Exponentiated power Lindley distribution

A new generalization of the Lindley distribution is recently proposed by Ghitany et al. [1], called as the power Lindley distribution. Another generalization of the Lindley distribution was introduced by Nadarajah et al. [2], named as the generalized Lindley distribution. This paper proposes a more generalization of the Lindley distribution which generalizes the two. We refer to this new generalization as the exponentiated power Lindley distribution. The new distribution is important since it contains as special sub-models some widely well-known distributions in addition to the above two models, such as the Lindley distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Least square estimation is used to evaluate the parameters. Three algorithms are proposed for generating random data from the proposed distribution. An application of the model to a real data set is analyzed using the new distribution, which shows that the exponentiated power Lindley distribution can be used quite effectively in analyzing real lifetime data.

Journal of Advanced Research (2015) 6, 895–905 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Exponentiated power Lindley distribution Samir K Ashour a b a,* , Mahmoud A Eltehiwy b Institute of Statistical Studies & Research, Department of Mathematical Statistics, Cairo University, Egypt Faculty of Commerce, Department of Statistics, South Valley University, Egypt A R T I C L E I N F O Article history: Received 15 July 2014 Received in revised form 11 August 2014 Accepted 17 August 2014 Available online 24 August 2014 Keywords: Lambert function Least square estimation Maximum likelihood estimation Order statistics Power Lindley distribution A B S T R A C T A new generalization of the Lindley distribution is recently proposed by Ghitany et al [1], called as the power Lindley distribution Another generalization of the Lindley distribution was introduced by Nadarajah et al [2], named as the generalized Lindley distribution This paper proposes a more generalization of the Lindley distribution which generalizes the two We refer to this new generalization as the exponentiated power Lindley distribution The new distribution is important since it contains as special sub-models some widely well-known distributions in addition to the above two models, such as the Lindley distribution among many others It also provides more flexibility to analyze complex real data sets We study some statistical properties for the new distribution We discuss maximum likelihood estimation of the distribution parameters Least square estimation is used to evaluate the parameters Three algorithms are proposed for generating random data from the proposed distribution An application of the model to a real data set is analyzed using the new distribution, which shows that the exponentiated power Lindley distribution can be used quite effectively in analyzing real lifetime data ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University Abbreviations: EPLD, Exponentiated power Lindley distribution; PLD, Power Lindley distribution; BGLD, Beta generalized Lindley distribution; GLD, Generalized Lindley distribution; LD, Lindley distribution; MW, Modified Weibull distribution; WD, Weibull distribution; EE, Exponentiated exponential distribution; Cdf, Cumulative distribution function; pdf, Probability density function; h(t), Hazard rate function; Q(p), Quantile function; E(Xr), The rth moment; MX(t), The moment generating function; MLE, Maximum likelihood estimator; LSE, Least square estimator; MSE, mean square error; L, Log-likelihood function; K–S, Kolmogorov– Smirnov test; AIC, Akaike information criterion; BIC, Bayesian information criterion * Corresponding author E-mail address: ashoursamir@hotmail.com (S.K Ashour) Peer review under responsibility of Cairo University Production and hosting by Elsevier Introduction Lindley [3], introduced a one-parameter distribution, known as Lindley distribution, given by its probability density function gt; hị ẳ h2 ỵ tịeht ; 1ỵh t > 0; h > 0: 1ị It can be seen that this distribution is a mixture of exponential (h) and gamma (2, h) distributions Its cumulative distribution function has been obtained as Gtị ẳ h ỵ ỵ ht ht e ; hỵ1 t > 0; h > 0: ð2Þ Ghitany et al [4] have discussed various properties of this distribution and