The LogGeneralized LindleyWeibull Distribution with Applications

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1 Georgia Southern University Digital Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of The LogGeneralized LindleyWeibull Distribution with Applications Broderick O. Oluyede Georgia Southern University, Fedelis Mutiso University of Washington Shujiao Huang University of Houston Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Oluyede, Broderick O., Fedelis Mutiso, Shujiao Huang "The LogGeneralized LindleyWeibull Distribution with Applications." Journal of Data Science, 13 (2): source: This article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Southern. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Southern. For more information, please contact
2 Journal of Data Science 13(215), The Log Generalized LindleyWeibull Distribution with Application Broderick O. Oluyede 1, Fedelis Mutiso 2, Shujiao Huang 3 1 Department of Mathematical Sciences, Georgia Southern University 2 Department of Biostatistics, University of Washington 3 Department of Mathematics, University of Houston Abstract: A new distribution called the log generalized LindleyWeibull (LGLW) distribution for modeling lifetime data is proposed. This model further generalizes the Lindley distribution and allows for hazard rate functions that are monotonically decreasing, monotonically increasing and bathtub shaped. A comprehensive investigation and account of the mathematical and statistical properties including moments, moment generating function, simulation issues and entropy are presented. Estimates of model parameters via the method of maximum likelihood are given. Real data examples are presented to illustrate the usefulness and applicability of this new distribution. Key words: Lindley distribution; LindleyWeibull distribution; Maximum likelihood estimation 1. Introduction The continuous one parameter Lindley distribution was introduced by Lindley (1958). Lindley used the distribution named after him to illustrate a difference between fiducial distribution and posterior distribution. Lindley distribution with the probability density function (pdf) f(x; θ) = θ2 (1+x)exp ( θx), x >, θ >, (1) 1+θ is a twocomponent mixture of an exponential distribution with scale parameter θ and gamma distribution with shape parameter 2 and scale parameter θ. The mixing proportion is p = θ/(θ + 1). Sankaran (197) derived the PoissonLindley distribution. In this case, Lindley distribution was chosen as the mixing distribution when the parameter of the Poisson distribution is considered random. The resulting PoissonLindley distribution provided a better fit to the empirical set of data considered than the negative binomial and Hermite distributions. Recently, Ghitany et al. (28, 211) studied various properties of Lindley distribution and the twoparameter weighted Lindley distribution with applications to survival data. Bakouch et al. (212) introduced an extension of the Lindley distribution that offers more flexibility in the modeling of lifetime data. Ghitany et al. (213) presented results on the twoparameter generalization referred to as the power Lindley distribution. See Krishna and Kumar (211) for additional results on reliability estimation of the Lindley distribution with progressive type II censored sample.
3 282 The Log Generalized LindleyWeibull Distribution with Application Because of having only one parameter, the Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. To increase the flexibility for modeling purposes it will be useful to consider further generalizations of this distribution. This paper offers a fiveparameter family of distributions which generalizes the Lindley distribution. There are several ways of generalizing a continuous distribution G(x), and they include KumaraswamyG, betag, McDonaldG, and gammag to mention a few. Kumaraswamy (198) distribution is given by G k (x) = 1 (1 x ψ ) ϕ, x 1, for ψ > and ϕ >. Replacing x by G(x) on the right hand side of the equation gives the KumaraswamyG family: G KG (x) = 1 (1 G ψ (x)) φ. The betag family of distributions (Lee et al., 27, Famoye et al., 25) among others is given by G BG (x) = 1 G(x) B(a, b) wa 1 (1 w) b 1 dw, for a > and b >. The McDonaldG family of distributions (Cordeiro et al., 212) is given by G 1 c (x) G McG (x) = B(ac 1, b) wac 1 1 (1 w) b 1 dw, for a, b and c >. The GammaG family of distributions (Zografos and Balakrishnan, 29, Pinho et al., 212) is G GG (x) = γ( θ 1 log(g (x)), α), Γ(α) for α, θ >, where G (x) = 1 G(x). We consider a further generalization of the generalized Lindley distribution via the TX family of distributions proposed by Alzaatreh et al. (213) to obtain the cumulative distribution function (cdf) of the log generalized LindleyWeibull distribution. The generalization (Alzaatreh et al., 213) is given by the following cdf: G(x) = W(F(x)) k(y)dy, where < W(F(x)) <, is a nondecreasing function of x, k(. ) is taken to be the generalized Lindley distribution of Zakerzadeh and Dolati (29) and F(x)is the Weibull cdf. The corresponding pdf g, is given by g(x) = f(x) k (W(F(x))), F (x) where W(F(x)) = ln(1 F(x)).
