Odd Log-Logistic Modified Weibull Distribution

Odd Log-Logistic Modified Weibull Distribution Abdus Saboor, Morad Alizadeh, Muhammad Nauman Khan , Indranil Ghosh and Gauss M. Cordeiro Abstract. We introduce and study a new distribution called the odd log– logistic modified Weibull (OLLMW) distribution. Various of its structural properties are obtained in terms of Meijer’s G–function, such as the moments, generating function, conditional moments, mean deviations, order statistics and maximum likelihood estimators. The distribution exhibits a wide range of shapes varying skewness and takes all possible forms of hazard rate function. We fit the OLLMW and some competitive models to two data sets and prove empirically that the new model has a superior performance among the compared distributions as evidenced by some goodness–of–fit statistics. Mathematics Subject Classification (2010). 60E05, 62E15, 62F10. Keywords. Maximum likelihood estimation, modified Weibull distribution, moment, odd log-logistic distribution, simulation study.

1. Introduction The Weibull distribution is a very popular distribution for modeling and analyzing lifetime data with monotonic hazard rates. On the other hand, for complex systems, the hazard rate function (hrf) can often be of non– monotonic shape, which the Weibull distribution cannot accommodate. To overcome such shortcomings, various generalizations of the classical Weibull distribution have been proposed by several authors in recent years; among them, the extended flexible Weibull [1], generalized modified Weibull [2], exponentiated Weibull [9], additive Weibull [11] and modified beta Weibull [6] distributions. Also, Sarhan and Zaindin [18] introduced the modified– Weibull (MW) distribution having three parameters β > 0, λ > 0 and γ > 0, with cumulative distribution function (cdf) and probability density function (pdf) given by γ G(x) = 1 − e−β x−λ x , x > 0, (1.1)

2

A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro

and

( ) γ g(x) = β + λγ xγ−1 e−β x−λ x , (1.2) respectively. It is worth noting that the MW distribution contains both the exponential (λ = 0) and Weibull (β = 0) distributions. ¯ Let G(x) be a parent continuous cdf and G(x) = 1 − G(x). The cdf of the odd log–logistic family (Gleaton and Lynch [5]) is defined by (for α > 0) F (x) =

G(x)α ¯ α. G(x)α + G(x)

(1.3)

An interpretation of (1.3) can be given as follows: take an OLL-G random variable (rv) X and a one-parameter log-logistic rv T with shape paα α rameter α)> 0 and ( ( cdf R(t) )= t /(1 + t ). Then, P (X ≤ x) = F (x) = G(x) G(x) G(x) R 1−G(x) = P T ≤ 1−G(x) . Since the function γ(x) = 1−G(x) is always monotonic and non-decreasing, this implies that (in distribution) T = γ(X). So, if X is OLL-G, then T = γ(X) is LL, holding for every continuous cdf G(x). The pdf corresponding to (1.3) can be expressed as ¯ α−1 α g(x) G(x)α−1 G(x) f (x) = , (1.4) { } ¯ α 2 G(x)α + G(x) where g(x) = dG(x)/dx is the baseline density function. In this paper, we introduce and study a new distribution having four parameters, so–called the odd log–logistic modified Weibull (OLLMW) distribution, which generalizes the MW distribution. Our motivation for introducing the OLLMW distribution is due to the simple analytic expressions of G(x) and g(x) for the MW distribution. Further, its density can be represented as an infinite linear combination of MW densities. The proposed distribution provides a wide range of shapes and skewness values, varied tail weights and shifting modes based on its additional parameter. It also accommodates most forms of hazard rates that are encountered in a variety of real-life problems. We now define the OLLMW distribution by taking G(x) in (1.3) to be the cdf (1.1) of the MW distribution. Hence, the cdf of the OLLMW distribution becomes ( γ )α 1 − e−β x−λ x F (x) = , x > 0. (1.5) α (1 − e−β x−λ xγ ) + e−αβ x−αλ xγ The OLLMW density function can follow from equations (1.1), (1.2) and (1.4) as γ ( γ )α−1 α (β + γ λ xγ−1 ) e−αβ x−αλ x 1 − e−β x−λ x f (x) = , (1.6) { }2 α (1 − e−β x−λ xγ ) + e−αβ x−αλ xγ where γ > 0 and α > 0 are shape parameters and λ > 0 and β > 0 are scale parameters. Henceforth, we denote by X a random variable having density function (1.6). If X ∼ OLLMW, then one can easily prove that the random variable

Odd Log-Logistic Modified Weibull Distribution

3

Table 1. Some special cases of the OLLMW distribution α 1 1 1 1 1 -

β 0 0 0 0 -

λ 0 0 -

γ 2 2 2 2

Reduced distribution Exponential distribution Weibull distribution Rayleigh distribution Linear failure rate distribution Exponential weibull distribution OLL exponential distribution [5] OLL weibull distribution [5] OLL Rayleigh distribution [5] OLL linear failure rate [5]

γ

Y = (eβ X+λ X − 1) has the LL distribution with shape parameter α and unity scale parameter. The survival function and hrf of X can be obtained from (1.5) and (1.6), respectively, as e−αβ x−αλ x S(x) = α (1 − e−β x−λ xγ ) + e−αβ x−αλ xγ

(1.7)

( γ )α−1 α (β + γ λ xγ−1 ) 1 − e−β x−λ x τ (x) = . α (1 − e−β x−λ xγ ) + e−αβ x−αλ xγ

(1.8)

γ

and

The remainder of this article is organized as follows. Some statistical functions of the OLLMW distribution are provided in Section 2, such as the quantile function (qf), moments, moment generating function (mgf), conditional moments, mean deviations, reliability curves and the density of the order statistics. The estimation of the parameters by the maximum likelihood method is investigated in Section 3. A simulation study is performed in Section 4 to show the accuracy of the maximum likelihood estimators (MLEs). In two applications to real data given in Section 5, we prove empirically, based on four goodness–of–fit statistics, that the proposed distribution provides the best fit than other four extended Weibull models. In Section 6, we offer some concluding remarks.

