On the Lindley family distribution: quantile function, limiting distributions of sample minima and maxima and entropies

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1 On the Lindley family distribution: quantile function, iting distributions of sample minima and maxima and entropies A. E. Bensid and H.Zeghdoudi Abstract. We propose in this paper an overview of several types of Lindley families distributions. More precisely, we give a treatment of the mathematical properties of new distributions named respectively gamma Lindley, Lindley Pareto, pseudo-lindley, quasi Lindley, two parameters Lindley and power Lindley distributions. The properties studied include the quantile function, entropy, and iting distribution of extreme order statistics. Simulations studies and data-driven applications are also reported. M.S.C. 2010: 60E05, 62H10. Key words: Lindley distribution; gamma Lindley distribution; pseudo Lindley distribution; quasi Lindley distribution, two parameters Lindley distribution, power Lindley distribution. 1 Introduction There are several distributions for modeling lifetime data. Among the known parametric models, the most popular are the gamma, lognormal and the Lindley distributions. The Lindley distribution is more popular than the gamma and lognormal distributions because it is flexible and modeling different types of lifetime data. Moreover, mixture models provide a mathematical-based, flexible and meaningful approach for the wide variety of classification requirements: Unsupervised or supervised classification, data features, a number of classes selection, etc. Fields in which mixture models have been successfully applied are numerous and specific software is now available. Although the Lindley distribution has drawn little attention in the statistical literature over the great popularity of the well-known exponential distribution, the considered form is closed [17], according to [12], the Lindley distribution is important for studying stress strength reliability modeling. Moreover, some researchers have proposed new classes of distributions based on modifications of the Lindley distribution, including also their properties. The main idea is always directed by embedding BSG Proceedings, Vol. 24, 2017, pp c Balkan Society of Geometers, Geometry Balkan Press 2017.

2 2 A. E. Bensid and H.Zeghdoudi former distributions to more flexible structures. Sankaran introduced in [19] the discrete Poisson-Lindley distribution by combining the Poisson and Lindley distributions. Ghitany et al. [9] investigated most of the statistical properties of the Lindley distribution, showing that this distribution may provide a better fitting than the exponential distribution. Mahmoudi and Zakerzadeh proposed in [16] an extended version of the compound Poisson distribution which was obtained by compounding the Poisson distribution with the generalized Lindley distribution. Louzada et al. proposed in [18] a complementary exponential geometric distribution by combining the geometric and the exponential distributions. That flexibility is the reason behind making the practitioners using it as the first choice to fit their data. Recently, Zeghdoudi and Nedar [25, 26, 28, 29], and Zeghdoudi and Lazri [24] introduced several distributions, called gamma Lindley, based on mixtures of gamma 2, θ and one-parameter Lindley θ distributions, pseudo Lindley, based on mixtures of gamma 2, θ and exponentialθ distributions, Lindley Pareto distributions. In this work, we give a treatment of the mathematical properties of new distributions named respectively Gamma Lindley, Lindley Pareto, pseudo-lindley, quasi Lindley, two parameters Lindley and power Lindley distributions. The studied properties include the quantile function, maximum likelihood estimation, and entropy. Simulation studies and data driven applications are also considered. 2 The quantile functions 2.1 The Lambert W function The Lambert W function has attracted a great deal of attention beginning with Lambert in 1758 and Euler in The name of Lambert W function has become a standard, after its implementation in the computer algebra system Maple in the 1980s and subsequent publication by Corless et al. [6] of a comprehensive survey of the history, theory and applications of this function. The Lambert W function is a multivalued complex function defined as the solution of the equation: 2.1 W zexpw z = z, where z is a complex number. If z is a real number such that z 1/e, then W z becomes a real function and there are two possible real branches. The real branch taking on values in, 1] is called the negative branch and denoted by W 1. The real branch taking on values in [ 1, is called the principal branch and denoted by W 0. Both real branches of W are depicted in Fig. 1. We emphasize that 2.1 has two possible solutions if z 1/e, 0], and a unique solution if z 0. For our results in this note, we shall use the negative branch W 1, which satisfies the following elementary properties: W 1 1/e = 1, W 1 z is decreasing as z increases to 0 and W 1 z as z 0. Lemma 2.1. Let a, b and c complex numbers. The solution of the equation z+ab z = c with respect to z C is z = c 1 logb W abc logb,