showed that in many ways that the pdf given by (1) provides a better model for some applications than the 2090-1232 ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University http://dx.doi.org/10.1016/j.jare.2014.08.005 896 S.K Ashour and M.A Eltehiwy exponential distribution Bakouch et al [5] obtained an extended Lindley distribution and discussed its various properties and applications Ghitany et al [6] developed a two-parameter weighted Lindley distribution and discussed its applications to survival data Nadarajah et al [2] obtained a generalized Lindley distribution and discussed its various properties and applications Merovci and Elbatal [7] use the quadratic rank transmutation map in order to generate a flexible family of probability distributions taking Lindley-geometric distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility and called it transmuted Lindley-geometric distribution Asgharzadeh et al [8] introduced a general family of continuous lifetime distributions by compounding any continuous distribution and the Poisson–Lindley distribution Oluyede and Yang [9] proposed a new four-parameter class of generalized Lindley (GLD) distribution called the beta-generalized Lindley distribution (BGLD) This class of distributions contains the betaLindley, GLD and Lindley distributions as special cases A two parameter power Lindley distribution (PLD), of which the Lindley distribution ‘Eq (1)’ is a particular case, has been suggested by Ghitany et al [1] They introduced a new extension of the Lindley distribution by considering the power transformation of the r.v Y = T1/b The pdf of the Y is readily obtained to be power Lindley distribution with parameters b and h and is defined by its probability density function pdf gy; hị ẳ h2 b1 ỵ yb ị b1 hyb y e ; hỵ1 y > 0; h; b > 0; ð3Þ It can easily be seen that at b = , Eq (3) reduces to the Lindley distribution From Eq (2), we see that the power Lindley distribution is a two-component mixture of Weibull distribution (with shape b and scale h), and a generalized gamma distribution (with shape parameters b and scale h), with mixing h proportion p ẳ 1ỵh gy; a; hị ẳ pf1 yị ỵ pịf2 yị; where p ẳ 4ị h , 1ỵh b f1 yị ẳ hbyb1 ehy ; y > 0; h; b > 0; ð5Þ and b f2 yị ẳ h2 by2b1 ehy ; y > 0; h; b > 0: ð6Þ In this paper, we introduce a new distribution with three parameters, referred to as the exponentiated power Lindley (EPLD) distribution, with the hope that it will attract many applications in different disciplines such as survival analysis, reliability, biology and others One of the main goals to introduce this new distribution was that it involves three distributions as sub-models Generally, the EPLD distribution generalizes the generalized Lindley (GLD) [2], power Lindley (PLD) [1] and Lindley (LD) [3] distributions The third parameter indexed to this distribution makes it more flexible to describe different types of real data than its sub-models The EPLD distribution, due to its flexibility in accommodating different forms of the hazard function, seems to be a suitable distribution that can be used in a variety of problems in fitting survival data The EPLD distribution is not only convenient for fitting comfortable bathtub-shaped failure rate data but also suitable for testing goodness-of-fit of some special submodels such as the PLD, GLD and LD distributions The new extension of the power Lindley distribution is most conveniently specified in terms of the cumulative distribution function:   !a hxb b ehx ; Fxị ẳ þ ð7Þ hþ1 for x > 0, h, b, a > and the corresponding probability density function (pdf) is given by   !aÀ1 Á ah2 bxbÀ1 À hxb b b ỵ xb ehx ỵ ehx fxị ẳ ; 8ị hỵ1 hỵ1 The corresponding hazard rate function is hxị ẳ ah2 b bxb1 ỵ xb ehx h ỵ 1ị   !a1 hxb b ehx 1ỵ fSxịg ; hỵ1 9ị where   !a hxb b ehx : Sxị ẳ ỵ hỵ1 10ị Note that Eq (8) has closed form survival functions and hazard rate functions For (b = 1) , (a = 1) and (a = b = 1) we have the pdfs of generalized Lindley distribution , power Lindley and Lindley distributions respectively As we shall see later, Eq (8) has the attractive feature of allowing for monotonically decreasing; monotonically increasing and bath tub shaped hazard rate functions while not allowing for constant hazard rate functions Another motivation for the new distribution in (Eq (7)) can be described as follows Consider