2. Structural properties of the OLLMW distribution Using the generalized binomial expansion, we can write ∞ ( ) ( ) ∑ γ α γ j 1 − e−β x−λ x = aj 1 − e−β x−λ x . j=0

( )( ) ∑∞ α i Here, aj = i=j (−1)i+j and i j (

1 − e−β x−λ x

γ

)α

+ e−αβ x−αλ x = γ

∞ ∑ j=0

( bj

1 − e−β x−λ x

γ

)j ,

4


( ) α where bj = aj + (−1) . Then, we obtain j [ ] ∑∞ ∞ −β x−λ xγ j [ ]j ∑ j=0 aj 1 − e −β x−λ xγ c 1 − e = F (x) = ∑∞ j −β x−λ xγ ]j j=0 j=0 bj [1 − e j

=

∞ ∑

cj Hj (x)

(2.1)

j=0

( γ )j where Hj (x) = 1 − e−β x−λ x , c0 = a0 /b0 and (for j ≥ 1) ) ( j ∑ −1 −1 br cj−r . aj − b0 cj = b0 r=1

By differentiating (2.1), the pdf (1.6) can be rewritten as f (x) =

∞ ∑

cj+1 hj+1 (x),

(2.2)

j=0

( ) γ ( γ )j where hj+1 (x) = (j + 1) β + λ γ xγ−1 e−β x−λ x 1 − e−β x−λ x is the density of the exponentiated MW (exp-MW) with power parameter j + 1 (for j ≥ 0). It is clear from (2.2) that f (x) can be expressed as an infinite linear combination of exp-MW densities and then several OLLMW properties can be deduced from the corresponding ones of the exp-MW distribution. In what follows, we discuss some properties of the new distribution and consider several associated statistical functions. 2.1. Asymptotic and Shapes In this section, we discuss the asymptotic and possible shapes of the pdf (1.6) and hrf (1.8). Proposition 1. The asymptotics of equations (1.5), (1.6) and (1.8) when x → 0 are given by F (x) ∼ (β x + λ xγ )α , f (x) ∼ α(β + λ γ xγ−1 )(β x + λ xγ )α−1 , τ (x) ∼ α(β + λ γ xγ−1 )(β x + λ xγ )α−1 . Proposition 2. The asymptotics of equations (1.5), (1.6) and (1.8) when x → ∞ are given by 1 − F (x) ∼ e−α(β x+λ x ) , γ

f (x) ∼ α(β + λ γ xγ−1 ) e−α(β x+λ x ) , γ

τ (x) ∼ α(β + λ γ xγ−1 ).

Odd Log-Logistic Modified Weibull Distribution HaL Λ=0.6, Γ=0.5, Α=1

HbL Λ=0.1, Γ=10, Α=0.6

1.0

Density function

0.6

0.4

Β = 0.1 Dotted Β = 0.4 Dashed Β = 0.8 Long Dashes Β = 1.0 Solid Β = 1.2 Thick

1.5 Density function


0.8

5

1.0

0.5 0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

1.4

0.5

1.0

HcL Λ=1.3, Γ=2, Α=1

HdL Β=0.1, Λ=1.2, Γ=1

2.0

Density function

1.0

Α = 0.2 Dotted Α = 0.5 Dashed Α = 1.0 Long Dashes Α = 1.2 Solid Α = 1.5 Thick

1.5 Density function


1.5

1.5

x

x

1.0

0.5

0.5

0.0

0.0 0.0

0.5

1.0 x

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

Figure 1. Plots of the OLLMW density function.

For a given set of parameters β, λ, γ and α, the form of (1.6) can be (x)] easily determined by examining the signs of the derivative d log[f . The dx mode and the minimum values are given by the points where this function vanishes. The plots (a), (b), (c) and (d) in Figure 1 show how the parameters affect the density shape, where the skewness, heavy tails and modality are pronounced. The plots in (a) indicate that the pdf of X is decreasing and decays more rapidly when β increases. The plots in (b) and (c) show that the parameter β controls the value of the mode, whereas the plots in (d) reveal the great influence of α on the pdf shape. In a similar manner, the forms of the hrf of X can be determined from the signs of the function d log[h(x)] . The dx plots (e), (f) and (g) in Figure 2 reveal great flexibility of the hrf of X, which can present increasing, decreasing, unimodal, v–shaped and bathtub–shaped forms. In the rest of this section, we obtain some statistical quantities for the OLLMW distribution.

