelihood estimate of Θ = (α, β, ω, a, b. Under the usual regularity conditions and that the parameters are in the interior of the parameter space, d but not on the boundary, we have: n( ˆΘ Θ N 5 (, I 1 (Θ, where

30 82 A New Class of Generalized Power Lindley Distribution I(Θ is the expected Fisher information matrix. The asymptotic behavior is still valid if I(Θ is replaced by the observed information matrix evaluated at ˆΘ, that is J( ˆΘ. The multivariate normal distribution with mean vector = (,,,, T and covariance matrix I 1 (Θ can be used to construct confidence intervals for the model parameters. That is, the approximate 1(1 η% two-sided confidence intervals for α, β, ω, a and b are given by ˆα ± Z η/2 I 1 αα( ˆΘ, ˆβ ± Zη/2 I 1 ββ ( ˆΘ, â ± Z η/2 I 1 aa ( ˆΘ and ˆb ± Z η/2 I 1 bb ( ˆΘ ˆω ± Z η/2 I 1 ωω( ˆΘ, respectively, where I 1 αα( ˆΘ, I 1 ββ ( ˆΘ, I 1 ωω( ˆΘ, I 1 aa ( ˆΘ and I 1 bb ( ˆΘ are diagonal elements of I 1 n ( ˆΘ = (ni ˆΘ 1 and Z η/2 is the upper (η/2 th percentile of a standard normal distribution. We can use the likelihood ratio (LR test to compare the fit of the BEPL distribution with its sub-models for a given data set. For example, to test α = ω = 1, the LR statistic is ω = 2[ln(L(â, ˆb, ˆβ, ˆα, ˆω ln(l(ã, b, β, 1, 1], where â, ˆb, ˆβ, ˆα and ˆω are the unrestricted estimates, and ã, b, and β are the restricted estimates. The LR test rejects the null hypothesis if δ > χ 2 ɛ, where χ 2 denote the upper 1ɛ% point of the ɛ χ2 distribution with 2 degrees of freedom. 7 Applications In this section, the BEPL distribution is applied to real data in order to illustrate the usefulness and applicability of the model. We fit the density functions of the beta-exponentiated power Lindley (BEPL, beta exponentiated Lindley (BEL, exponentiated power Lindley (EPL [4], beta power Lindley (BPL, power Lindley (PL, and Lindley (L distributions. We provide examples to illustrate the flexibility of the BEPL distribution in contrast to other models including the BEL, BPL, PL, L, beta-weibull (BW [26], beta-exponential (BE [15] and Weibull (W distributions for data modeling purposes. The pdf of the BW distribution [26] is given by f BW (x; α, λ, a, b = αλ α B(a, b xα 1 exp( b(λx α [1 exp( (λx α ] a 1,

31 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 83 for x >, α >, λ >, a >, b >. When α = 1, the beta exponential pdf is obtained, [15]. Estimates of the parameters of the distributions, standard errors (in parentheses, Akaike Information Criterion (AIC = 2p 2 log(l, Consistent Akaike Information Criterion (AICC = AIC + 2p(p+1, Bayesian Information Criterion (BIC = p log(n 2 log(l, where L = L( ˆΘ is the value of the n p 1 likelihood function evaluated at the parameter estimates, n is the number of observations, and p is the number of estimated parameters are obtained. The first data set represents the maintenance data with 46 observations reported on active repair times (hours for an airborne communication transceiver discussed by Alven [27], Chhikara and Folks [28] and Dimitrakopoulou et al. [29]. It consists of the observations listed below:.2,.3,.5,.5,.5,.5,.6,.6,.7,.7,.7,.8,.8, 1., 1., 1., 1., 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2., 2., 2.2, 2.5, 2.7, 3., 3., 3.3, 3.3, 4., 4., 4.5, 4.7, 5., 5.4, 5.4, 7., 7.5, 8.8, 9., 1.3, 22., The second data set represents the remission times (in months of a random sample of 128 bladder cancer patients reported in Lee and Wang ([3]. See the table below. Table 7.1: Cancer Patients Data, Lee and Wang [3] The third data set consists of the number of successive failures for the air conditioning system of each member in a fleet of 13 Boeing 72 jet airplanes (Proschan [31]. The data is presented in Table 7.2.