the two parameters power Lindley distribution [1] specified by the cumulative distribution function:   ! hxb b eÀhx ; ð11Þ FpL xị ẳ ỵ hỵ1 for x > and h, b > Suppose X1, X2 , , Xa are independent random variables distributed according to (Eq (11)) and represent the failure times of the components of a series system, assumed to be independent Then the probability that the system will fail before time x is given by PrðmaxðX1 ;X2 ; ;Xa Þ xị ẳ PrX1 xịPrX2 xị PrXa xị ẳ FpL xịFpL xị   !a hxb b FpL xị ẳ ỵ ehx : hỵ1 So, Eq (7) gives the distribution of the failure of a series system with independent components From Eqs (4)–(6), it follows that cumulative distribution function, Eq (7), can be represented as   X a aÀi aÀi i Fxị ẳ xịFGG2 xị; p pịi FW i iẳ0 b where FW xị ẳ ehx Þ is the cumulative distribution function of Weibull random variable with shape b and scale h Exponentiated power Lindley distribution 897 b w(y) = [F(y)]aÀ1 The weight function is an increasing or decreasing if a > or a < respectively Therefore, if Y is decreasing pdf then for a > 1, The pdf of X also is a decreasing pdf If Y has a unimodal pdf, then Mode(X) > ( ( ( 0; a ; È É II 12 < b < 1; h P g1 ðbÞ; a , pffiffiffiffiffiffiffiffiffiffi 12 b1bị , where g1 bị ẳ b III {b = 1, h P 1, a = 1}; IV {b P 1, h > 0, a < 1} b Unimodal if I {b = 1, h > 0, a > 1}; II {b = 1, < h < 1, a = 1}; III {b > 1, h > 0, a P 1} À c Decreasing–increasing–decreasing if 12 < b < 1; < h < g1 bị; a ẳ 1Þ Fig illustrates some of the possible shapes of the density function of exponentiated power Lindley distribution for selected values of the parameters (h, b, a) The behavior of h(x) at x = and x = 1, respectively, are given by Shapes In this section, we discuss the shape characteristics of the pdf f(x) in Eq (8) of the EPL(h, b, a) distribution The behavior of f(x) at x = and x = 1, respectively, are given by > < 1; if b < or a < 1; h2 f0ị ẳ hỵ1 ; if b ¼ and a ¼ ; fð1Þ ¼ 0: > : 0; if b > or a > 1; h0ị ẳ > < 1; if b < or a 1; h2 ; > hỵ1 if b ¼ and a ¼ 1; : 0; if b > or a P 1; if b < or a 1; > < 0; h2 ; if b ẳ and a ẳ 1; h1ị ẳ hỵ1 > : 1; if b > or a P 1; If X has the pdf (8), the pdf of X is a weighted version of the pdf of Y in Eq (3) and the weight function in this case is 1.5 f ( x , , 0.2 , 9) f ( x , , , 1) f ( x , , , 5) 0.8 f ( x , , , 2) 0.6 f ( x , , , 1) f ( x , 1.5 , , 1.5) 0.4 0.5 0.2 0 x 4 x 0.12 f ( x , , 85 , 1) f ( x , 35 , , 1) 0.1 0.08 0.06 Fig x Probability density function of the EPLD(h, b, a) for selected values of the parameters 898 S.K Ashour and M.A Eltehiwy 1.5 h ( x , , , 1.5) h ( x , , , 1) h ( x , , , 1) h ( x , 10 , , 8) 0.5 2 x 10 x 10 h ( x , 1.2 , 88 , 1) h ( x , 1.5 , , 1) h ( x , , , 05) 0.9 h ( x , , , 09) 0.8 0.7 Fig 2 x 10 x Hazard rate function of the EPLD(h, b, a) for selected values of the parameters Ghitany et al [1] discussed and proved the cases in which the hazard rate function of the power Lindley distribution is decreasing, increasing and decreasing–increasing–decreasing It follows that the HRF h(x) in Eq (9) of the EPLD distribution is (a) Decreasing if È É I È0 < b 12 ; h > 0; a ; É ð2bÀ1Þ2 II 12 < b < 1; h P g2 bị; a , where g2 bị ẳ 4bð1ÀbÞ ; (b) Increasing if {b P 1, h > 0, a P 1} È (c) Decreasing–increasing–decreasing if 12 < b < 1; < h < g2 ðbÞ; a ¼ 1g Fig shows the HRF h(x) of the exponentiated power Lindley distribution for some choices of values of the parameters (h, b, a) Stochastic orders Theorem Let X1 $ EPLD(h1, b, a1) and X2 $ EPLD (h2, b, a2) If h1 = h2, and a1 > a2 (or if a1 = a2 and h1 P h2), then X1 LrX2 and hence X1 hrX2 and X1 stX2 Proof f1 xị a1 h21 h2 ỵ 1ịb1 xb1 b b ẳ ỵ xb1 ịeh2 x Àh1 x Þ f2 ðxÞ a2 h22 ðh1 þ 1Þb2 xb2 À1   !a1 À1 h1 xb1 b eh1 x 1ỵ h1 ỵ 1ị   !1Àa2 h2 xb2 b eÀh2 x 1À 1ỵ ; h2 ỵ 1ị thus @ f xị ẳ ln @x f2 ðxÞ Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behavior Suppose Xi is distributed according to Eqs (7) and (8) with common shape parameter b and parameters hi and for i = 1, Let Fi denote the cumulative distribution of Xi and let fi denote the probability density function of Xi A random variable X1 is said to be smaller than a random variable X2 in the I Stochastic order ðX 6st X Þ if F1(x) P F2(x) for all x II Hazard rate order ðX 6hr X Þ if h1(x) P h2(x) for all x III Likelihood ratio order ðX 6Lr X Þ if ff 12 ðxÞ decreases in x ðxÞ The following results due to Shaked and