6

A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro HfL Β=0.9, Λ=1.2, Α=2.3

HeL Β=0.1, Λ=0.8, Α=1.3 6

Γ = 0.1 Dotted Γ = 0.4 Dashed Γ = 1.0 Long Dashes Γ = 1.3 Solid Γ = 3.0 Thick

1.5

5 Hazard rate function

Hazard rate function

2.0

1.0

0.5

Γ = 0.5 Dotted Γ = 0.9 Dashed

4 3 2 1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0

1.4

0.0

0.2

0.4

0.6

x

0.8

1.0

1.2

1.4

x

HgL Β=2.3, Λ=10, Α=0.3 6 Γ = 3.4 Solid Γ = 12 Thick

Hazard rate function

5 4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

x

Figure 2. Plots of the OLLMW hrf.

2.2. Quantile function The qfs are in widespread use in general statistics to obtain mathematical properties of a distribution and often find representations in terms of lookup tables for key percentiles. For simulating the OLLMW model, let u ∼ U (0 , 1 ). Then, the solution of non-linear equation [ β x + λ x + log

has pdf (1.6).

]

1

(1 − u) α

γ

1

1

u α + (1 − u) α

=0

(2.3)


7

2.3. Moments and generating function We now obtain representations for the moments and mgf of X on the basis of the result developed in [10]: ∫ ∞ k (2π)1−(q+p)/2 q1/2 p η−1/2 xη−1 e−θ x es x dx= (−s)η 0 ( ) ( p )p ( θ )q 1 − u+η , u = 0, 1, ..., p − 1 q,p p × Gp,q − , (2.4) v/q , v = 0, 1, ..., q − 1 s q where ℜ(η), ℜ(θ), ℜ(s) < 0 and k is a rational number such that k = p/q, where p and q ̸= 0 are natural co-prime integers. Using the linear combination (2.2) and (2.4)and expanding the binomial in hj+1 , the rth order moment of X can be expressed in terms of the Meijer’s G–functions (assuming that γ = p/q with p and q ̸= 0 natural co-prime integers) ( ) j ∞ ∑ ∑ j (2π)1−(q+p)/2 q 1/2 pr+1/2 E(X r ) = β (−1)k (j + 1) cj+1 k (β(k + 1))r+1 j=0 k=0 (( ) )p ( )q p λ(k + 1) 1 − u+r+1 , u = 0, 1, . . . , p − 1 q,p p × Gp,q v/q, v = 0, 1, . . . , q − 1 β(k + 1) q ( ) j ∞ ∑ ∑ j (2π)1−(q+p)/2 q 1/2 pr+γ−1/2 + λγ (−1)k (j + 1) cj+1 (β(k + 1))r+γ k j=0 k=0 (( ) )p ( )q u+r+γ p λ(k + 1) , u = 0, 1, . . . , p − 1 1 − p × Gq,p . p,q v/q, v = 0, 1, . . . , q − 1 β(k + 1) q (2.5) The rth order negative moment of X can readily be determined by replacing r with −r in (2.5). In a similar manner, the mgf of X is given by ( ) j ∞ ∑ ∑ j (2π)1−(q+p)/2 q 1/2 p 1/2 M (t) = β (−1)k (j + 1) cj+1 k β(k + 1) − t j=0 k=0 (( ) )p ( )q p λ(k + 1) 1 − u+1 , u = 0, 1, . . . , p − 1 q,p p × Gp,q v/q, v = 0, 1, . . . , q − 1 β(k + 1) − t q ( ) j ∞ ∑ ∑ j (2π)1−(q+p)/2 q 1/2 p γ−1/2 + λγ (−1)k (j + 1) cj+1 γ k ((n + ρ) λ − t) j=0 k=0 (( ) )p ( )q u+γ λ(k + 1) p 1 − , u = 0, 1, . . . , p − 1 p × Gq,p . p,q v/q, v = 0, 1, . . . , q − 1 β(k + 1) − t q 2.4. Order statistics Let X1 , . . . , Xn be a random sample from the OLLMW family. Denote the random variables in the ascending order by X1:n ≤, . . . , ≤ Xn:n . The pdf of

8


Xi:n (David and Nagarajah [4]) can be expressed as n−i

fi:n (x) = K f (x) F i−1 (x) {1 − F (x)}

=

∞ ∑ n−i ∑

mj,r,k hr+k+1 (x),

r,k=0 j=0

where K = n!/[(i − 1)! (n − i)!], hr+k+1 (x) denotes the exp-MW density function with power parameter parameter r + k + 1, mj,r,k =

(−1)j n! (r + 1) cr+1 fj+i−1,k , (i − 1)! (n − i − j)! j! (r + k + 1)

and ck is defined by Equation (2.2). Here, the quantities fj+i−1,k are obtained recursively by fj+i−1,0 = cj+i−1 and (for k ≥ 1) 0 fj+i−1,k = (k c0 )

k ∑

−1

[m(j + i) − k] cm fj+i−1,k−m .