4 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 283 The main objective of this article is to construct and explore the properties of the fiveparameter log generalized LindleyWeibull (LGLW) distribution. The beauty of this model is the fact that it not only generalizes the generalized Lindley distribution but also exhibits the desirable properties of increasing, decreasing, and bathtub shaped hazard function. The model provides a better fit to data in the sense that it leads to more accurate results and prediction, which should facilitate better public policy in a wide range of areas including but not limited to medicine and environmental health, genetics, reliability, survival analysis and timeto event data analysis. The outline of this paper is as follows: In section 2 some generalized Lindley distributions including the new LGLW distribution are introduced. This section also includes some properties such as the behavior of the hazard function, reverse hazard function and submodels of the log generalized LindleyWeibull distribution. Section 3 contains the moment generating function, moments, distribution of functions of log generalized LindleyWeibull random variables and simulation. Measures of uncertainty are given in section 4. Section 5 contains the estimation of parameters via the maximum likelihood estimation technique. Fisher information and asymptotic confidence intervals are also presented in section 5. We end with applications in section 6 and concluding remarks in section 7. Generalizations of the Lindley Distribution In this section, we present further generalizations of the Lindley distribution. First, we discuss some generalizations that are in the literature, or in preparation. 2.1 Generalized Lindley Distribution Let V 1 and V 2 be two independent random variables distributed according to gamma(α, θ) and gamma(α + 1, θ), respectively. That is, V 1 ~GAM(α, θ) and V 2 ~GAM(α + 1, θ). For β, θ consider the random variable X = V 1 with probability, and X = V θ+β 2 with probability β. It is θ+β easy to verify that the density function of X is given by f GL (x; α, θ, β) = θ f θ+β g α (x; α, θ) + β f θ+β g α+1 (x; α + 1, θ) (2) which may be written as f GL (x; α, θ, β) = θ2 (θx) α 1 (α+βx)e θx, x >, β, α, θ > (3) (θ+β)γ(α+1) where f gα (x) is the gamma pdf with parameters α and θ, that is, f gα (x; α, θ) = θα x α 1 e θx, (4) Γ(α) for x >, α, θ >. See Zakerzadeh and Dolati, (29) for additional details. The distribution contains the Lindley distribution as particular case, where α = β = 1. When β =, equation (3) reduces to the density function of the gamma distribution with the parameters α and θ. The case
5 284 The Log Generalized LindleyWeibull Distribution with Application α = 1 and β =, reduces to ordinary exponential distribution. In general, if V i ~GAM(α i, θ i ), i = 1,2,, and X = v i with probability p i, and i p i = 1, then θ f X (x) = p i i i Γ(α i ) xαi 1 e θix I (,) (x) (5) Clearly, the generalized Lindley (GL) distribution is a special case of (5). 2.2 Exponentiated Lindley Distribution A generalization of the Lindley distribution due to Nadarajah et al. (211) is the two parameter Exponentiated Lindley distribution with cumulative distribution function (cdf) and probability density function (pdf) given by and F GL (x; θ, α) = [1 1+θ+θx 1+θ e θx ] α, (6) f GL (x; θ, α) = θ2 α 1+θ+θx [1 1+θ 1+θ e θx ] α 1 (1 + x)e θx, (7) for x >, α >, and θ >, respectively. 2.3 BetaGeneralized Lindley Distribution A further generalization of the Lindley distribution, although not studied in this paper is the betageneralized Lindley (BGL) distribution, (Oluyede and Yang, 214). The four parameter betageneralized Lindley (BGL) cdf is given by F BGL (x; α, θ, a, b) = 1 B(a,b) G(x;θ,α) ta 1 (1 t) b 1 dt, (8) where G(x; θ, α) = {1 1+θ+θx 1+θ e θx } α, for x, α >, θ >, a >, b >. The corresponding pdf is given by f BGL (x; α, θ, a, b) = αθ2 (1 + x)e θx B(a, b)(1 + θ) 1 + θ + θx {1 1 + θ {1 {1 1+θ+θx 1+θ e θx } α } b 1 aα 1 e θx } (9) for x, α >, θ >, a >, b >. If α = 1, we obtained the betalindley (BL) distribution. If a = b = α = 1, we obtain the Lindley distribution. See Yang and Oluyede (214) for additional details on the Exponentiated Kumaraswamy Lindley distribution. 2.4 The Log Generalized LindleyWeibull Distribution
6 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 285 In this section, we introduce a new generalization of the Lindley distribution via the Weibull model and study its mathematical and statistical properties GeneralizationThe Model Based on a continuous baseline cdf F(x) and survival function F (x) = 1 F(x), with pdf f(x), Zografos and Balakrishnan (29) defined the cdf G ZB (x) = 1 log (1 F(x)) tδ 1 e t dt, δ > and x R. (1) Γ(δ) Along the same lines, Ristić and Balakrishnan (211) proposed an alternative gammagenerator given by the cdf and pdf and G RB (x) = 1 1 Γ(δ) log(f(x)) tδ 1 e t dt, δ > and x R, (11) g RB (x) = 1 Γ(δ) [ log(f(x))]δ 1 f(x), δ > and x R. (12) Now, we consider a generalizations of the generalized Lindley distribution given by Zakerzadeh and Dolati (29) via the Weibull distribution. The generalization is given by the following cdf (Alzaatreh et al., 213): log (1 F W (x)) G LGLW (x) = f GL (y) dy, (13) where g GL (x) is the generalized Lindley pdf and F W (x) is the Weibull cdf. The pdf of the log generalized LindleyWeibull (LGLW) distribution is given by g LGLW (x; α, β, θ, γ, c) = f W (x;γ,c) f F W(x;γ,c) GL( ln(1 F W (x; γ, c)) ; α, β, θ),(14) where the survival function F W(x; γ, c) = 1 F W (x; γ, c) = e (x γ )c, for x >, γ >, and c >. The wellknown hazard function of the Weibull distribution is given by h W (x; γ, c) = f W (x;γ,c) F W (x;γ,c) = c γ (x γ )c 1. It follows therefore that the fiveparameter LGLW cdf is given by G LGLW (x) = ( x γ )c where Γ(s, x) = θ 2 (θy) α 1 (α + βy)e θy dy (β + θ)γ(α + 1) 1 = (β + θ)γ(α) {θ[γ(α) Γ(α, u)] + β [Γ(α) Γ(α + 1, u)]}, α x The corresponding pdf is given by t s 1 e t dt is the upper incomplete gamma function and u = θ ( x γ )c. cθ α+1 cα 1 g LGLW (x; α, θ, β, γ, c) = γ(β + θ)γ(α + 1) (x γ ) {α + β ( x c} γ ) e θ(x γ )c, for x >, and θ, α, c, γ, β >. The graphs of the LGLW pdf, g LGLW are given in Figure 1 for selected values of the parameters α, θ, β, γ, and c. Note that the parameters β, γ, and θ are scale parameters, and α, c are shape parameters. The graphs show that the pdf of the LGLW distribution can be right skewed or decreasing for the selected values of the model parameters.