Shanthikumar [10] are well known for establishing stochastic ordering of distributions X1 6Lr X2 ) X1 6hr X2 ) X1 6st X2 : The EPLD is ordered with respect to the strongest ‘‘likelihood ratio’’ ordering as shown in the following theorem:     b1 h1 xb1 À1 À b2 h2 xb2 b1 b2 ỵ h1 xb1 h2 xb2 x  b ỵb  b1 1 x b1 b2 ị ỵ b1 x b2 xb2 ỵ xb1 ỵ 1ịxb2 ỵ 1ị b a1 1ịh21 b1 xb1 1 ỵ xb1 ÞeÀh1 x     b b ðh1 þ 1Þ À þ hh11xþ11 eÀh1 x þ ða2 À 1Þh22 b2 xb2 À1 ð1 þ xb2 ÞeÀh2 x    : b b ðh2 þ 1Þ À þ hh22xþ12 eÀh2 x b @ Case (i) if h1 = h2, and a1 > a2, then @x ln ff12 ðxÞ < This means ðxÞ that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2 @ Case (ii) if h1 P h2, and a1 = a2 = a, then @x ln ff1 ðxÞ ðxÞ < This means that X1 6Lr X2 and hence X1 6hr X2 and X1 6st X2 h Moments Theorem The rth moments E(Xr) of a exponentiated power Lindley random variable X, is given by Exponentiated power Lindley distribution 899 a EXr ị ẳ EXr ị ẳ Ci;k hỵ1     C rỵbkỵ1ị C rỵbkỵ2ị b b 5; ỵ rỵkb rỵbkỵ1ị rỵbkỵ2ị rỵkb h b 1ị :i ỵ 1ịC b ị h b ị :i ỵ 1ịC b ị where Ci;k ¼ P1 P1 i¼0  k¼0 aÀ1 i Theorem Let X have an exponentiated power Lindley distribution Then the moment generating function of X, MX(t), is     C jỵbkỵ1ị C jỵbkỵ2ị b b a MX tị ẳ Ci;k;j 4 jỵkb   ỵ  jỵkb  5; jỵbkỵ1ị jỵbkỵ2ị hỵ1 i ỵ 1ịC b ị h b i ỵ 1ÞCð b Þ hb    k i h 1ịi hỵ1 k Proof iẳ0 kẳ0  jẳ0    k i h tj 1ịi hỵ1 j! k a1 i Proof EXr ị ẳ Z xr fxịdx; MX tị ẳ Z Z etx fxịdx;   !a1 ah2 bxrỵb1 hxb b b ỵ xb ehx ỵ ehx dx hỵ1 hỵ1   ! Z rỵb1 aÀ1 ah bx hxb b b ¼ eÀhx eÀhx ỵ dx hỵ1 hỵ1   !a1 Z rỵ2b1 ah bx hxb b b ỵ ehx ehx ỵ dx: 12ị hỵ1 hỵ1 EXr ị ẳ MX tị ẳ Z ẳ Z ỵ Z   !a1 ah2 bxb1 hxb b b ỵ xb etx ehx ỵ ehx dx hỵ1 hỵ1   !aÀ1 ah2 bxbÀ1 tx Àhxb hxb b eÀhx e e 1ỵ dx hỵ1 hỵ1 Using the following binomial h   iaÀ1 b hxb eÀhx given by ỵ hỵ1 series expansion of   !aÀ1 X   i  aÀ1 hxb hxb b b 1ỵ ẳ eihx ; ehx 1ịi ỵ hỵ1 h ỵ i iẳ0 Eq (12) takes the following form k Z b1ỵrỵkb   X i h ah bx b eÀhx iỵ1ị dx h ỵ h ỵ i k iẳ0 kẳ0  k Z 2b1ỵrỵkb    X X aÀ1 i h ah bx b 1ịi ehx iỵ1ị dx ỵ hỵ1 hỵ1 i iẳ0 kẳ0 k EXr ị ẳ   X aÀ1 Let t = x and Ci;k ¼ P1 i¼0      k P aÀ1 i h 1ịi kẳ0 k hỵ1 i EX ị ẳ Ci;k rỵkb ah2 t b htiỵ1ị e dt ỵ Ci;k hỵ1 Z expansion of Eq (13) takes the following form MX tị ẳ a1 X ỵ ! i 1ị i iẳ0 a1 X X i kẳ0 k ! 1ịi i iẳ0 X kẳ0 ! h hỵ1 ! i  k k Z h hỵ1 k Z ah2 bxb1ỵkb tx hxb iỵ1ị e e dx hỵ1 ah2 bx2b1ỵkb tx hxb iỵ1ị e e dx: hỵ1 Using the following expansion of etx given by j j X tx j¼0 j! : Eq (13) can be rewritten as follows: rỵbkỵ1ị ah2 t b hỵ1 series h ii hxb and using the following binomial series expansion of ỵ hỵ1 given by !i X k   i hxb h 1ỵ ẳ xkb : hỵ1 hỵ1 k kẳ0 etx ¼ Eq (12) can be rewritten as follows: Z Using the following binomial h   iaÀ1 b hxb eÀhx given by ỵ hỵ1 1ịi b   !aÀ1 ah2 bx2bÀ1 tx Àhxb hxb b eÀhx e e 1ỵ dx: 13ị hỵ1 hỵ1   !a1 X   i  aÀ1 hxb hxb b b i ehx 1ị ỵ 1ỵ ẳ eihx ; hỵ1 hỵ1 i iẳ0 h ii hxb and using the following binomial series expansion of ỵ hỵ1 given by !i X k   i hxb h 1ỵ ẳ xkb ; h ỵ hỵ1 k kẳ0 r P1 P1 P1 where Ci;k;j ẳ htiỵ1ị e dt; MX tị ẳ a1 X     C rỵbkỵ1ị C rỵbkỵ2ị b b a EXr ị ẳ Ci;k 4 rỵkb    ỵ  rỵkb   5: rỵbkỵ1ị rỵbkỵ2ị hỵ1 h b i ỵ 1ịC b ị h b ; i ỵ 1ịC b ị ỵ X a1 b X i k¼0 k ! i i¼0 Let t = x  k h tj j! hỵ1 i 1ị i iẳ0   C rỵbkỵ1ị b a a r EX ị ẳ Ci;k  rỵkb   ỵ Ci;k rỵbkỵ1ị C b ị 1ị hỵ1 h ỵ b i ỵ 1ị h   C rỵbkỵ2ị b  rỵkb  ; rỵbkỵ2ị ị b h i ỵ 1ịC b ị ! 1ịi ! X i kẳ0 k h hỵ1 ! and Ci;k;j ẳ k X j Z t ah2 bxb1ỵkbỵj hxb iỵ1ị e dx j! hỵ1 jẳ0 h hỵ1 k X j Z t ah2 bx2b1ỵkbỵj hxb iỵ1ị e dx: j! hỵ1 jẳ0 P1 P1 P1 iẳ0 kẳ0 jẳ0  a1 i   i 1ịi k Eq (13) can be rewritten as follows: MX tị ẳ Ci;k;j Z kbỵj ah2 t b htiỵ1ị e dx ỵ Ci;k hỵ1 Z bkỵ1ịỵj ah2 t b hỵ1 ehtiỵ1ị dx; 900 S.K Ashour and M.A Eltehiwy  jỵbkỵ1ị  Algorithm I (mixture form of the Lindley distribution) C b a MX tị ẳ Ci;k;j  jỵkb   jỵbkỵ1ị hỵ1 i ỵ 1ịC b ị hb  ỵ jỵbkỵ2ị b  C a Ci;k;j  jỵkb  ; jỵbkỵ2ị hỵ1 i ỵ 1ịC b ị hb     C jỵbkỵ1ị C jỵbkỵ2ị b b a MX tị ẳ Ci;k;j 4 jỵkb   ỵ  jỵkb  5: jỵbkỵ1ị jỵbkỵ2ị hỵ1 i þ 1ÞCð b Þ h b À1 ði þ 1ÞCð b Þ hb à Quantile function Let X denotes a random variable with the probability density function (Eq (8)) The quantile function, say Q(p), defined by F(Q(p)) = p is the root of the equation # n o hẵQpịb exp hẵQpịb ẳ p1=a ; 1ỵ 1ỵh Generate Ui $ Unifrom(0, 1), i = 1, , n; Generate Vi $ Exponential(h), i = 1, n; Generate Gi $ Gamma(2, h), i = 1, n; 1=a 1=b h , then set Xi ¼ Vi , otherwise, If Ui p ¼ 