m=1

Thus, one can easily obtain the moments, generating function and incomplete moment of the OLLMW order statistics. In the rest of this section, we shall use the following lemma. Lemma 1. Let ∫ J(x; r, θ) =

x r

y f (y) dy =

0

∫

j,l=0 k=0 x

×

j ∞ ∑ ∑

( ) j (−1)k+l l (j + 1) cj+1 {β(k + 1)} l! k

(

) γ y l+r β + λγ y γ−1 e−λ(k+1) y dy,

r = 1, 2, . . . ,

0

where θ = (β, γ, λ, α). Then, we have J(x; r, θ) = {

j ∞ ∑ ∑ j,l=0 k=0

(j + 1) cj+1

( ) j (−1)k+l l {β(k + 1)} l! k

β q xp (l+r+1) p (2 π)(q−1)/2 ( ) 1−l−r , . . . , p−l−r−1 ,− (λ(k + 1))q xp −l−r q,p p , p p × Gp,p+q p−l−r−2 0 , −l−r−1 , l+r qq p p ,..., p ×

λ xp (l+r+γ) (2π)(q−1)/2 ( )} −l−r−γ+1 2−l−r−γ , , . . . , p−l−r−γ ,− (λ(k + 1))q xp q,p p p p × Gp,p+q . 0 , −l−r−γ , l+r+γ−1 , . . . , p−l−r−γ−1 qq p p p +

) − Proof. For an arbitrary function g(x), we have e = g(x) . 0 Letting k = p/q, where p ≥ 1 , q ≥ 1 are natural co-prime numbers, and (

−g(x)

G1,0 0,1


9

using the identity ( ) ∫ x t 1,0 p/q − y G0,1 λ(k + 1)y dy 0 0 ( ) p−t−1 1−t , , . . . , , − q xp (t+1) (λ(k + 1))q xp −t q,p p p = G , p −t−1 0 , p , pt , . . . , p−t−2 qq p(2π)(q−1)/2 p,p+q p the proof follows from Equation (13) of [3].

2.5. Conditional moments and mean deviations In connection with lifetime distributions, it is important to determine the conditional moments E(X r |X > t), r = 1, 2, · · · , which are of interest in predictive inference. The rth conditional moment of X is given by [ ] ∫ t 1 E(X r |X > t) = E(X r ) − x r f (x) dx S(t) 0 ( γ )α γ 1 − e−β x−λ x + e−αβ x−αλ x [E(X r ) − J(t; r, θ)] . = e−αβ x−αλ xγ The mean deviations provide useful information about the characteristics of a population and it can be calculated from the first incomplete moment. Indeed, the amount of dispersion in a population may be measured to some extent by the totality of the deviations from the mean and median. The mean deviations of X about the mean µ = E(X) and about the median M can be expressed as δ = 2µF (µ) − 2m(µ) and η = µ − 2m(M ), where F (µ) is obtained from (1.5) and ∫ z m(z) = x f (x)dx = J(z; 1, θ). 0

2.6. Reliability curves The Bonferroni and Lorenz curves have various applications in economics, reliability, insurance and medicine. The Bonferroni curve BF [F (x)] of X is given by ∫ x 1 J(x; 1, θ) BF [F (x)] = yf (y)dy = . E(X)F (x) 0 E(X)F (x) The associated Lorenz curve of X is given by LF [F (x)] =

J(x; 1, θ) . E(X)

The scaled total time on test transform of a distribution function F [17] is ∫t 1 defined by SF [F (t)] = E(X) F¯ (y)dy. It is important for the ageing proper0 ties of the underlying distribution and can be applied to solve geometrically some stochastic maintenance problems. The influence of some parameters on the reliability curves can be investigated numerically using several plots by fixing the other parameters. Since the number of possible combinations is large and the function J(x; 1, θ) is so complicated to be dealt analytically, we

10 A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro do not provide any numerical study on the effects of the model parameters on the Bonferroni and Lorenz curves.

3. Estimation, inference and goodness-of-fit statistics Estimating the unknown parameters of a distribution is an essential issue in applied statistics. Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The MLEs enjoy desirable properties and can be used when constructing confidence intervals and also in test statistics. The normal approximation for these estimators in large sample distribution theory is easily handled either analytically or numerically. In this section, we investigate the MLEs of the parameters of the new distribution and provide four goodnessof-fit statistics to compare the densities fitted to any data set. 3.1. Maximum likelihood estimation By using the log-likelihood of the sample in conjunction with the NMaximize command in the symbolic computational package Mathematica, we can estimate the unknown parameters of the new distribution. Given the observed values x1 , . . . , xn , from the OLLMW distribution, the MLEs of the parameters in θ = (α, β, λ, γ)T can be obtained by maximizing directly the log-likelihood function given by n ( ) ∑ ℓ(θ) = n log(α) + log β + γλxγ−1 i i=1

+

n ∑

(−αβxi − αλxγi ) + (α − 1)

i=1 n ∑

−2

n ∑

) ( γ log 1 − e−βxi −λxi

i=1

) } {( γ γ α + e−αβxi −αλxi . log 1 − e−βxi −λxi

i=1

Alternatively, the components of the nonlinear log-likelihood system ∂ℓ(θ) ∂θ = 0 of equations are given by )α ( ) ( −βxi −λxγ −βxi −λxγ n n i i log 1 − e 1 − e ∑ ∑ ∂ℓ(θ) n (βxi + λxγi ) − 2 =− ( γ )α γ ∂α α i=1 e−αβxi −αλxi + 1 − e−βxi −λxi i=1 +

n ∑ i=1

(

log 1 − e

−βxi −λxγ i

)