3 On the Lindley family distribution 3 Figure 1: Lambert W function graph where W denotes the Lambert W function. For details of the proof see [11]. The quantile function of X is Q X u = F 1 X u, 0 < u < 1. In the following, we give the explicit expressions for Q X in terms of the Lambert W function. θ 1+θ, the quantile function of the Gamma Lindley Theorem 2.2. For any θ 0, distribution X is 1 + θ Q X u = θ1 + θ θ θu 1 θ W θ θ e 1+θ 1+θ θ where W 1 denotes the negative branch of the Lambert W function., 0 u 1, Proof. For any fixed θ 0, θ 1+θ, let u 0, 1. We have to solve the equation F X x = u with respect to x, for x 0. We have to solve the following equation: 2.2 [θ [1 + θ θ] x θ] e θx = 1 + θ1 u. Multiplying by e θ 1+θ θ 1+θ θ θ [1 + θ θ] x θ 1 + θ θ at both sides of equation 2.2, we obtain: [ ] e θ[1+θ θ]x+1+θ 1 + θu 1 1+θ θ = 1 + θ θ e 1+θ 1+θ θ. From equation 2.3, we see that θ[1+θ θ]x+1+θ 1+θ θ is the Lambert W function of the real argument 1+θu 1 1+θ e 1+θ θ. Then, we have 1+θ θ 1 + θu W 1 + θ θ e 1+θ 1+θ θ = θ [1 + θ θ] x θ, 0 < u < θ θ Moreover, for any θ, 0 and x 0 it is immediate that θ[1+θ θ]x+1+θ 1+θ θ 0, and it can also be checked that [ ] θ [1 + θ θ] x θ e θ[1+θ θ]x+1+θ 1 + θu 1 1+θ 1+θ θ = 1 + θ θ 1 + θ θ e 1+θ θ 1 e, 0, since u 0, 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, equation 2.4 becomes 1 + θu 1 W θ θ e which completes the proof. 1+θ 1+θ θ = θ [1 + θ θ] x θ, 1 + θ θ

4 4 A. E. Bensid and H.Zeghdoudi 2.2 The quantile function of the Lindley Pareto distribution Theorem 2.3. The quantile function of the L-P distribution X is x γ = 1θ 1θ 1 LambertW 1, γ 1θ + 1 exp θ + 1 k, 0 < γ < 1, where θ,, k > 0 and LambertW denotes the negative branch of the Lambert W function W z expw z = z, and z is a complex number. Proof. For any fixed θ,, k > 0, let γ 0, 1. We have to solve the equation F X x = γ with respect to x, for x >, namely: θ x k 2.5 k xk + 1 exp θ k 1 = γ 1 θ + 1. Multiplying by e θ+1 both sides of equation 2.5, we obtain: θ θ k xk + 1 exp k xk + 1 = γ 1 θ + 1 e θ+1, and we see that 1 + xk k θ γ 1 θ + 1 e θ+1. Thus, we have 2.6 LambertW γ 1 θ + 1 e θ+1 = is the Lambert W function of the real argument θ k xk + 1. Moreover, for any θ,, k > 0 and x > it is immediate that 1 + θ x k > 1 and it k can also be checked that θ x k + 1 e θ +1 k k xk = γ 1 θ + 1 e θ+1 1 e, 0 since γ 0, 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, equation 2.6 becomes LambertW 1, γ 1θ + 1e θ+1 θ = k xk + 1. Again, solving for x, we get x γ = 1θ 1θ 1 LambertW 1, γ 1 θ + 1 e θ+1 k. 2.3 The quantile function of the pseudo Lindley distribution Theorem 2.4. For any θ 0, 1, the quantile function of the pseudo-lindley distribution X is Q X u = θ 1 θ W 1 e u 1, 0 < u < 1, where W 1 denotes the negative branch of the Lambert W function.