1ỵh 1=b set Xi ẳ Gi ; i ẳ 1; n Algorithm II (mixture form of the power Lindley distribution) Generate Ui $ Unifrom(0, 1), i = 1, , n; Generate Yi $ Weibul(h, b), i = 1, n; Generate Si $ GG(2, h, b), i = 1, n; 1=a h If Ui p ẳ 1ỵh , then set Xi = Yi, otherwise, set Xi = Si,i = 1, n " ð14Þ for < p < Substituting Z(p) = À À h À h[Q(p)]b, one can rewrite Eq (14) as Algorithm III (inverse cdf) Generate Ui $ Unifrom(0, 1), i = 1, , n; Set ZpịexpfZpịg ẳ ỵ hịexp1 hị1 p1=a ị; & Xi ẳ for < p < So, the solution for Z(p) is Zpị ẳ W1 ỵ hịexp1 hị1 p1=a ịị;  i'1=b 1 h W ỵ hịexp1 À hÞ À U1=a ; i h h ð15Þ where W(.) denotes the Lambert W function for < p < 1, where W(.) is the Lambert W function, see Corless et al [11] for detailed properties Inverting Eq (15), one obtains !1=b 1 1=a Qpị ẳ W1 ỵ hịexp1 hị1 p ịị ; h h ð16Þ for < p < The particular case of Eq (16) for (a = b = 1) has been derived recently by Jodra´ [12] Algorithm IV Generate Ui $ Unifrom(0, 1), i = 1, , n; Solve numerically the following equation in v (0, 1):   À vi ẵh ỵ ln vi ẳ 0: h ỵ 1ị U1=a i Set Xi ẳ Generation algorithms Here, we consider simulating values of a random variable X with the probability density function in Eq (8) Let U denote a uniform random variable on the interval (0, 1) One way to simulate values of X is to set ! È É hxb exp Àhxb ¼ U1=a ; 1ỵ 1ỵh and solve for X, i.e use the inversion method Using Eq (16), we obtain X as & X¼ ' À Áà 1=b 1  W ỵ hịexp1 hị À U1=a ; h h where W[.] denotes the Lambert W function We propose three different algorithms for generating random data from the exponentiated power Lindley distribution: ÀÀ ln vi Á1=b h Maximum likelihood estimation of parameters Let x1, , xn be a random sample of size n from EPLD Then, the log-likelihood function is given by La; b; hị ẳ n X ln fxi ị; iẳ1 ẳ nẵln aị ỵ ln bị ỵ lnhị lnh ỵ 1ị n n X X ln ỵ xbi ỵ b 1ị ln xi ị ỵ iẳ1 iẳ1 n n X X ln Ai b; hị: h xbi ỵ a 1ị iẳ1 i¼1 h   i b hxb where Ai ðb; hị ẳ ỵ hỵ1i ehxi ; i ¼ 1; ; n ð17Þ Exponentiated power Lindley distribution 901 ^ ^ The MLEs ^h; b; a of h, b, a are then the solutions of the following non-linear equations: n @ nh ỵ 2ị X La;b;hị ẳ xb @h hh ỵ 1ị iẳ1 i " # n a 1ị X Ai;h b;hị ỵ ẳ 0; ỵ hị2 iẳ1 Ai b;hị 18ị n n @ n X xbi ln ðxi Þ X Lða; b; hị ẳ ỵ ỵ ln xi ị b @b b iẳ1 x ỵ iẳ1 n n X X Ai;b b; hị ẳ 0; h xbi :ln xi ị ỵ a 1ị Ai b; hị iẳ1 iẳ1 19ị n @ n X La; b; hị ẳ ỵ ln Ai b; hị ẳ 0; @a a iẳ1 20ị where n n n X @2L X xbi ln ðxi Þ2 n X x2b b i ln ðxi Þ ¼ À À hx ln ð x Þ À i i @b2 b2 iẳ1 xbi ỵ xbi ỵ iẳ1 iẳ1 h i n ða À 1Þ Ai ðb; hÞ:A ðb; hÞ À Ai;b b; hị X i;b ỵ ; ẵAi b; hị2 i¼1 " # @2L ¼ Àn À @h2 h h ỵ 1ị2 h i n a 1ị Ai b; hịA b; hị ẵAi;h b; hị X i;h ; ỵ ẵAi b; hị2 iẳ1 n X @2L Ai;h;a b; hị ẳ ; @h@a Ai b; hị iẳ1 From (20) we can obtain the MLE of a as a function of (b, h), say ^aðb; hị, where n : ln Ai b; hị iẳ1 @ L Àn ¼ ; @a2 a n X @2L Ai;b;a b; hị ẳ ; @b@a Ai b; hị iẳ1 b @Ai b; hị Ai;h b; hị ẳ ẳ ehxi h1 ỵ hị ỵ h þ hxbi À xbi ; @h " À Á# h xbi ỵ @Ai b; hị b hxbi : Ai;b b; hị ẳ ẳ hxi e ln xi ị @b hỵ1 ^ ab; hị ẳ Pn The second derivatives of L can be derived as follows: ð21Þ Putting ^aðb; hị in (17), we obtain n X gb; hị ẳ Lẵ^ab; hị; b; h ẳ C n ln ẵ ln Ai b;hị n X @2L ẳ xbi ln xi ị @h@b iẳ1 n X a 1Þ Ai ðb; hÞAi;h;b ðb; hÞ À Ai;h ðb; hÞAi;b b; hị ỵ ; ẵAi b; hị2 iẳ1 where iẳ1 n X ỵ b Ai;h2 b; hị ẳ ln Ai b; hị ỵ nẵln bị ỵ lnhị lnh ỵ 1ị iẳ1 n X n n X X ln ỵ xbi ỵ b 1ị ln xi ị h xbi : iẳ1 iẳ1 22ị i¼1 ^MLE ; ^hMLE , can be obtained Therefore, the MLE of b, h, say b by maximizing (22) with respect to b and h For the three parameters exponentiated power Lindley distribution EPLD(h, b, a), all the second order derivatives exist Thus we have the inverse dispersion matrix is 20 13 ^ b11 V b12 V b13 h h V 6B C B b B^C b22 V b23 C A5; @ b A $ N4@ b A; @ V V 21 a ^ a 20 VÀ1 V11 6B ¼ ÀE4@ V31 b31 V b32 V 13 @2 L V13 C7 B @h A5 ¼ ÀE@ @2 L V33 @h@a @2 L @h@a @2L @h@b @2L V23 ¼ @b@a V12 ¼ @2L ; @h@a @2L V33 ¼ : @a V13 ẳ b hxi ỵ x2b i e h ỵ 1ị3 b Ai;b2 b; hị ẳ hxbi ln xi ị2 ehxi (" " b Ai;b;a b; hị ẳ hxbi ln xi ịehxi " Ai;h;a b; hị ẳ @2 L @a2 " ! h ỵ 1ị2 !# hxbi 1ỵ ; hỵ1 ! # " #) h2 x2b 2hxbi b i ỵ hxi ; hỵ1 hỵ1 # hxbi ỵ h ; hỵ1 ! # hxbi ; ỵ1 hỵ1 h ỵ 1Þ2 b xbi eÀhxi b C A: Eq (23) is the variance covariance matrix of the EPLD(h, b, a) @2L @h2 @2L V22 ¼ @b ! b33 V 23ị V11 ẳ 2xbi ehxi " Ai;h;b b;hị ¼ xbi eÀhxi ln ðxi Þ ! !