−2

n ∑ i=1

γ

e−αβxi −αλxi (−βxi − λxγi ) ( γ γ )α = 0, e−αβxi −αλxi + 1 − e−βxi −λxi

γ n n n ∑ ∑ ∑ ∂ℓ(θ) e−βxi −λxi xi 1 = (−1 + α) −αx + γ + i −βx −λx i i ∂β 1−e β + γλx−1+γ i i=1 i=1 i=1 ( ) γ γ γ −1+α −βx −λx −αβx −αλx −βx −λx i i i n −e i i αx + e i 1−e αxi ∑ i −2 = 0, ( γ γ )α e−αβxi −αλxi + 1 − e−βxi −λxi i=1


11

γ n n n ∑ ∑ ∑ γxi−1+γ ∂ℓ(θ) e−βxi −λxi xγi γ = (−1 + α) + −αx + γ i ∂λ 1 − e−βxi −λxi β + γλx−1+γ i i=1 i=1 i=1 ( )−1+α γ γ γ n −e−αβxi −αλxi αxγ + e−βxi −λxi 1 − e−βxi −λxi αxγi ∑ i −2 = 0, ( γ γ )α e−αβxi −αλxi + 1 − e−βxi −λxi i=1 γ n n ∑ ∑ ∂ℓ(θ) −e−αβxi −αλxi αλ log (xi ) xγi =− αλ log (xi ) xγi − 2 ( )α γ −αβxi −αλxi + 1 − e−βxi −λxγ ∂γ i i=1 i=1 e

+ (α − 1)

γ n ∑ e−βxi −λxi λ log (xi ) xγ

i

−βxi −λxγ i

+

n ∑ λx−1+γ + γλ log (xi ) x−1+γ i i −1+γ β + γλx i i=1

1−e ) ( γ −1+α γ n e−βxi −λxi 1 − e−βxi −λxi αλ log (xi ) xγi ∑ = 0. −2 ( γ )α γ e−αβxi −αλxi + 1 − e−βxi −λxi i=1 i=1

By solving the equations above simultaneously, we can obtain the MLEs of the parameters of the OLLMW distribution. Some numerical iterative techniques may be used for estimating these parameters and the global maxima of the log-likelihood can be investigated by setting different starting values for the parameters. The existence and uniqueness of these MLEs is intimately related to the observations xi ’s. Then, it will be very difficult to establish some necessary and sufficient conditions for these properties. In fact, the analytical proof that these MLEs exist is impossible in generality. We can obtain sufficient conditions (depending on the observations) for very special cases, e.g., just considering one unknown parameter with the other parameters known, which is of limited practical use. Certainly, there is no sense of fitting the OLLMW model with only unknown parameter to a data set. However, based on some simulation studies in Table 2, we conclude that these estimators are consistent in accordance with first-order asymptotic theory. The information matrix will be required for interval estimation. The elements of the 4 × 4 total observed information matrix J(θ) = {Jrs (θ)} (r, s = α, β, λ, γ) can be obtained from the authors upon request. Under standard regularity conditions, the asymptotic distribution of (θˆ − θ) is N4 (0, I(θ)−1 ), where I(θ) = E{J(θ)} is the expected total information matrix. The approximate multivariate normal N4 (0, J(θ)−1 ) distribution, where J(θ)−1 is ˆ can be used the inverse observed information matrix evaluated at θ = θ, to set up approximate confidence intervals for the model parameters. Further, we can evaluate the maximum values of the unrestricted and restricted log-likelihoods to obtain likelihood ratio (LR) statistics for testing some submodels listed in Table 1 in the classical way. 3.2. Goodness-of-Fit Statistics To check the goodness-of-fit of the fitted models, we compute four goodnessof-fit statistics using the computational package Mathematica, namely the Anderson-Darling (A∗ ), Cramér-von Mises (W ∗ ), Liao-Shimokawa (L-S) and

12 A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro Kolmogrov-Simnorov (K-S) statistics with their p-values, to compare the fitted models. These statistics can determine how closely the distribution fit the empirical distribution of the data. The distribution with better fit than the others will be the one having the smallest statistics and largest p-values. These statistics are given by ] ( )[ n ∑ 2.25 0.75 1 A∗ = + + 1 −n − (2i − 1) log (zi (1 − zn−i+1 )) , n2 n n i=1 ] ( ) [∑ )2 n ( 0.5 2i − 1 1 ∗ W = +1 zi − + , n 2n 12n i=1 ] [ i−1 i − zi , zi − , K-S = Max n n [ ] n 1 ∑ Max ni − zi , zi − i−1 n √ √ L-S = . n i=1 zi (1 − zi ) where zi = cdf(yi ) and the yi, s are the ordered observations.

4. Simulation study In order to assess the performance of the MLEs, a small simulation study is performed using the statistical software R through the package (stats4), command mle. However, one can also perform in SAS by using the PROC NLMIXED procedure. The number of Monte Carlo replications is 20, 000. For maximizing the log-likelihood function, we use the MaxBFGS subroutine with analytical derivatives. The evaluation of the estimates is performed based on the following quantities for each sample size. The empirical mean squared errors (MSEs) are calculated using the R package from the Monte Carlo replications. The MLEs are determined for each simulated data, say ˆ i , γî ) for i = 1, 2, · · · , 20, 000. The biases and MSEs are calculated (ˆ αi , βî , λ by 20000 ∑ 1 biash (n) = (hî − h), 20000 i=1 and M SEh (n) =

20000 ∑ 1 (hî − h)2 , 20000 i=1

for h = α, β, λ, γ. We consider the sample sizes n = 100, 200 and 500 and different parameter values. The empirical results are given in Table 2. The figures in this table indicate that the estimates are quite stable and, more importantly, they are close to the true values for these sample sizes. Furthermore, the MSEs and biases of the MLEs decrease when the sample size increases, as expected.