7 286 The Log Generalized LindleyWeibull Distribution with Application Some LGLW Submodels In this subsection, we present some submodels of the LGLW distribution for selected values of the parameters c, α, γ, β and θ. If c = γ = 1, then g LGLW (x; α, θ, β) = θα+1 x α 1 (α+βx)e θx. This is the generalized (β+θ)γ(α+1) Lindley distribution, denoted by GENLIN(α, θ,\beta). If c = α = γ = β = 1, then g LGLW (x; θ) = θ2 (1 + 1+θ x)e θx, which is the Lindley distribution and is denoted by LIN(θ) for x, θ >. If c = 1 and λ = θ, then g γ(λx) γ LGLW(x) = α (β+λγ)γ(α+1) (λ) {α + β x (x)} γ e λx. If c = α = β = 1, then g LGLW (x) = (1 + x ) γ(1+θ) γ e θ( If α = β = 1, then g LGLW (x) = cθ2 γ(1+θ) (x γ )c 1 {1 + ( x γ )c } e θ( If γ = β = 1, then g LGLW (x) = θ2 x γ ). x γ )c. cθα+1 (1+θ)Γ(α+1) xcα 1 (α + x c )e θxc. θ α+1 If c = 1, then g LGLW (x) = γ(β+θ)γ(α+1) (x γ )α 1 {α + β ( x γ If c = α = 1, then g LGLW (x) = {1 + β γ(β+θ) (x γ θ2 x )} e θ( γ ). If α = 1, then g LGLW (x) = cθ2 γ(β+θ) (x γ )c 1 {1 + β ( x γ )c } e θ( Shape For the LGLW pdf, the first derivative of log (g LGLW (x)) is d cθ {( x dx ln(g γ )c } LGLW(x)) = d 2 x )} e θ( γ ). x γ )c. β + { c(β θ)α β( 1 + c)} ( x γ )c cα 2 + α x {α + β ( x. γ )c } Therefore, g LGLW (x) has a unique model at x, where x is the solution of the equation dx ln{g LGLW(x)} =. That is, cθ {( x γ )c } which implies that 2 β + { c(β θ)α β( 1 + c)} ( x γ )c cα 2 + α x {α + β ( x =, γ )c }
8 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 287 cθ {( x 2 c} γ ) β + { c(β θ)α β( 1 + c)} ( x c γ ) cα 2 + α =. Solving for x in the above equation gives the mode. That is, x = [ {c(β θ)α+β( 1+c)} 2cθβ + { c(β θ)α β( 1+c)}2 4(βcθ)( cα 2 +α) 2cθβ 1 c ] γ.(15) Note that ln(g LGLW (x)/ x) < x > x and ln ( g LGLW (x) ) > x < x x, where x is given by equation (15) above. When α = β = γ = c = 1, x = (1 θ)+ (θ 1)2 2θ = (1 θ)+ (θ+1)2 4θ, (16) 2θ which is the same result for the generalized Lindley distribution given by Zakerzadeh and Dolati (29) Hazard and Reverse Hazard Functions In this section, we present the hazard and reverse hazard functions of the LGLW distribution. Graphs of the hazard function for selected values of the model parameters are given in Figure 2 and 3. The hazard and reverse hazard functions of the LGLW distribution are given by η LGLW cθ α+1 h(x) = g LGLW(x) G LGLW (x) = γ(β + θ)γ(α + 1) (x γ )cα 1 {α + β ( x γ )c } e θ( γ )c 1 1 (β + θ)γ(α) {θ[γ(α) Γ(α, u)] + β, α [Γ(α) Γ(α + 1, u)]} and τ LGLW (x) = g x LGLW(x) cθα+1 ( G LGLW (x) = γ )cα 1 {α + β ( x γ )c } e θ( x γ )c γα {θ[γ(α) Γ(α, u)] + β [Γ(α) α Γ(α + 1, u)]} for θ, α, c, γ, β >, and u = θ ( x γ )c, respectively. We obtain η LGLW (x) = g LGLW(x) and (x) and apply Glaser s Lemma (198) to the LGLW pdf: cθ α+1 g LGLW (x) = γ(β+θ)γ(α+1) (x γ )cα 1 {α + β ( x γ )c } e θ( γ )c, (17) for x >, c, θ, α, γ, β >. Note that cθ α+1 ( x γ )cα 1 (cα 1) {α + β ( x γ )c } e θ(x γ )c g LGLW (x) = γ(β + θ)γ(α + 1)x + c 2 θ α+1 ( x γ )cα 1 β ( x γ )c e θ(x γ )c γ(β + θ)γ(α + 1)x x x g LGLW (x)
9 288 The Log Generalized LindleyWeibull Distribution with Application and η LGLW (x) = c 2 θ α+1 ( x γ )cα 1 (cα 1) {α + β ( x γ )c } θ ( x γ )c e θ(x γ )c, γ(β + θ)γ(α + 1)x cθ {( x 2 γ )c } β + { c(β θ)α β( 1 + c)} ( x γ )c cα 2 + α x {α + β ( x. γ )c } When c = γ = 1, we have η LGLW (x) = α α2 αβx+θx(α+βx), which is the same result given x(α+βx) by Zakerzadeh and Dolati (29). Now, 2c 2 θ {( x γ )c } β η LGLW (x) = x 2 {α + β ( x γ )c } 2 { c(β θ)α β( 1 + c)} ( x γ )c c + x 2 {α + β ( x γ )c } cθ {( x 2 γ )c } β + { c(β θ)α β( 1 + c)} ( x γ )c cα 2 + α x 2 {α + β ( x γ )c } {cθ {( x 2 γ )c } β + { c(β θ)α β( 1 + c)} ( x γ )c cα 2 + α} β ( x γ )c c x 2 {α + β ( x. γ )c } If c = γ = 1, then η LGLW (x) = α3 +2α 2 βx+αβ 2 x 2 α 2 2αβx, which is the same result obtained x 2 (α+βx) 2 by Zakerzadeh and Dolati (29). The graphs of the hazard and reverse hazard functions are given below for different values of parameters α, θ, β, γ, and c. For the selected values of the parameters α, θ, β, γ, and c, the graphs of the hazard function are decreasing, increasing and bathtub shaped. Moments and Distribution of Functions of Random Variables This section deals with the moment generating function, moments and related functions of LGLW distribution. The mean, standard deviation, coefficients of variation, skewness and kurtosis can be readily computed. Distributions of functions of the LGLW random variables are also presented. 3.1 Moments In this section, we obtain the moments of the LGLW distribution and its submodels. The k th noncentral moment for the LGLW distribution is