# hxbi hxb hxbi ỵ h : ỵ ỵ i hxbi hỵ1 hỵ1 h ỵ 1ị By solving this inverse dispersion matrix, these solution will yield the asymptotic variance and co-variances of these ML ^ and ^ h; b a By using Eq (23), approximately estimators for ^ 100(1 À a)% confidence intervals for h,b and a can be determined as ^ h Ỉ Za2 qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi ^ Ỉ Za V b11 b b22 ^ b33 ; a a Ỉ Z V V 2 902 S.K Ashour and M.A Eltehiwy where Za2 is the upper ath percentile of the standard normal distribution Order statistics Suppose X1, X2, , Xn is a random sample from Eq (8) Let X(1) < X(2) < Á Á Á < X(n) denote the corresponding order statistics It is well known that the probability density function and the cumulative distribution function of the kth order statistics, say Y = X(k), are given by  n   n X nÀJ   X X nÀJ n J n nÀJ I F yịf1 Fyịg ẳ 1ị FJỵI yị; I J J¼k J J¼k I¼0 respectively, for k = 1, 2, , n It follows from Eqs (7) and (8) that    !akỵjị1 b nk  nk ah2 bn!yb1 ỵ yb ịehy X hyb b 1ịj ỵ ehy ; h ỵ 1k 1ị!n kị! jẳ0 hỵ1 j and FY yị ẳ Jẳk Iẳ0 J I  !aJỵIị hxb hxb 1ị ỵ e : hỵ1 I !2 i ; nỵ1 24ị with respect to the unknown parameters Therefore, in the case of EPL distribution the least square estimators of h, b and a, ^LSE and ^ say ^ hLSE ; b aLSE , respectively, can by obtained by minimizing the following equation (" ! #a )2 n X hxbiị hxb i iị Qh; b; aị ẳ 1ỵ ; 25ị e hỵ1 nỵ1 iẳ0 (" n X hxbðiÞ (" n X hxbðiÞ  The qth moment of Y can be expressed as an! h ỵ 1Þðk À 1Þ!ðn À kÞ!     qỵbrỵ1ị qỵbrỵ2ị C C b b  þ  qþrb  5; Aj;i;r 4 qþrb  qþbðrþ1Þ qỵbrỵ2ị i ỵ 1ịC b ị h b i þ 1ÞCð b Þ h b ! (" n X 1ỵ " 1ỵ hxbiị ! hxbiị #a )" ! #a1 hxbiị hxb i iị 1ỵ e 1ỵ e nỵ1 hỵ1 hỵ1 iẳ0 13  b h xiị ỵ hxb A5 ¼ 0; ð27Þ Â 4hxbðiÞ e ðiÞ ln xðiÞ @ hỵ1 iẳ0  n X nJ   X n nJ F xiị iẳ0 )" ! #a1 hxbiị hxb i iị 1ỵ e 1ỵ e nỵ1 hỵ1 hỵ1 iẳ0 n h   io hxb 26ị e iị h1 ỵ hị ỵ h ỵ hxbiị xbiị ẳ 0; and fY yị ẳ with respect to h, b and a To minimize equation (Eq (25)) with respect to h, b and a We differentiate it with respect to these parameters, which leads to the following equations n! fY yị ẳ Fk1 yịf1 FðyÞgnÀk fðyÞ ðk À 1Þ!ðn À kÞ!  nÀk  X nk n! ẳ 1ịj Fk1ỵj yịfyị; k 1ị!n kị! jẳ0 j FY yị ẳ Qh; b; aị ẳ n X hxbiị hxbiị ! #a hxbiị e hỵ1 ! #a hỵ1 hxbiị e #a ) i nỵ1 " ln ỵ hxbiị ! hỵ1 hxbiị e # ẳ 0: 28ị By solving this nonlinear system of Eqs (26)–(28), this solution ^LSE and ^ will yield the LSE estimators ^ hLSE ; b aLSE EYq ị ẳ where Aj;i;r ẳ  r h 1ịiỵj hỵ1 Pnk P1 P1 jẳ0 iẳ0 rẳ0  nk j  ak ỵ jị 1ị i   i r Least square estimation In this section, we provide the regression based method estimators of the unknown parameters, which was originally suggested by Swain et al [13] to estimate the parameters of Beta distributions The method can be described as follows: Suppose X1, X2, , Xn be a random sample of EPLD(h, b, a) exponentiated power Lindley distribution with cdf F(x), and suppose that X(i), i = 1, 2, , n denote the ordered sample It is well known that  À Áà i E F xiị ẳ E P X xiị ẳ : nỵ1 (See, Johnson et al 14) The least square estimators (LSES) are obtained by minimizing Data analysis In this section we provide a data analysis in order to assess the goodness-of-fit of EPL model with respect to maximum flood levels data to see how the new model works in practice The data have been obtained from Dumonceaux and Antle [15] We fit the EPL distribution to the real data set and compare its fitting with some usual survival distributions Namely, i The modified Weibull (MW) distribution [16] with pdf given by À Á b fðxÞ ẳ h ỵ abxb1 ehxax ; x > 0; h; b; a > 0; where a and b are the shape parameters and h is the scale parameter ii The exponentiated exponential (EE) distribution [17] with pdf given by À b1 fxị ẳ hbehx ehx ; x > 0; h; b > 0; where h is the scale parameter and b is a shape parameter (see Table 1) iii The Weibull (W) distribution with pdf given by b fxị ẳ hbxb1 ehx ; x > 0; h; b > 0: Exponentiated power Lindley distribution Since the power Lindley (PLD), generalized Lindley (GLD) and Lindley (LD) distributions are special cases of the exponentiated power Lindley distribution, we fit them to these data as well The analysis of least square estimates for the unknown parameters in the seven fitted distributions by using the method of least squares, is defined The LSE(s) of the unknown parameter(s), coefficient of determination (R2) and the corresponding Mean square error