13

5. Applications In this section, we fit some well-known distributions and the new distribution to two data sets, one in hydrology and another in reliability, and then select the best model among them. 5.1. Data fitting In two applications, the OLLMW model is compared with the following distributions • The classical Weibull (W) distribution with density function k ( x )k−1 −(x/λ)k f (x) = e , x > 0, k > 0, λ > 0 . λ λ • The Transmuted–Weibull(TW) [13] with density function ( ) x η ( )η−1 x η ηe−( σ ) σx 2λe−( σ ) − λ + 1 f (x) = , σ > 0,η > 0,λ > 0. σ • The MW [18] with density function ( ) k f (x) = λ + β k xk−1 e−λ x−β x , x > 0 , λ, β, k > 0 , • The Marshall-Olkin exponential–Weibull (MO-EW) [16] with density function ( ) c α a + b c x−1+c e−(ax+bx ) f (x) = [ ]2 , λ, β, k, α > 0; 1 − (1 − α) e−(ax+bxc ) • The Transmuted modified–Weibull (TMW) [15] with density function ( ) ( ) k k f (x) = λ + βkxk−1 e−λx−βx 1 − α + 2αe−λx−βx The applications are to the “Flood” and “Aeroplane communication” data sets given below: (i) The first data set reported in [14] represents the exceedances of flood peaks (in m3 /s) of the Wheaton River near Carcross in Yukon Territory, Canada and consist of 72 exceedances for the years 1958-1984 rounded to one decimal place. (ii) The second data set refers to 46 observations reported on active repair times (hours) for an airborne communication transceiver at page 156 of [12]. The maximum likelihood method is used for estimating the parameters of all distributions. The MLEs with their standard errors (computed by inverting the observed information matrices) are given in Tables 3 and 5 for both data sets. Further, for both data sets, we compute the goodness-of-fit statistics for each distribution, which are listed in Tables 4 and 6. The estimated pdf and cdf for the OLLMW distribution are plotted in Figures 3 and 4 for the flood and Aeroplane communication data, respectively.

500

200

Sample Size n 100

α 0.5 0.5 0.7 0.9 1 1.5 0.5 0.5 0.7 0.9 1 1.5 0.5 0.5 0.7 0.9 1 1.5

Actual β 0.5 0.5 0.8 0.7 1.5 2 0.5 0.5 0.8 0.7 1.5 2 0.5 0.5 0.8 0.7 1.5 2

Value λ γ 2 4 3 5 4 3 6 4 0.9 0.6 0.6 0.8 2 4 3 5 4 3 6 4 0.9 0.6 0.6 0.8 2 4 3 5 4 3 6 4 0.9 0.6 0.6 0.8 α ˆ −0.4173 −0.7738 0.4891 0.1883 0.1780 −0.0841 .0717 0.5182 0.3165 0.1375 0.1258 0.0343 −0.04609 −0.0512 −0.0730 −0.1023 −0.0783 0.0078

Bias ˆ βˆ λ −0.4198 0.3554 0.3239 −0.2145 −0.2460 −0.6227 0.9799 −0.5091 −0.4983 −0.4292 −0.3634 −0.4059 0.3617 0.0499 0.1845 0.0843 0.1594 0.0505 −0.0493 0.1309 0.4753 0.0866 0.1737 0.2243 −0.02898 −0.0368 −0.1110 −0.0355 −0.0527 0.0467 −0.0208 −0.0986 −0.0519 −0.0169 −0.0691 0.0660 γˆ −0.3932 −0.3426 0.4821 0.0563 −0.5454 −0.2203 0.0729 0.1153 −0.3298 −0.0318 0.2424 −0.1509 −0.0475 0.0023 −0.02271 −0.0235 0.0026 −0.0853

α ˆ 0.0518 0.0180 0.0154 0.0481 0.4271 0.9537 0.0216 0.0084 0.0065 0.0181 0.0646 0.4015 0.0094 0.0041 0.0039 0.0086 0.0270 0.1368

Table 2. Biases and MSEs of the MLEs in the OLLMW model. MSE ˆ βˆ λ 0.0460 0.0538 0.0426 0.0978 0.1207 0.1065 0.0222 0.0442 0.0282 0.0917 0.0183 0.0737 0.0230 0.0248 0.0229 0.0457 0.0591 0.0441 0.0105 0.0203 0.0131 0.0281 0.0089 0.0367 0.0114 0.0111 0.0102 0.0229 0.0315 0.0241 0.0053 0.0102 0.0071 0.0123 0.0040 0.0149 γˆ 0.953 0.626 0.167 0.1138 0.4956 0.572 0.3132 0.5781 0.1587 0.0955 0.1472 0.432 0.0842 0.0182 0.0364 0.0217 0.0154 0.01156