32 84 A New Class of Generalized Power Lindley Distribution Table 7.2: Air conditioning system data Estimates of the parameters of BEPL distribution (standard error in parentheses, Akaike Information Criterion, Consistent Akaike Information Criterion and Bayesian Information Criterion are given in Table 7.3 for the active repair time data, in Table 7.4 for the cancer patient data and in Table 7.5 for the air conditioning system data. Table 7.3: Estimates of Models for Repair Times Data Estimates Statistics Model α β ω a b 2 log L AIC AICC BIC BEPL(α, β, ω, a, b (.2992 ( ( ( ( PL(α, β, 1, 1, (.7424 (.116 L(1, β, 1, 1, (.499 BL(1, β, 1, a, b (.3449 (.255 (.3284 BW(α, β,, a, b (.1821 ( ( (.1214 W(α, β,, 1, (.9576 (.5138 BE(1, β,, a, b (.319 (.1793 (1.771 For the repair times data set, the LR statistic for the hypothesis H : P L(α, β, 1, 1, 1 against H a : BEP L(α, β, ω, a, b, is ω = 1.7. The p-value is <.5. Therefore, there is a significant difference between PL and BEPL distributions. A LR test of H : L(1, β, 1, 1, 1 vs H a : BEP L(α, β, ω, a, b shows that ω = 2.7, and p-value= <.1. Therefore, there is

33 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 85 a significant difference between L and BEPL distributions. There is also a significant difference between PL and L distributions where ω = 1. with a p-value of <.1. Moreover, the values of the statistics AIC and AICC are smaller for the BEPL distribution and show that the BEPL distribution is a better fit than its sub-models for the repair times data, however a comparison of BEPL and BW distributions shows that the four parameter BW distribution is slightly better. The asymptotic covariance matrix of MLEs for BEPL model parameters, which is the FIM I 1 n ( ˆΘ, is given by and the 95% two-sided asymptotic confidence intervals for α, β, ω, a and b are given by.8792± , ± , ± , ± and ± , respectively. Plots of the fitted densities and the histogram of the repair time data are given in Figure 7.1.

34 86 A New Class of Generalized Power Lindley Distribution Figure 7.1: Plot of the fitted densities for the Repair Times Data Table 7.4: Estimates of Models for Cancer Patient Data Estimates Statistics Model α β ω a b 2 log L AIC AICC BIC BEPL(α, β, ω, a, b (.2657 (.258 (.39 (.199 (.2251 BPL(α, β, 1, a, b (.2299 ( ( ( PL(α, β, 1, 1, (.472 (.37 L(1, β, 1, 1, (.499 BW(α, β,, a, b (.2368 (.4177 ( ( W(α, β,, 1, (.676 (.93 For the cancer patients data, the LR statistics for the test of the hypotheses H : P L(α, β, 1, 1, 1 against H a : BEP L(α, β, ω, a, b and H : L(1, β, 1, 1, 1 against H a : BEP L(α, β, ω, a, b are (p value =.4956 <.5 and

35 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede (p value =.459 <.1, respectively. Consequently, we reject the null hypothesis in favor of the BEPL distribution and conclude that the BEPL distribution is significantly better than the PL and L distributions. However, there is no significant difference between the BPL and BEPL distributions based on the LR test. Also, based on the values of the statistics: AIC, AICC and BIC, we conclude that the BPL distribution is the better fit for the cancer patient data. The BPL distribution is also slightly better that the BW distribution based on the values of these statistics. Plots of the fitted densities and the histogram for the cancer patient data are given in Figure 7.2. Figure 7.2: Plot of the fitted densities for the Cancer Patients Data