of the distributions mentioned before are given in Table It is clear that the exponentiated power Lindley (EPLD) distribution provides better fit than the other distributions Another check is to compare the respective coefficients of determination for these regression lines We have supporting evidence that the coefficient of determination of (EPLD) is 0.975, which is higher than the coefficient of determination (R2) of (PLD), (GLD), (LD), (EE), (MW) and (WD) distributions Hence the data point from the exponentiated power Lindley distribution (EPLD) has better relationship and hence this distribution is good model for life time data As a second application, we analyze a real data set on the active repair times (h) for an airborne communication transceiver The data are given in Table 3, and their source is Jorgensen [18] In order to compare distributions we consider the ^ ^hÞ values, the Akaike information criteÀLOG ¼ À log Lð^a; b; rion (AIC) and Bayesian information criterion (BIC), which are defined, respectively, by À2LOG + 2q and À2LOG + qlog(n), ^ ^hÞ are the MLEs vector, q is the number of paramwhere ð^a; b; eters estimated and n is the sample size The best distribution corresponds to lower ÀLOG, AIC and BIC values Table shows the values of the AIC, BIC and ÀLOG, and also the Kolmogorov–Smirnov statistic with their p values Table shows the parameter MLEs according to each one of the seven fitted distributions The values of AIC, BIC, ÀLOG and K–S statistic with their p value in Table 4, indicate that the EPLD distribution is a strong competitor to other distributions commonly used in literature for fitting lifetime data, moreover being the best fitting considering AIC, BIC, ÀLOG and K–S criterion Simulation study We used a simulation study to investigate the performance of the accuracy of point and interval estimates of the EPL(a, b, h) The following steps are as follows: Specify the values of the parameters a, band h; Specify the sample size n; Use Algorithm IV to generate a random sample with size n from EPL(a, b, h) a Calculate the MLE of the three parameters and the inverse of the Fisher matrix b Calculate the squared deviation of the MLE from the exact value of each parameter c Calculate a 95% CI for each parameter Table Maximum flood levels data from Dumonceaux and Antle [15] 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265 903 Table Estimated parameters of EPLD, PLD, GLD, LD and WD distributions Distribution h b a MSE R2 EPLD PLD GLD LD EE MW WD 11.465 50.001 10.884 2.353 9.011 0.0365 49.025 0.774 4.69 – – 24.385 0.05 4.69 131.759 – 36.675 – – 0.219 – 0.00224 0.003 0.00244 0.036 0.00277 0.0034 0.003 0.975 0.964 0.970 0.551 0.967 0.963 0.964 Table Active repair times (h) 0.50 1.00 1.50 3.00 5.40 0.60 1.00 1.50 3.30 7.00 0.60 1.00 2.00 4.00 7.50 0.70 1.00 2.00 4.00 8.80 0.70 1.10 2.20 4.50 9.00 0.70 1.30 2.50 4.70 10.20 0.80 1.50 2.70 5.00 22.00 0.80 1.50 3.00 5.40 24.50 Repeat steps 2–3, N times; Calculate the mean square error (MSE), the average of the confidence interval widths, and the coverage probability for each parameter The MSE associated with MLE of the parameter #, MSE#, is MSE# ¼ N  2 1X #^i À # ; N i¼1 where #^i is the MLE of # using the ith sample, i = 1, 2, , N, and # = a, b, h Coverage probability is the proportion of the N simulated confidence intervals which include the true parameter # Table Comparison criterion Model AIC BIC ÀLOG K–S statistic p-Value EPLD PLD GLD LD EE MW WD 186.5721 195.8854 199.8218 198.5826 194.9158 195.4046 195.0227 191.6387 199.2631 203.1995 201.2715 198.2936 200.4713 198.4005 90.2861 95.9427 97.9109 98.7913 95.4579 94.7023 95.5114 0.0909 0.1596 0.1410 0.1907 0.1334 0.1631 0.1540 0.8627 0.4092 0.4362 0.3424 0.4896 0.4004 0.3782 Table Parameters MLES Distribution h b a EPLD PLD GLD LD EE MW WD 3.5472 0.5867 0.3588 0.4242 0.2678 0.3037 0.2688 0.2901 0.7988 – – – 1.7269 0.9604 30.8299 – 0.7460 – 1.1137 0.0106 – 904 S.K Ashour and M.A Eltehiwy Table MSE, coverage probability, and average width a b h n MSEa MSEb MSEh CPa AWa CPb AWb CPh AWh 1.5 1 25 50 75 100 1.053 0.365 0.206 0.141 0.035 0.015 0.009 0.006 0.060 0.029 0.016 0.012 0.957 0.954 0.952 0.955 4.126 1.430 0.808 0.552 0.955 0.953 0.949 0.951 0.669 0.458 0.369 0.318 0.949 0.954 0.956 0.949 0.886 0.609 0.493 0.425 0.1 25 50 75 100 0.285 0.092 0.056 0.040 0.058 0.024 0.015 0.012 