5 On the Lindley family distribution 5 Proof. For any fixed θ 0, 1, let u 0, 1. F X x = u with respect to x, for x 0, namely: θxe θx = 1 u. Multiplying by e both sides of equation 2.7, we obtain: θx e +θx = u 1e. We have to solve the equation From 2.8, we see that + θx is the Lambert W function of the real argument u 1e. Then, we have 2.9 W e u 1 = + θx. Moreover, for any θ, 0 and x 0 it is immediate that + θx 0 and it can also be checked that + θx e +θx = u 1e 1 e, 0, since u 0, 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, 2.9 becomes which in turn implies the result. W 1 e u 1 = + θx, 2.4 The quantile function of the quasi Lindley distribution Theorem 2.5. For any θ, > 1 e 1, the quantile function of the quasi Lindley distribution X is where Q X u = 1 θ θ 1 θ W 1 u exp 1, 0 < u < 1, W 1 denotes the negative branch of the Lambert W function. Proof. For any fixed θ, > 1 e 1, let u 0, 1. We have to solve the equation F X x = u with respect to x, for x > 0, namely: θx exp θx = 1 u + 1. Multiplying by exp 1 + both sides of the equation 2.10, we obtain: θx exp θx = u exp 1 +. From this equation, we see that θx is the Lambert W function of the real argument u exp 1 +. Then, we have W [u exp 1 + ] = θx. Moreover, for any θ, > 1 e 1 and x > 0, it is immediate that θx > 0, and it can also be checked that θx exp θx = u exp 1 + e, 0, since u 0, 1, > 1 e 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, equation 2.11 becomes W 1 [u exp 1 + ] = θx, which in turn implies the result.

6 6 A. E. Bensid and H.Zeghdoudi = 0.5 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 1.5 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 1 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 2 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod The quantile function of the two parameters Lindley distribution Theorem 2.6. For any θ, > 0, the quantile function of the two parameter Lindley distribution X is Q X u = θ + θ 1 [ θ W 1 1 u θ + exp θ + ], 0 < u < 1, where W 1 denotes the negative branch of the Lambert W function. Proof. For any fixed θ, > 0, let u 0, 1. We have to solve the equation F X x = u with respect to x, for x > 0, namely the equation: 2.12 θ + + θx = 1 u θ + expθx. Multiplying by 1 and adding , we obtain: 1 u θ + expθx θ+ θx u θ + expθx = θ. to both sides of equation From this, we see that θx θ+ is the Lambert W function of the real argument 1 1 u θ + exp θ+. Then, we have 2.13 θx + θ + = W [ 1 1 u θ + exp θ + ], 0 < u < 1. Moreover, for any θ, > 0 and x > 0, it is immediate that θx + θ+ also be checked that θx + θ + exp > 0, and it can θx + θ + = 1 1 u θ + exp θ + 1 e, 0, since u 0, 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, 2.13 becomes [ W u θ + exp θ + ] = θx + θ +, which in turn implies the result.

7 On the Lindley family distribution 7 = 0.5 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 1.5 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 1 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod = 2 θ = 0.1 θ = 0.5 θ = 5 Q Q 2 = median Q mean mod The quantile function of the power Lindley distribution Theorem 2.7. For any, > 0, the quantile function of the power Lindley distribution X is [ Q X u = ] 1 W 1 [ 1 u + 1 exp + 1], 0 < u < 1, where W 1 denotes negative branch of the Lambert W function. Proof. For any fixed, > 0, let u 0, 1. We have to solve the equation F X x = u with respect to x, for x > 0, namely 2.14 x 1 u + 1 exp x = + 1. From equation 2.14, we obtain: 2.15 x = + 1 W 1 u + 1 exp + 1. From equation 2.15, we see that x is the Lambert W function of the real argument 1 u + 1 exp + 1. Then, we have 2.16 x = W [ 1 u + 1 exp + 1], 0 < u < 1. Moreover, for any θ, > 0 and x > 0 it is immediate that θx + θ+ also be checked that x exp x + 1 = 1 u + 1 exp + 1 > 0 and it can 1 e, 0, since u 0, 1. Therefore, by taking into account the properties of the negative branch of the Lambert W function, equation 2.16 becomes x = W 1 [ 1 u + 1 exp + 1], which in turn implies the result. = 0.5 = 0.1 = 0.5 = 5 Q Q 2 = median Q mean = 1 = 0.1 = 0.5 = 5 Q Q 2 = median Q mean