36 88 A New Class of Generalized Power Lindley Distribution Table 7.5: Estimates of Models for Air Conditioning System Data Estimates Statistics Model α β ω a b 2 log L AIC AICC BIC BEPL(α, β, ω, a, b (.276 (.212 ( (.2146 (.1512 BPL(α, β, 1, a, b (.573 (.1658 (.4284 (.8737 BEL(1, β, ω, a, b (.194 ( (.623 (.649 BL(1, β, 1, a, b (.972 (.538 (.232 PL(α, β, 1, 1, (.316 (.165 L(1, β, 1, 1, (.111 BW(α, β,, a, b (.1114 (.7861 (4.738 (.2421 W(α, β,, 1, (.54 (.97 BE(1, β,, a, b (.184 (.864 (.3651 For the air conditioning system data, the LR statistics for the test of the hypotheses H : BL(1, β, 1, a, b against H a : BEP L(α, β, ω, a, b is 16.5 (p value =.263 <.1. Consequently, we reject the null hypothesis in favor of the BEPL distribution and conclude that the BEPL distribution is significantly better than the BL distribution. The LR test statistics for the test of the hypotheses H : BL(1, β, 1, a, b against H a : BEL(1, β, ω, a, b is 15.8 (p value =.74 <.1, so that the null hypothesis of BL model is rejected in favor of the alternative hypothesis of BEL model. The BPL distribution is also significantly better than the PL and BL models based on the LR test. However, there is no significant difference between the BPL and BEPL distributions, as well as between the BEL and BEPL distributions based on the LR test. The sub-models: BPL and BEL are better fits than the BEPL distribution for the air conditioning system data. Also, the values of the statistics: AIC, AICC and BIC, points to the BEL distribution, so we conclude that the BEL distribution is the better fit for the air conditioning system data. The BEL distribution also compares favorably with the BW distribution based on the values of these statistics. Plots of the fitted densities and the histogram for the air conditioning system data are given in Figure 7.3. Based on the values of these statistics, we conclude that the BEPL distribution and its sub-models can provide good fits for lifetime data. In the first data set, the BEPL distribution performed better than the BL, PL, L, BE, and Weibull distributions. The four parameter BW distribution was slightly better based on the values of AIC, AICC and BIC. In the second data set,

37 Mavis Pararai, Gayan Warahena-Liyanage and Broderick O. Oluyede 89 the BPL distribution performed better than the other models including the beta Weibull distribution. In the third data set, the BEL distribution as well as the BPL distribution seem to be the better fits, and the BEL distribution compares favorably with the BW distribution. The BEPL and its sub-models including the BEL and BPL distributions can provide better fits than other common lifetime models. Figure 7.3: Plot of the fitted densities for the Air Conditioning System Data 8 Concluding Remarks We have developed and presented the mathematical properties of a new class of distributions called the beta-exponentiated power Lindley (BEPL distribution including the hazard and reverse hazard functions, monotonicity properties, moments, conditional moments, reliability, entropies, mean deviations, Lorenz and Bonferroni curves, distribution of order statistics, and max-

38 9 A New Class of Generalized Power Lindley Distribution imum likelihood estimates. Applications of the proposed model to real data in order to demonstrate the usefulness of the distribution are also presented. ACKNOWLEDGEMENTS.The authors are grateful to the referees for some useful comments on an earlier version of this manuscript which led to this improved version. References [1] D.V. Lindley, Fiducial distributions and Bayes Theorem, Journal of the Royal Statistical Society, Series B, 2, (1958, [2] M.E. Ghitany, B. Atieh and S. Nadarajah, Lindley Distribution and Its Applications, Mathematics and Computers in Simulation, 78(4, (28, [3] H. Zakerzadeh and A. Dolati, Generalized Lindley Distribution, Journal of Mathematical Extension, 3(2, (29, [4] G. Warahena-Liyanage and M. Pararai, A Generalized Power Lindley Distribution with Applications, Asian Journal of Mathematics and Applications, 214(Article ID ama169, (214, [5] M. Sankaran, The Discrete Poisson-Lindley Distribution, Biometrics, 26(1, (197, [6] A. Asgharzedah, H.S. Bakouch and H. Esmaeli, Pareto Poisson-Lindley Distribution with Applications, Journal of Applied Statistics, 4(8, (213. [7] S. Nadarajah, H.S. Bakouch and R. Tahmasbi, A Generalized Lindley Distribution, Sankhya B 73, (211, [8] M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan and L.J. Al-Enezi, Power Lindley distribution and associated inference, Computational Statistics and Data Analysis, 64, (213, 2-33.

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