0.017 0.008 0.005 0.004 0.962 0.956 0.955 0.951 1.118 0.362 0.218 0.156 0.953 0.953 0.955 0.955 0.866 0.594 0.481 0.414 0.926 0.934 0.940 0.945 0.481 0.339 0.277 0.239 0.6 25 50 75 100 0.187 0.062 0.038 0.026 0.009 0.004 0.002 0.001 0.063 0.025 0.015 0.012 0.960 0.954 0.953 0.955 0.734 0.244 0.150 0.100 0.955 0.955 0.956 0.954 0.336 0.229 0.185 0.159 0.942 0.954 0.952 0.953 0.888 0.608 0.492 0.426 0.8 0.2 10 25 50 75 100 0.450 0.201 0.123 0.094 0.003 0.002 0.001 0.001 1.471 0.506 0.295 0.198 0.958 0.952 0.955 0.951 5.150 1.669 0.947 0.645 0.932 0.938 0.942 0.952 0.135 0.092 0.074 0.064 0.947 0.948 0.947 0.951 3.941 2.505 1.976 1.679 0.88 1.2 25 50 75 100 0.078 0.036 0.024 0.014 0.036 0.015 0.009 0.007 0.089 0.038 0.024 0.017 0.962 0.957 0.955 0.951 0.160 0.056 0.033 0.018 0.950 0.952 0.952 0.948 0.674 0.457 0.368 0.318 0.955 0.952 0.954 0.952 1.075 0.735 0.594 0.510 0.9 1.5 25 50 75 100 0.151 0.075 0.050 0.036 0.063 0.025 0.016 0.012 0.175 0.065 0.040 0.029 0.963 0.959 0.956 0.950 0.652 0.231 0.137 0.072 0.953 0.951 0.959 0.954 0.875 0.596 0.481 0.414 0.960 0.953 0.953 0.955 1.415 0.945 0.759 0.652 0.05 2 25 50 75 100 0.064 0.028 0.020 0.014 0.144 0.061 0.037 0.028 0.060 0.026 0.016 0.012 0.962 0.956 0.950 0.949 0.135 0.049 0.030 0.017 0.952 0.947 0.953 0.949 1.342 0.916 0.740 0.638 0.943 0.945 0.950 0.951 0.883 0.608 0.492 0.424 0.09 25 50 75 100 0.072 0.033 0.019 0.012 0.148 0.061 0.038 0.028 0.092 0.038 0.024 0.017 0.965 0.955 0.955 0.950 0.161 0.057 0.032 0.018 0.951 0.952 0.949 0.947 1.344 0.916 0.738 0.636 0.949 0.952 0.949 0.954 1.076 0.734 0.593 0.509 The simulation study is used when N = 10,000, the sample sizes are 25, 50, 75, 100, and the parameters values (a, b, h) = (1.5, 1, 1), (1, 2, 0.1), (1, 0.6, 2), (0.8, 0.2, 10), (1, 0.88, 1.2), (1, 0.9, 1.5), (0.05, 2, 2), (0.09, 3, 1) Some of the selected values of (a, b, h) give increasing, decreasing, increasing–decreasing–increasing, bath tub hazard shapes, respectively as shown in Fig Table presents the MSE, Coverage probability (CP#), and average width (AW) of 95% confidence intervals of each parameter As it was expected, this table shows that the MSEs of the estimates decrease as the sample size increases, that the coverage probabilities are very close to the nominal level of 95%, and that the average widths decrease as the sample size increases subject distribution can be used to model reliability data We derived the maximum likelihood estimates of the parameters and their variance covariance matrix A real data application of the EL distribution shows that it could provide a better fit than a set of usual statistical distributions considered in lifetime data analysis Finally, we examined the accuracy of the maximum likelihood estimators of the EPL(a, b, h) parameters as well as the coverage probability and average width of the confidence intervals for the parameters using simulation Conflict of Interest The authors have declared no conflict of interest Compliance with Ethics Requirements Conclusion In this study we have proposed a new family of distributions called exponentiated power Lindley distribution (EPLD) We get the probability density functions for generalized Lindley, Power Lindley, and Lindley distributions as special cases from ELPD Some mathematical properties along with estimation issues are addressed The hazard rate function behavior of the exponentiated power Lindley distribution shows that the This article does not contain any studies with human or animal subjects References [1] Ghitany ME, Al-Mutairi DK, Balakrishnan N, Al-Enezi LJ Power Lindley distribution and associated inference Comput Stat Data Anal 2013;64:20–33 Exponentiated power Lindley distribution [2] Nadarajah S, 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(W) distribution with pdf given by À b fxị ẳ hbxb1 ehx ; x > 0; h; b > 0: Exponentiated power Lindley distribution Since the power Lindley (PLD), generalized Lindley (GLD) and Lindley (LD) distributions... new family of distributions called exponentiated power Lindley distribution (EPLD) We get the probability density functions for generalized Lindley, Power Lindley, and Lindley distributions as... class of generalized Lindley (GLD) distribution called the beta-generalized Lindley distribution (BGLD) This class of distributions contains the betaLindley, GLD and Lindley distributions as special