10 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 289 E(x k ) = γ k cθ α+1 γ(β + θ)γ(α + 1) (x γ ) cα+k 1 {α + β ( x γ ) c } e θ(x γ )c Let y = ( x γ )c, then x = γy 1 c, and dx = 1 dy c γy1 c 1. Now, E(X k γ k cθ α+1 ) = (β + θ)γ(α + 1) {α k yα+ c 1 e θy dy + β y α+k c +1 1 e θy dy}. Let u = θy, then du dy du = θ and dy =, so that θ E(X k ) = {α ( u α+ k θ ) c 1 e u γ k θ α+1 (β + θ)γ(α + 1) du θ + β (u θ ) α+ k c +1 1 du = γk θ 1 k cγ(α+ k c ) [α + (β+θ)γ(α+1) βθ 1 (α + k )]. (18) c The mean of LGLW distribution is γθ 1 1 c } θ E(X) = (β + θ)γ(α + 1) Γ (α + 1 c ) {α + βθ 1 (α + 1 c )}. If c = 1, then E(X) = γ {α + β+θ βθ 1 (α + 1)}. 3.2 Moment Generating Function Let X denote a random variable with pdf g LGLW (x). The moment generating function (MGF) of X, M(t) = E(exp (tx)), is given by tj θ 1 j cγ j M X (t) = j! (β + θ)γ(α + 1) {αγ (α + j c ) + βθ 1 Γ (α + j c + 1)}. j= t j Note that M X (t) = j= j! E(Xj ), where E(X j ) is given by equation (18). 3.3 Distribution of Functions of Random Variables In this section, distributions of functions of random variables are presented. Recall the LGLW pdf is cθ α+1 cα 1 g LGLW (x) = γ(β + θ)γ(α + 1) (x γ ) {α + β ( x c} γ ) e θ(x γ )c, for x >, α, β, θ, γ, c >. Pdf of Y = ( X γ )c : Let y = ( x γ )c, then x = γy 1 c and dx dy = 1 c γy1 c 1. The pdf of Y = ( X γ )c is given by dx.
11 29 The Log Generalized LindleyWeibull Distribution with Application θ α+1 f Y (y) = (β + θ)γ(α + 1) yα 1 (α + βy)e θy, for y >, α, β, θ >, which is the GENLIN(α, θ, β). Pdf of W = X c : Let w = x c, then x = w 1 c and dx = 1 dw c w1 c 1. The pdf of W = X c is given by θ α+1 w α 1 f W (w) = γ(β + θ)γ(α + 1) γ cα 1 {α + β (w e θ( γ γc)} c), for w >, α, θ, β, γ, c >. If γ = 1, the two pdf s above are the same. 1 1 Pdf of V = θ ( X γ )c : Let v= θ ( x γ )c, then x = γ ( v ) c, and dx = γ θ dv c (v) c 1 1. The pdf of V = θ θ θ ( X γ )c is given by θ α α 1 f V (v) = (β + θ)γ(α + 1) (v θ ) {α + β ( v θ )} e v, for v >, α, θ, γ, β >. 3.4 Simulation The density of generalized Lindley (GL) distribution can be written in terms of the gamma density function as f(x; α, θ, β) = θ f β+θ g(x; α, θ) + β f β+θ g(x; α + 1, θ). (19) To generate a random data X i, i = 1,, n, from GL(α, θ, β), Zakerzadeh and Dolati (29), provided the following algorithm; 1. Generate U i, i = 1,, n, from U(,1) distribution. 2. Generate V 1i, i = 1,, n, from the gamma(α, θ). 3. Generate V 2i, i = 1,, n, from the gamma(α + 1, θ). 4. If U i θ, then set X β+θ i = V 1i ; otherwise set X i = V 2i, i = 1,, n. Now given γ and c, we can generate random data Y i, i = 1,, n where Y i = 1 γx c i ~LGLW(α, β, θ, γ, c). 2. Uncertainty Measures 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 can be shown to be a good measure of randomness or uncertainty. In this section, we present Renyi entropy, generalized entropy and sentropy for the LGLW distribution. w
12 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang Generalized Entropy Generalized entropy (GE) is widely used to measure inequality trends and differences. It is primarily used in income distributions. Kleiber and Kotz (23) derived Theil index for GB2 distribution and SinghMaddala model. The generalized entropy (GE) I(α ) is defined as: where I(α ) = v α μ α 1 α (α 1), α,1, (2) ν α = E(X α ) = x α g LGLW (x)dx γ α θ 1 α c α = Γ (α + (β + θ)γ(α + 1) c ) {α + βθ 1 (α + α c )}, and γθ 1 1 c μ = (β + θ)γ(α + 1) Γ (α + 1 c ) {α + βθ 1 (α + 1 c )}. The bottomsensitive index is I( 1), and the topsensitive index is I(2). The mean logarithmic deviation (MLD) index is given by: I() = lim α I(α ) = log(μ) ν, (21) where = ν = log(x)dg LGLW (x) = (log(x))g LGLW (x) dx cθ α+1 (log(x)) ( x cα 1 γ(β + θ)γ(α + 1) γ ) {α + β ( x c} γ ) e θ(x γ )c dx. Let y = θ ( x γ )c, then log(x) = log(γ) 1 log(θ) + 1 log (y). Now, we have c c θ ν = (β + θ)γ(α + 1) {log(γ) + 1 c log(y) 1 1 log(θ)} yα c{α + βθ 1 y}e y dy c θ = (β + θ)γ(α + 1) [{log ( γ θ 1 ) αγ (α 1 c + 1) + α c Γ (α 1 c + 1)} c + {log ( γ θ 1 ) βθ 1 Γ (α 1 c c Therefore, the MLD index reduces to I() = log(γ) + (1 1 ) log(θ) log(β + θ) log(γ(α + 1)) c + log (Γ (α + 1 c )) + log {α + βθ 1 (α + 1 c )} βθ 1 + 2) + Γ (α 1 c c + 2)}].