14 A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro


15

Table 3. MLEs of the parameters (standard errors in parentheses) for flood data Distributions W(k, λ) TW(η, σ, λ) MW(λ, β, k) MO–EW(a, b, c, α) TMW( λ, β, k, α) OLLMW(β, λ, γ, α)

Estimates 0.901166 (0.0855573) 0.901184 (0.10919) 0.041204 (0.101567) 0.035949 (0.111317) 0.041208 (0.102764) 0.081514 (0.0182428)

9.127211 (2.51255) 11.63293 (3.22782) 0.070387 (0.104638) 0.091176 (0.17193) 0.070372 (0.120072) 0.000119 (0.000703)

0.000103 (0.388014) 0.812037 (0.403678) 0.802195 (0.364772) 0.812044 (0.404524) 2.398056 (1.36316)

1.141779 (0.933519) 0.0001001 (0.392157) 0.800609 (0.108328)

Table 4. Goodness-of-fit statistics for flood data Distributions W(k, λ) TW(η, σ, λ) MW(λ, β, k) MO–EW(a, b, c, α) TMW( λ, β, k, α) OLLMW(λ, β, k, α, ρ)

A∗ 0.853663 0.853677 0.826604 0.812298 0.826617 0.511073

W∗ 0.149942 0.149945 0.144744 0.140813 0.144747 0.081330

L-S 2.61154 2.61151 2.58983 2.63243 2.58979 2.40375

K-S 0.105206 0.105205 0.104685 0.106525 0.104684 0.094502

p-value 0.402882 0.4029 0.409108 0.387379 0.409125 0.541086

Table 5. MLEs of the parameters (standard errors in parentheses) for Aeroplane communication data Distributions W(k, λ) TW(η, σ, λ) MW(λ, β, k) MO–EW(a, b, c, α) TMW( λ, β, k, α) OLLMW(β, λ, γ, α)

Estimates 0.898582 (0.0957596) 0.979166 (0.104738) 2.4 × 10−8 (0.517408) 1.01 × 10−6 (0.098555) 0.0010002 (1.92486) 0.000278 (0.0081809)

2.996281 (0.673386) 5.212067 (1.35319) 0.333747 (0.566711) 3.740617 (1.11635) 0.197561 (1.95379) 0.613379 (0.116901)

0.673446 (0.303747) 0.898582 (0.136771) 0.297722 (0.086219) 0.979106 (0.156513) 0.187583 (0.302812)

98.99996 (113.255) 0.673443 (0.303748) 5.829999 (9.02651)

Table 6. Goodness-of-fit statistics for Aeroplane communication data Distributions W(k, λ) TW(η, σ, λ) MW(λ, β, k) Mo–EW(a, b, c, α) TMW( λ, β, k, α) OLLMW(β, λ, γ, α)

A∗ 0.903242 0.730471 0.903242 0.726767 0.730521 0.442864

W∗ 0.121764 0.096532 0.121764 0.0833892 0.0965499 0.0661108

L-S 2.50635 2.3909 2.50635 2.46086 2.39136 1.82769

K-S 0.120438 0.116084 0.120438 0.124384 0.1161 0.0940646

p-value 0.516987 0.564905 0.516987 0.475071 0.564722 0.810375

16 A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro 0.08

1.0

0.8 0.06

0.6 0.04

0.4 0.02

0.2

10

20

30

40

50

0

60

10

20

30

40

50

60

Figure 3. Left panel: The estimated OLLMW density superimposed on the histogram for flood data. Right panel: The estimated OLLMW cdf and empirical cdf. 1.0

0.30

0.25

0.8

0.20

0.6 0.15

0.4 0.10

0.2 0.05

5

10

15

20

0

5

10

15

20

25

Figure 4. Left panel: The estimated OLLMW density superimposed on the histogram for Aeroplane communication data. Right panel: The estimated OLLMW cdf and empirical cdf. The variance-covariance matrices of the MLEs of the OLLMW distribution for flood and Aeroplane communication data are given, respectively, by   0.0117350  0.0003374   −0.0000353 0.0557179

and



81.4778874  0.0369614   1.0348956 −2.7172794

0.0003374 0.0003327 −0.0000072 0.0121303

−0.0000353 −0.0000072 0.0000004 −0.0009413

0.0557179 0.0121303  , −0.0009413  1.8582021

0.0369614 0.0000669 0.0004298 −0.0014014

1.0348956 0.0004298 0.0136659 −0.0346084

 −2.7172794 −0.0014014  . −0.0346084  0.0916950

Note that the diagonal entries of the above matrices are the variances of the MLEs of β, λ, γ and α, respectively.


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It can be observed from Tables 4 and 6 that the OLLMW distribution gives the smallest values for all statistics and the largest p-values. Accordingly, we can conclude that the OLLMW distribution provides the best fit to the current data sets among the compared distributions.

6. Conclusions There has been a growing interest among statisticians and applied researchers in constructing flexible lifetime models in order to improve the modeling of survival data. As a result, significant progress has been made towards the generalization of the traditional Weibull model. In this paper, we propose a four–parameter model named the odd log-logistic modified Weibull (OLLMW) distribution, which is obtained by applying the odd log-logistic generalized technique to the modified Weibull model. The new model extends several important lifetime distributions. We study some of its statistical properties and obtain explicit expressions for the positive and negative moments, conditional moments, mean deviations and reliability curves. The proposed distribution as applied to two real data sets turned out to provide better fits than other competing lifetime models. Further properties and applications of the new family can be considered in the future research. In particular, the following topics are interesting and still remain as open problems: (i) Discuss the Bayesian analysis of the new distribution. (ii) Introduce and study a new bivariate odd log-logistic modified Weibull distribution.