8 8 A. E. Bensid and H.Zeghdoudi = 1.5 = 0.1 = 0.5 = 5 Q Q 2 = median Q mean = 2 = 0.1 = 0.5 = 5 Q Q 2 = median Q mean Limiting distributions of sample minima and maxima If X 1,..., X n is a random sample, and if X = X X n denotes the sample n n X EX mean then by the usual central it theorem approaches the standard V arx normal distribution as n. Sometimes one would be interested in the asymp- totic of the extreme values X n:n = maxx 1,..., X n and X 1:n = minx 1,..., X n. We can derive the asymptotic distributions of the sample minimum X 1:n and the sample maximum X n:n by using [1, Th , 8.3.3, and 8.3.6], so we have the following results. 3.1 Limiting distributions of sample minima and maxima of the gamma Lindley distribution The asymptotic distributions of the sample minimum X 1:n and the sample maximum X n,n of the Gamma Lindley distribution are { P {an X n:n b n x} D exp exp θx, P {c n X 1:n d n x} D 1 exp x, where the norming constants a n, b n, c n > 0 and d n are given in [14, Th ]. Proof. For all x > 0, by using l Hospital rule, it follows that, F tx t 0 F t = ftx x t 0 ft θ 2 + θ θxt + 1e θxt t 0 = x. 1 + θ θ 2 + θ θt + 1e θt 1 + θ Also, 1 F t+x t 1 F t t θ+ θθt+x+1+θ exp θx+t 1+θ θ+ θθt+1+θ exp θt 1+θ θ+ θθt+x+1+θ exp θx+t t θ+ θθt+1+θ exp θt θ+ θθt+x+1+θ exp θx t θ+ θθt+1+θ, and hence t 1 F t+x 1 F t = exp θx.

9 On the Lindley family distribution Limiting distributions of sample minima and maxima of the Lindley Pareto distribution The asymptotic distributions of the sample minimum X 1:n and the sample maximum X n,n of the Lindley Pareto distribution are { P {an X n:n b n x} D exp exp θ x k, P {c n X 1:n d n x} D 1 exp x 2k, where the norming constants a n, b n, c n > 0 and d n are given in [14, Th ]. Proof. By using the l Hospital rule, it follows that, for all x > 0, Also F tx t 0 F t ftx t 0 ft 1 F t+x t 1 F t t kθ 2 expθ θ+1 2k tx2k 1 exp θ tx k = x 2k. t 0 kθ 2 expθ θ+1 2k t2k 1 exp θ t k t 1 θ+1 k k +t+x k θ exp θ t+xk k 1 1 θ+1 k k +t k θ exp θ tk k 1 k +t+x k θ exp θ k t+xk t k, k +t k θ whence t 1 F t+x 1 F t = exp θ x k. 3.3 Limiting distributions of sample minima and maxima of the pseudo Lindley distribution The asymptotic distributions of the sample minimum X 1:n and the sample maximum X n,n of the pseudo Lindley distribution are { P {an X n:n b n x} D exp exp θx, P {c n X 1:n d n x} D 1 exp x, where the norming constants a n, b n, c n > 0 and d n are given in [14, Th ]. Proof. By using the l Hospital rule, it follows that, for all x > 0, Also F tx t 0 F t 1 F t + x t 1 F t t 0 ftx ft t 0 +θt+x x θ 1+θtx exp θtx θ 1+θt exp θt +θt + θt + x x + θt exp θt + x exp θt = x exp θx = exp θx.

10 10 A. E. Bensid and H.Zeghdoudi 3.4 Limiting distributions of sample minima and maxima of the quasi Lindley distribution The asymptotic distributions of the sample minimum X 1:n and the sample maximum X n,n of the quasi Lindley distribution are { P {an X n:n b n x} D exp exp θx, P {c n X 1:n d n x} D 1 exp x, where the norming constants a n, b n, c n > 0 and d n are given in [14, Th ]. Proof. By using the l Hospital rule, it follows that for all x > 0, we have Also, F tx t 0 F t 1 F t + x x 1 F t ftx t 0 ft t 0 θ+txθ +1 exp θtx θ+tθ +1 exp θt t 0 +txθ exp θtx +tθ exp θt = x. 1++θt+x θ+1 exp θt + x x 1++θt θ+1 exp θt θt + x exp θx = exp θx. x θt 3.5 Limiting distributions of sample minima and maxima of the two parameters Lindley distribution The asymptotic distributions of the sample minimum X 1:n and the sample maximum X n,n of the wo Parameters Lindley distribution are { P {an X n:n b n x} D exp exp θx, P {c n X 1:n d n x} D 1 exp x, where the norming constants a n, b n, c n > 0 and d n are given in [14, Th ]. Proof. By using the l Hospital rule, it follows that, for all x > 0, Also, F tx t 0 F t 1 F t + x t 1 F t t 0 ftx ft t 0 θ 2 1+tx θ+ θ 2 1+t θ+ θ++θt+x θ+ t θ++θt θ+ exp θtx = x. exp θt exp θt + x exp θt θ + + θt + x exp θx = exp θx. t θ + + θt