13 292 The Log Generalized LindleyWeibull Distribution with Application θ (β+θ)γ(α+1) [{log ( γ 1 αγ (α θc) 1 + 1) + α c c Γ (α 1 c +1) } {log ( γ 1) βθ 1 Γ (α 1 + 2) + c θc and Theil index is: βθ 1 I(1) = lim α 1 I(α ) = μ ν 1 log(μ) = θ α 1 c Γ (α + 1 c ) {α + βθ 1 (α + 1 c )} Γ (α + α c ) {α + βθ 1 (α + α c )} c Γ (α 1 c +2) }], (22) log(γ) + (1 1 ) log(θ) log(β + θ) log(γ(α + 1)) c + log (Γ (α + 1 c )) + log {α + βθ 1 (α + 1 c )}. (23) The generalized entropy for the submodels can be readily obtained as well. 4.2 Renyi Entropy An entropy of a random variable X is a measure of variation of the uncertainty. A popular entropy measure is Rényi entropy (1961). If X has the pdf f(. ), then Rényi entropy is defined by I R (b) = 1 log( 1 b gb (x)dx), where b > and b 1. Suppose X has the LGLW pdf, then for any real number b >, and b 1, b g LGLW (x) dx = cθ α+1 = { γ(β + θ)γ(α + 1) } j= [ cθ α+1 {α + β ( x γ )c } { γ(β + θ)γ(α + 1) } b b! j! (b j)! βj α b j b ( x γ ) b(cα 1) e θb(x γ )c ] dx ( x γ ) b(cα 1)+cj e θb(x γ )c 1 Let ν = θb ( x γ )c, and dx = 1 ( ν ) c 1 γ. Now, making the substitution, we have dν c θb θb b g LGLW cθ α+1 (x)dx = { γ(β + θ)γ(α + 1) } b b! j! (b j)! βj α b j γ c (θb) bα+b c j 1 cγ (bα b + j + 1 ). (24) c c Now, for any real number b >, and b 1, Rényi entropy is given by I R (b) = b 1 b log ( cθ α+1 γ(β + θ)γ(α + 1) ) j= dx.
14 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang b j b! log { j! (b j)! (β α ) α b γ c (θb) bα+b c j 1 cγ (bα b c + j + 1 c )}, j= for α, β, γ, c >. By taking the limit as b 1 and using L Hospital s rule, we obtain Shannon entropy (1948). Rényi entropy for the submodels can be readily obtained. 4.3 sentropy The sentropy is a one parameter generalization of the Shannon entropy and is defined by H s (g LGLW ) = 1 [1 g s s 1 LGLW(x) dx], s >, and s 1. (25) Now, if s R + and s 1, 1 cθ { α+1 H s (g LGLW ) = 1 [ γ(β+θ)γ(α+1) }s ( s j= j ) βj α s j ]. (26) s 1 ( x γ )s(cα 1)+cj e θs(x γ )c dx The integral in equation (26) follow directly from the result of the integral in Rényi entropy with s in place of b. For s=1, H s (g LGLW ) = E[log(g LGLW (X))], which is Shannon entropy (1948). Maximum Likelihood Estimation in the LGLW Distribution In this section, we obtain estimates of the parameters of the LGLW distribution. Methods of maximum likelihood (ML) estimation and asymptotic confidence intervals for the model parameters are presented. 5.1 Maximum Likelihood Estimators Suppose x = (x 1, x 2,, x n ) is a random sample of size n from the LGLW distribution. The loglikelihood function is given by: l(α, β, γ, c, θ) = n ln c + n(α + 1) ln θ n ln γ n ln(β + θ) n ln Γ(α + 1) + n( cα + 1) ln γ + (cα 1) ln x i n + ln {α + β ( x i i=1 θ ( x i i=1. (27) γ )c } n γ )c n i=1 The partial derivatives of l with respect to the parameters are: l α = n ln θ nγ (α+1) Γ(α+1) nc ln γ + c ln n i=1 n 1 x i + i=1, (28) α+β( x i γ )c
15 294 The Log Generalized LindleyWeibull Distribution with Application l and c = n c l = n + ( n β β+θ l = n + n( cα+1) γ γ γ l = n(α+1) n θ θ β+θ x i γ )c i=1, (29) α+β( x i γ )c n β( x i γ )c c i=1 + 1 θ γ (x i i=1, (3) n γ{α+β( x i γ )c } (x i γ )c n γ )c i=1, (31) x i γ )c ln{β( x i n nα ln γ + α ln x n i=1 i + β( γ )} n i=1 θ ( x i γ )c ln {θ ( x i i=1 )}. (32) γ {α+β( x i γ )c } The MLE of the parameters α, β, γ, θ and c, say α, β, γ, θ, and c are obtained by solving the equations l l l l l =, =, =, =, and =. There is no closed form solution, so these α β γ θ c equations must be solved numerically to obtain the MLE of the parameters α, θ, β, γ and c, Note that, if α, β, γ and c are known, it follows from equation (29) that n θ = β. (33) n i=1 ( x i γ ) c α+β( x i γ ) c When β, θ, γ, and c are known, it follows from equation (31) that α = θ + θ n β+θ n (x i i=1 γ )c 1. (34) When α, θ, γ, and c are known, it follows from equation (31) that n β = θ. (35) n(α+1) θ 5.2 Fisher Information n ( x i i=1 γ )c Let θ = (θ 1, θ 2, θ 3, θ 4, θ 5 ) = (α, β, θ, γ, c), and g LGLW (x; θ) the LGLW pdf. If log(g LGLW (x; θ)) is twice differentiable with respect to θ, and under certain regularity conditions, Fisher information matrix (FIM) is the 5 5 matrix whose elements are: I(θ) = E θ [ 2 log(g LGLW (X;θ)) θ i θ j ]. (36) The second and mixed partial derivatives of the loglikelihood function used to obtain the observed Fisher information matrix can be readily computed. 5.3 Asymptotic Confidence Intervals The 5 5 observed information matrix J(θ) = 2 l(θ) θ θ T can be used for interval estimation of α, β, θ, γ, and c, and for test of hypothesis on these parameters. Under conditions that are