Acknowledgements The research of first author has been supported in part by the Higher Education Commission of Pakistan under NRPU project No. 3104.

References [1] M. Bebbington, C.D. Lai, R. Zitikis, A flexible Weibull extension, Reliability Engineering and System Safety, 92 (2007) 719–726. [2] J.M.F. Carrasco, M.M. Edwin Ortega, G.M.Cordeiro, A generalized modified Weibull distribution for lifetime modeling, Computational Statistics and Data Analysis, 53 (2008) 450–462. [3] G.M. Cordeiro, E.M.M. Ortega, A.J. Lemonte, The exponential–Weibull lifetime distribution, Journal of Statistical Computation and Simulation, 84 (2014) 2592–2606. [4] H.A. David, H.N. Nagaraja, Order statistics, John Wiley & Sons, Inc. (1970). [5] J.U. Gleaton, J.D. Lynch, Properties of generalized log-logistic families of lifetime distributions, Journal of Probability and Statistical Science 4, no. 1 (2006) 51–64. [6] M.N. Khan, The modified beta Weibull distribution, Hacettepe Journal of Mathematics and Statistics, 44 (2015) 1553–1568. [7] Y.L. Luke, The Special Functions and Their Approximations. San Diego: Academic Press, (1969).

18 A. Saboor, M. Alizadeh, M. N. Khan, I. Ghosh and G. M. Cordeiro [8] C.S. Meijer, On the G-function I–VIII, Proc. Kon. Ned. Akad. Wet, 49 (1946) 227–237, 344–356, 457–469, 632–641, 765–772, 936–943, 1063–1072, 1165–1175. [9] G.S. Mudholkar, D. K. Srivastava, M. Friemer, The exponentiated Weibull family: A reanalysis of the bus-motor-failure data, Technometrics, 37 (1995), 436– 445. [10] A. Saboor, S.B. Provost, M. Ahmad, The moment generating function of a bivariate gamma-type distribution, Applied Mathematics and Computation, 218(24) (2012) 11911–11921. [11] M. Xie, C.D. Lai, Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function, Reliability Engineering and System Safety, 52 (1995) 87–93. [12] W.H. Von Alven, Reliability Engineering by ARINC. Prentice-hall, New Jersey, 1964. [13] G.R. Aryal, C.P. Tsokos, Transmuted Weibull Distribution: A Generalization of the Weibull Probability Distribution, European Journal of Pure and Applied Mathematics, 4 (2011), 89–102. [14] V. Choulakian, M.A. Stephens, Goodness-of-fit for the generalized Pareto distribution. Technometrics, 43 (2001), 478-484. [15] M.S. Khan, R. King, Transmuted Modified Weibull Distribution: A Generalization of the Modified Weibull Probability Distribution, European Journal of Pure and Applied Mathematics, 6 (2013), 66–88. [16] Pog´ any, T. K., Saboor, A. and Provost, S. The Marshall-Olkin exponential Weibull distribution, Hacettepe Journal of Mathematics and Statistics, 44 (2015), 1579–1594 . [17] S. Pundir, S. Arora, K. Jain, Bonferroni curve and the related statistical inference, Statistics and Probability Letters, 75 (2005) 140-150. [18] A.M. Sarhan, M. Zain-din, Modified Weibull distribution, Applied Sciences, 11 (2009), 123–136.

Appendix A. Meijer G–function The symbol Gm,n p,q (·| ·) denotes Meijer’s G−function ([8]), which is defined in terms of a Mellin–Barnes integral as ∏m ∏n I ( a ,··· ,a ) 1 1 j=1 Γ(bj − s) j=1 Γ(1 − aj + s) p ∏ ∏ Gm,n z = z s ds, p,q q b1 , · · · , b q 2πi C j=m+1 Γ(1 − bj + s) pj=n+1 Γ(aj − s) where 0 ≤ m ≤ q, 0 ≤ n ≤ p and the poles aj , bj are such that no pole of Γ(bj − s), j = 1, m coincides with any pole of Γ(1 − aj + s), j = 1, n; i.e. ak − bj ̸∈ N, while z ̸= 0, C being a suitable integration contour, see [7, p. 143] and [8] for more details. The G-function’s Mathematica code reads MeijerG[{{a1 , ..., an }, {an+1 , ..., ap }}, {{b1 , ..., bm }, {bm+1 , ..., bq }}, z]. Abdus Saboor Department of Mathematics Kohat University of Science & Technology Kohat, Pakistan e-mail: [email protected]

Odd Log-Logistic Modified Weibull Distribution Morad Alizadeh Department of Statistics Persian Gulf University Bushehr, Iran e-mail: [email protected] Muhammad Nauman Khan Department of Mathematics Kohat University of Science & Technology Kohat, Pakistan e-mail: [email protected] Indranil Ghosh Department of Mathematics and Statistics University of North Carolina Wilmington, USA e-mail: [email protected] Gauss M. Cordeiro Department of Statistics Federal University of Pernambuco Recife, Brazil e-mail: [email protected]

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