11 On the Lindley family distribution Limiting distributions of sample minima and maxima of the power Lindley distribution The asymptotic distribution of the sample minimum X 1:n of the power Lindley distribution is P {c n X 1:n d n x} D 1 exp x, where the norming constants c n > 0 and d n are given in [14, Th ]. Proof. By using the l Hospital rule, it follows that, for all x > 0, F tx t 0 F t t 0 ftx ft tx tx 1 exp tx t t t 1 exp t 1 + tx x 1 exp tx t t exp t = x. 1 F t+x Also, t 1 F t 4 Entropies t 1+ t+x +1 exp x+t 1+ t +1 exp t = 0. In many fields of science such as communications, physics and probability, entropy is an important concept to measure the amount of uncertainty associated with a random variable X. Several entropy measures and information indices are available but among them the most popular entropy measure called Rényi entropy is defined as Iζ = 1 1 ζ log f ζ xdx, for ζ > 1. As well, the Shannon entropy is defined by E { lnfx}. This is a special case derived from ζ 1 I ζ. 4.1 Rényi entropy of the Gamma Lindley distribution The Gamma Lindley distribution with parameters θ and is defined by fx = θ2 + θ θx + 1e θx, x, θ, > θ Then we have ζ f ζ xdx = θ2ζ e +θ θ ζ 1 + θ ζ E ζ ζ + θ θ,

12 12 A. E. Bensid and H.Zeghdoudi where ζ E ζ = u ζ e + θ θ 0 ζ +θ θ u du. Thus, the Rényi entropy of the GaLθ, distribution is given by [ I ζ = 1 1 ζ log θ 2ζ ζ e +θ θ ζ 1 + θ ζ E ζ ] ζ. + θ θ 4.2 Rényi entropy of the Lindley Pareto distribution The Lindley Pareto distribution with parameters θ,, and k is defined by its density function fx = kθ2 e θ x k θ + 1 2k x2k 1 exp θ, x >. Then f ζ xdx = k ζ θ 2ζ e ζθ x k x ζ2k 1 exp θζ dx θ+1 ζ 2ζk = kζ θ 2ζ e ζθ θ+1 ζ ζ x ζ2k 1 x k exp θζ dx, and substituting x = y i.e y = x, the above expression reduces to = kζ θ 2ζ e ζθ θ+1 ζ ζ k = ζ θ 2ζ e ζθ Γ 2kζ ζ+1 k θ+1 ζ ζ 1 f ζ xdx = k ζ 1 θ 2ζ e ζθ 1 θ 1 ζ k ζ 2kζ ζ+1 k y ζ2k 1 exp θζy k dy,θζ kθζ 2kζ ζ+1 k θ+1 ζ ζ 1 Γ 2kζ ζ+1 k, θζ. Thus the Rényi entropy of the LP θ,, k distribution is given by I ζ = 1 1 ζ ln θ 1 ζ k k ζ θ 2ζ e ζθ ζ 2kζ ζ+1 k Γ 2kζ ζ+1 k θ+1 ζ ζ 1 k,θζ, ζ > 0, ζ Rényi entropy of the pseudo Lindley distribution The pseudo Lindley distribution with parameters θ and is defined by its density function θ 1 + θx e θx fx; θ, =, x, θ, > 0. Then f ζ x; θ, dx = ζ θ 1 + θx ζ exp ζθx dx,

13 On the Lindley family distribution 13 and by substituting x = 1 θ z and using power series expansion a+bn = the above expression reduces to f ζ x; θ, dx = 1 θ = 1 θ = 1 θ = 1 θ ζ θ θ θ θ 1 + z ζ exp ζzdz R + ζ ζ ζ ζ 1 z ζ exp ζzdz ζ 1 ζ 1 Γ ζ ζ ζ. z ζ exp ζzdz Thus, the Rényi entropy of the P slθ, distribution is given by I ζ = 1 1 ζ ln 1 ζ θ ζ 1 Γ ζ θ ζ ζ. 4.4 Rényi entropy of the quasi Lindley distribution n a b n, The quasi Lindley distribution with parameters and θ is defined by its density function θ + xθ fx;, θ = exp θx ; x > 0, θ > 0, > Then ζ θ f ζ x;, θdx = + xθ ζ exp ζθx dx, + 1 and substituting x = 1 θ z and using the power series expansion a+bn = the above expression reduces to f ζ x;, θdx = = 1 θ = 1 θ = 1 θ 1 θ ζ θ + 1 θ + 1 θ + 1 θ z ζ exp ζz dz R + ζ ζ ζ ζ z ζ exp ζz dz ζ ζ Γ ζ ζ ζ. z ζ exp ζz dz n a b n,