16 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 295 fulfilled for parameters in the interior of the parameter space but not on the boundary, the asymptotic distribution of θ θ can be approximated by N 5 (, J(θ ) 1 ). Thus, the multivariate normal N 5 (, J(θ ) 1 ) distribution can be used to construct approximate confidence intervals and confidence regions for the parameters. In fact, the asymptotic 1(1 η)% confidence intervals for α, β, θ, γ and c are given by α ± Zη I 1 αα (θ ), β ± Zη I 1 ββ (θ ), θ ± Zη I 1 θθ (θ ), γ ± Zη I 1 γγ (θ ), and c ± Zη I 1 cc (θ ), where Zη is the (1 η )th quantile of the standard normal distribution. The likelihood ratio (LR) statistic is useful for testing the goodnessoffit of the LGLW model and for comparing it with other submodels such as generalized Lindley (GL) and Lindley (L) distributions. We can easily check if the fit using LGLW model is statistically superior to a fit using the GL model for a given data set by computing w = 2{l(α, β, θ, γ, c ) l(α, β, θ, 1,1)}, where α, β, θ, γ, and c are the unrestricted MLEs and α, β, and θ are the restricted estimates. Also, the LR statistic is asymptotically distributed under the null model as χ 2 2. Further, the LR 2 test rejects the null hypothesis if ω > ξ η, where ξ η denotes the upper 1η% point of the χ 2 distribution. Applications The maximum likelihood estimates (MLEs) of the parameters are obtained via the subroutine NLP in SAS. The maximum likelihood estimates (MLEs) of the parameters are obtained via the subroutine NLP in SAS. The estimates (standard error in parenthesis), 2 Log Likelihood, Akaike Information Criteria (AIC), Consistent Akaike Information Criterion (AICC), Bayesian Information Criterion (BIC), SS and KS values are given, where AIC = 2p 2 ln L, AICC = AIC + 2 p(p+1), BIC = p ln n 2 ln L, and KS = max {G n p 1 LGLW(x (j) ) j 1, j G 1 j n n n LGLW(x (j) )}, where L(θ ) = L is the value of the likelihood function evaluated at the estimated parameters, n is the number of observations, and p is the number of estimated parameters are given in Tables 1, 2 and 3. Probability plots (Chambers et al., 1983) are also presented in Figure 4, Figure 5 and Figure 6. For the probability plot, we plotted the estimated cdf G LGLW (x (j) ; α, β, θ, γ, c ) against j.375 n+.25, j = 1,2,, n, where x (j) are the ordered values of the observed data. We also computed a measure of closeness of each plot to the diagonal line. This measure of closeness is given by the sum of squares SS = n j=1 n+.25 )]2. [G LGLW (x (j) ; α, β, θ, γ, c ) ( j.375 This first data (Aarset, 1987) consists of the times to failure of 5 devices put on life test at time. The data are:
17 296 The Log Generalized LindleyWeibull Distribution with Application The results are given in Table 1, and plots are given in Figure 4. The estimated covariance matrix for the LGLW distribution is given by: E E 1 1.5E E E E 1 1.5E E E E E E E E E E E E E 11 ( 2.72E E E E E 8 ) The 95% asymptotic confidence intervals are: α ± 1.96(.1341), θ.176 ± 1.96(.5), β.161 ± 1.96(.6), γ ± 1.96(.1), c ± 1.96(.28). The LR test statistics of the hypotheses H : LGLW(α, θ, 1,1, c) vs H a : LGLW(α, θ, β, γ, c), H : LGLW(α, θ, β, γ, 1) vs H a : LGLW(α, θ, β, γ, c) and H : LGLW(1, θ, β, γ, c) vs H a : LGLW(α, θ, β, γ, c) are 43.2 (pvalue <.1), 34.9 (pvalue <.1) and 4.5 (pvalue <.1). We conclude LGLW(α, θ, β, γ, c) distribution is significantly better than the submodels. Also, LGLW(α, θ, β, γ, c) distribution gives the smallest AIC, AICC, BIC, SS and KS values. Consequently, we conclude that the LGLW(α, θ, β, γ, c) distribution is the best model for Aarset data. The second data set given by Murthy et al. (24) consists of the failure times of 2 mechanical components. The data are: The results and plots are given in Table 2 and Figure 5. The estimated covariance matrix for the LGLW distribution is given by: ( ) The LR test statistics of the hypotheses H : LGLW(α, θ, β, γ, 1) vs H a : LGLW(α, θ, β, γ, c) and H : LGLW(1, θ, β, γ, c) vs H a : LGLW(α, θ, β, γ, c) are 4.9 (pvalue =.27) and 11.8 (pvalue <.1). Therefore, we conclude that the LGLW(α, θ, β, γ, c) distribution is significantly better than the LGLW(α, θ, β, γ, 1) and LGLW(1, θ, β, γ, c) submodels. Also, note that the LGLW(α, θ, β, γ, c) distribution gives the smallest SS, KS values and second smallest AIC, AICC, BIC values when compare to gamma distribution. We conclude that the LGLW(α, θ, β, γ, c) distribution is a reasonably good model for the failure times data.
18 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 297 The third example consists of prices ( 1 4 dollars) of 428 new vehicles for the 24 year (Kiplinger's Personal Finance, Dec 23). The data are given in Table 3. The results and plots are given in Table 4 and Figure 6. The estimated covariance matrix for the LGLW distribution is given by: E E E E E 1 5.5E E E E 1 6.2E E E E E E E 7 ( E E E ) Plots of the fitted densities and the histogram, observed probability vs predicted probability, and empirical survival function are given in Figure 6. The LR test statistics of the hypotheses H : LGLW(α, θ, β, γ, 1) vs H a : LGLW(α, θ, β, γ, c) and H : LGLW(1, θ, β, γ, c) vs H a : LGLW(α, θ, β, γ, c) are 46. (pvalue <.1) and 12.6 (pvalue =.4), respectively. We conclude that the LGLW(α, θ, β, γ, c) distribution is significantly better than the submodels. Also, gamma distribution gives the smallest AIC, AICC, BIC, SS, KS values followed by the LGLW(α, θ, β, γ, c) distribution. Consequently, the gamma and LGLW(α, θ, β, γ, c) distributions are good models for prices of 24 new cars and trucks data. Concluding Remarks In line with results on generalized distributions and following the contents of the TX class of distributions (Alzaatreh et al., 213), we derive and present the mathematical and statistical properties of a new generalized Lindley distribution called log generalized LindleyWeibull (LGLW) distribution. This distribution contains several submodels including Lindley distribution and the generalized Lindley distribution of Zakerzadeh and Dolati (29). The hazard rate function of the LGLW distribution can be decreasing, decreasing or bathtub shaped. Moments and distributions of functions of random variables from the LGLW distribution are derived. Uncertainty measures including generalized entropy, Rényi and Shannon entropies are obtained. We discuss maximum likelihood estimation and hypotheses tests of the model parameters. The LGLW distribution permits testing the goodnessoffit of Lindley and generalized Lindley distribution by taking these distributions as submodels. Asymptotic confidence intervals for the parameters of the LGLW distribution are given. We fit the LGLW distribution and its submodels to three real data sets to demonstrate the potential importance, practical relevance and applicability of this model in lifetime analysis and other areas. Acknowledgments The authors thank the associate editor and anonymous referees for careful review of the manuscript and their suggestions and comments which lead to the improvement of the quality of the paper.
19 298 The Log Generalized LindleyWeibull Distribution with Application Table 1: LGLW Estimates of Models for Aarset Data Estimates Statistics α Θ β γ c 2 Log Likelihood AIC AICC BIC SS KS GLW(α, θ, β, γ, c) (.1341) (.5) (.6) (.1) (.2) GLW(α, θ, 1,1, c) (.15333) (.4351)   ( ) GLW(α, θ, β, γ, 1) (.9441) (.116) (.142) (.549)  GLW(1, θ, β, γ, c) (.2363) (.17642) (.266) (.15219) λ k Weibull(λ, k) ( ) ( ) α Β Gamma(α, β) ( ) (.2753) Table 2: LGLW Estimates of Models for Failure Times Data Estimates Statistics α Θ β γ c 2 Log Likelihood AIC AICC BIC SS KS GLW(α, θ, β, γ, c) (1.5231) (2.948) (1.2885) (.23) (.775) GLW(α, θ, β, γ, 1) E (1.62E18) (3.64E12) (1.28E25) (1.32E1)  GLW(1, θ, β, γ, c) E (.63) (1.19E5) (.1446) (.3491) λ k Weibull(λ, k) (.226) (.329) α Β Gamma(α, β) (2.731) (27.786)
20 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 299 Table 3: Prices of 24 New Cars and Trucks Data
21 3 The Log Generalized LindleyWeibull Distribution with Application Table 4: LGLW Estimates of Models for Prices of 24 New Cars and Trucks Data Estimates Statistics α Θ β γ c 2 Log AIC AICC BIC SS KS Likelihood GLW(α, θ, β, γ, c) (.1886) (.3) (.3) (.2) (.8) GLW(α, θ, β, γ, 1) E E (4.73E17) (4.29E1) (2.68E24) (3.1E1)  GLW(1, θ, β, γ, c) E (.3) (.18) (.5) (.7) λ k Weibull(λ, k) (.11) (.11) Α Β Gamma(α, β) (.35) (.13)
22 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 31 Figure 1: Plot of the pdf of LGLW distribution
23 32 The Log Generalized LindleyWeibull Distribution with Application Figure 2: Plot of Hazard Function
24 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 33 Figure 3: Plot of Hazard Function
25 34 The Log Generalized LindleyWeibull Distribution with Application Figure 4: Fitted Densities, Observed Probabilities, and Empirical Survival Functions Plots for Aarset Data
26 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 35 Figure 5: Fitted Densities, Observed Probabilities, and Empirical Survival Functions Plots for Failure Times Data