14 14 A. E. Bensid and H.Zeghdoudi Thus, the Rényi entropy of the QL, θ distribution is given by Iζ = 1 1 ζ log 1 ζ θ ζ Γ ζ θ + 1 ζ ζ. 4.5 Rényi entropy of the two parameters Lindley distribution The two parameters Lindley distribution with parameters and θ is defined by its density function Then fx;, θ = θ2 1 + x θ + exp θx ; θ f ζ 2 ζ x;, θdx = θ + x > 0, θ > 0, > θ. 1 + x ζ exp ζθx dx, and substituting x = 1 z and by using power series expansion a+bn = the above expression reduces to f ζ x;, θdx = = 1 = 1 1 θ 2 ζ θ + θ 2 θ + θ 2 θ z ζ ζ ζ exp ζθ z dz ζ z exp ζθ z dz ζ z exp ζθ z dz. n a b n, If > 0, we have f ζ x;, θdx = 1 θ 2 θ + ζ ζ Γ. ζθ Thus, the Rényi entropy of the two parameter LD, θ distribution is Iζ = 1 1 ζ log 1 θ 2 ζ ζ Γ θ +. If θ < < 0, we have Iζ = 1 1 ζ log 1 θ 2 ζ θ + ζ Γ ζθ ζθ, 1 2n, n N.

15 On the Lindley family distribution Rényi entropy of the power Lindley distribution The power Lindley distribution with parameters and is defined by its density function fx;, = x x 1 exp x ; x > 0, > 0, > 0. Then f ζ 2 ζ x;, θdx = 1 + x ζ x ζ 1 exp ζx dx, + 1 and substituting x = z 1 the above expression reduces to f ζ x;, θdx = where y = 1. If y > ζ 1, then and using power series expansion a + b n = = ζ f ζ x;, θdx = ζ n a b n, ζ z z ζ 1 1 z exp ζzdz. ζ z +ζy y exp ζzdz, ζ ζ Γ + ζy y ζ +ζy y. Thus, the Rényi entropy of the P ower Lindley P L, distribution is Iζ = 1 1 ζ log 1 2 ζ ζ Γ + ζy y + 1 ζ +ζy y. In the opposite case, we infer Iζ = 1 1 ζ log 1 2 ζ ζ Γ ζy + y + 1 ζ ζy+y.

16 16 A. E. Bensid and H.Zeghdoudi 5 Illustration According to [13, pp. 204, 263] - we consider two series of real data. The first one represents the failure times mm for a sample of fifteen electronic components in an acceleration life test: 1.4, 5.1, 6.3, 10.8, 12.1, 18.5, 19.7, 22.2, 23, 30.6, 37.3, 46.3, 53.9, 59.8, The second set of data are the number of cycles to failure for cm specimens of year, tested at a particular strain level: 15, 20, 38, 42, 61, 76, 86, 98, 121, 146, 149, 157, 175, 176, 180, 180, 198, 220, 224, 251, 264, 282, 321, 325, 653. The results of these data are presented in the following table Table 1 Comparison between distributions Data Distribution θ γ LL K S AIC BIC Serie1 power Lindley P sl n=15 GaLD m= QLD s=20.06 T wop LD Gamma W eibull Lognormal Serie2 power Lindley P sl n=25 GaL m= QLD s = T wop LD Gamma W eibull Lognormal According to Table 1, we can observe that the pseudo Lindley distribution provides the smallest -LL, AIC, and BIC values, as compared to gamma Lindley, Lindley Pareto, quasi Lindley, two parameters Lindley, gamma, Weibull, Lognormal and power Lindley distributions, and hence best fits the data among all the models considered. 6 Conclusions We introduce the new distributions gamma Lindley, Lindley Pareto, pseudo Lindley, quasi Lindley, two parameter Lindley and power Lindley. Then, we study some of their properties, as: the function quantile, the iting distributions of sample minima and maxima, and the Rényi entropy. Our future concerns involve using censured data. We will be able in our future research to propose and investigate other distributions, as: - the size biased gamma Lindley distribution; - the size biased pseudo Lindley distribution. Acknowledgements. The authors acknowledge the editors of BSGP for the constant encouragements to finalize the paper.

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