27 36 The Log Generalized LindleyWeibull Distribution with Application Figure 6: Fitted Densities, Observed Probabilities, and Empirical Survival Functions Plots for Prices of 24 New Cars and Trucks Data
28 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 37 References [1] Aarset, M. V. (1987). How to Identify Bathtub Hazard Rate, IEEE Transactions on Reliability, 36(1), [1] Aarset, M. V. (1987). How to Identify Bathtub Hazard Rate, IEEE Transactions on Reliability, 36(1), [2] Alzaatreh, A., Lee, C., and Famoye, F. (213). A New Method for Generating Families of Continuous Distributions, Metron, 71(1), [3] Bakouch, H. S., AlZahrani, B. M., AlShomrani, A. A., Marchi, V. A. A., and Louzada, F. (212). An Extended Lindley Distribution, Journal of the Korean Statistical Society, 41, [4] Chambers, J., Cleveland, W., Kleiner, B. and Tukey, J. (1983). Graphical Methods for Data Analysis, Chapman and Hall, London. [5] Cordeiro, G. M., Hashimoto, E., M., Ortega, E. M. M, and Pascoa, M. A., R. (212). The McDonald Extended Distribution: Properties and applications, AstA Advances in Statistical Analysis, 96(3), , 212. [6] Eugene, N., Lee, C., and Famoye, F. (22). Betanormal distribution and its applications, Communications in Statistics: Theory and Methods, 31, [7] Famoye, F., Lee, C., and Olumolade, O. (25). The BetaWeibull Distribution, Journal of Statistical Theory and Applications, [8] Ghitany, M. E., Alqallaf, F., AlMutairi, D. K., and Husain, H. A. (211). A TwoParameter Weighted Lindley Distribution and its Applications to Survival Data, Mathematics and Computers in Simulation, 81, [9] Ghitany, M. E., AlMutairi, D. K., Balakrishnan, N., and AlEnezi, L. J. (213). Power Lindley Distribution and Associated Inference, Computational Statistics and Data Analysis, 64, [1] Ghitany, M. E., Atieh, B., and Nadarajah, S. (28). Lindley Distribution and its Application, Mathematics and Computers in Simulation, 78(4), [11] Glaser, R. E. (198). Bathtub and related failure rate characterizations, Journal of American Statistical Association, 75, [12] Kleiber, C., and Kotz, S. (23). Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, New York.
29 38 The Log Generalized LindleyWeibull Distribution with Application [13] Krishna, H., and Kumar, K. (211). Reliability Estimation in Lindley Distribution with Progressively Type II Right Censored Sample, Mathematics and Computers in Simulation, 82, [14] Kumaraswamy, P., (198). Generalized Probability Density Function for DoubleBounded Random Process, Journal of Hydrology, 46, [15] Lindley, D. V. (1958). Fiducial Distributions and Bayes' Theorem, Journal of the Royal Society, Series B, 2, [16] Murthy, D. N. P., Xie, M., and Jiang, R. (24). Weibull models, Wiley series in probability and statistics, John Wiley and Sons. [17] Nadarajah, S., Bakouch, H. S., and Tahmasbi, R. (211). A generalized Lindley Distribution, Technical Report, School of Mathematics, University of Manchester, UK. [18] Oluyede, B. O., and Yang, T. (214). A New class of Generalized Lindley Distribution with Applications, To appear in the Journal of Statistical Computation and Simulation. [19] Pinho, L. G. B., Cordeiro, G. M., and Nobre, J. S. (212). The GammaExponentiated Weibull Distribution, Journal of Statistical Theory and Applications, 11(4), [2] Renyi, A. (1961). On Measures of Entropy and Information, Berkeley Symposium on Mathematical Statistics and Probability, 1(1), [21] Sankaran, M. (197). The Discrete PoissonLindley distribution, Biometrics, 26, [22] Shannon, E. (1948). A Mathematical Theory of Communication, The Bell System Technical Journal, 27 (1), , [23] Ristić, M. M., and Balakrishnan, N. (211). The GammaExponentiated Exponential Distribution, Journal of Statistical Computation and Simulation, 82(8), [24] Yang, T., and Oluyede, B. O. (214). The Exponentiated Kumaraswamy Lindley Distribution, Submitted. [25] Zakerzadeh, H., and Dolati, A. (29). Generalized Lindley Distribution, Journal of Mathematical Extension, 3(2), [26] Zografos, K., and Balakrishnan, N. (29). On Families of beta and Generalized Gamma Generated Distribution and Associated Inference, Statistical Methodology, 6, Received January 14, 215; accepted March 1, 215.
30 Broderick O. Oluyede, Fedelis Mutiso, Shujiao Huang 39 Broderick O. Oluyede Department of Mathematical Sciences Georgia Southern University, GA 346, USA Fedelis Mutiso Department of Biostatistics University of Washington, Seattle, WA , USA Shujiao Huang Department of Mathematics University of Houston, Houston, TX , USA
31 31 The Log Generalized LindleyWeibull Distribution with Application