A new extension of Weibull distribution: Properties and different methods of estimation

Journal of Computational and Applied Mathematics 336 (2018) 439–457

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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

A new extension of Weibull distribution: Properties and different methods of estimation Mazen Nassar a , Ahmed Z. Afify b, *, Sanku Dey c , Devendra Kumar d a

Department of Statistics, Faculty of Commerce, Zagazig University, Egypt Department of Statistics, Mathematics and Insurance, Benha University, Egypt Department of Statistics, St. Anthony’s College, Shillong-793001, Meghalaya, India d Department of Statistics, Central University of Haryana, Mahendergarh, India b c

article

info

Article history: Received 14 September 2017 MSC: 60E05 62F10 Keywords: Weibull distribution Moments Quantile function Stochastic ordering Stress–strength reliability Parameter estimation

a b s t r a c t The Weibull distribution has been generalized by many authors in recent years. Here, we introduce a new generalization of the Weibull distribution, called Alpha logarithmic transformed Weibull distribution that provides better fits than some of its known generalizations. The proposed distribution contains Weibull, exponential, logarithmic transformed exponential and logarithmic transformed Weibull distributions as special cases. Our main focus is the estimation from frequentist point of view of the unknown parameters along with some mathematical properties of the new model. The proposed distribution accommodates monotonically increasing, decreasing, bathtub and unimodal and then bathtub shape hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, percentile based estimators, least squares estimators, weighted least squares estimators, maximum product of spacings estimators and compare them using extensive numerical simulations. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The potentiality of the distribution is analyzed by means of two real data sets. © 2017 Elsevier B.V. All rights reserved.

1. Introduction In recent years, researchers proposed various ways of generating new continuous distributions in lifetime data analysis to enhance its capability to fit diverse lifetime data which have a high degree of skewness and kurtosis. These extended distributions provide greater flexibility in modeling certain applications and data in practice. Due to the computational and analytical facilities available in programming softwares such as R, Maple and Mathematica, it is easy to tackle the problems involved in computing special functions in these extended distributions. A detailed survey of methods for generating distributions were discussed by Lee et al. [1] and Jones [2]. When modeling monotonic hazard rates, one may prefer to adopt the exponential, gamma, Weibull and generalized exponential over other distributions. However, in case of non-monotone hazard rates such as the bathtub-shaped hazard rates or upside down bathtub-shaped hazard rates, the aforementioned distributions are not realistic or reasonable. The studies reveal that the most realistic hazard rate is bathtub-shaped which occurs in most real life systems. The lifetime models that present upside-down bathtub shaped failure rates are very useful in survival analysis. For greater details, readers may refer to Kotz and Nadarajah [3].

*

Corresponding author. E-mail address: [email protected] (A.Z. Afify).

https://doi.org/10.1016/j.cam.2017.12.001 0377-0427/© 2017 Elsevier B.V. All rights reserved.

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In the recent past, many generalizations of Weibull distribution have been attempted by researchers. Notable among them are: Mudholkar and Srivastava [4] introduced exponentiated Weibull distribution, Marshall and Olkin [5] obtained extended Weibull distribution, Xie et al. [6] proposed modified Weibull distribution, Lee et al. [7] proposed beta Weibull distribution, Bebbington et al. [8] studied the flexible Weibull distribution, Cordeiro et al. [9] proposed Kumaraswamy Weibull distribution, Zhang and Xie [10] proposed the truncated Weibull distribution and complementary extended Weibull power series class of distributions proposed by Cordeiro and Silva [11] and the references cited therein. In this article, we propose a new three-parameter generalization of the Weibull distribution, referred to as Alpha logarithmic transformed Weibull distribution (ALTW). The proposed distribution contains several lifetime distributions, such as Weibull, exponential, logarithmic transformed exponential [12] and logarithmic transformed Weibull distributions as special cases. We are motivated to introduce the ALTW distribution because (i) it contains a number aforementioned of known lifetime sub models; (ii) The ALTW distribution exhibits monotone as well as non-monotone hazard rates which makes this distribution to be superior to other lifetime distributions, which exhibit only monotonically increasing/decreasing, or constant hazard rates. (iii) It is shown in Section 2 that the ALTW distribution can be viewed as a mixture of Weibull distribution introduced by Weibull [13]; (iv) it can be viewed as a suitable model for fitting the skewed data which may not be properly fitted by other common distributions and can also be used in a variety of problems in different areas such as public health, biomedical studies, industrial reliability and survival analysis; and (v) The ALTW distribution outperforms several of the well-known lifetime distributions with respect to two real data examples. Comprehensive comparisons of estimation methods for other distributions have been performed in the literature: see Kundu and Raqab [14] for generalized Rayleigh distributions, Alkasasbeh and Raqab [15] for generalized logistic distributions, Mazucheli et al. [16] for weighted Lindley distribution, do Espirito Santo and Mazucheli [17] for Marshall–Olkin extended Lindley distribution and Dey et al. [18–24] for Two-parameter Rayleigh distribution, Weighted exponential distribution, twoparameter Maxwell distribution, Exponentiated Chen distribution, Dagum distribution, transmuted Rayleigh distribution and Two parameter exponentiated-Gumbel distribution, respectively. The final motivation of the paper is to show how different frequentist estimators of this distribution perform for different sample sizes and different parameter values and to develop a guideline for choosing the best estimation method for the ALTW distribution, which we think would be of interest to applied statisticians. The rest of the paper is organized as follows. In Sections 2 and 3, we introduce the ALTW distribution, and discuss some properties of this distribution. Section 4 describes five frequentist methods of estimation. In Section 5, a simulation study is carried out to compare the performance of these methods of estimation for the proposed model. In Section 6, the usefulness of the ALTW distribution is illustrated by means of two real data sets. Finally, Section 7 offers some concluding remarks. 2. Model description Pappas et al. [25] proposed a new method for generating distributions with the following cumulative distribution function (CDF) and probability density function (PDF)

{ F (x) =

1−

log[α − (α − 1)G(x)] log(α )

G(x)

if α > 0, α ̸ = 1 if α = 1

(1)

and f (x) =

⎧ ⎨

(α − 1)g(x) log(α )[α − (α − 1)G(x)]

if α > 0, α ̸ = 1

(2)

if α = 1

⎩

g(x)

where G(x) is the baseline CDF and g(x) = dG(x)/dx. They used the CDF in (1) to introduce a new generalization of the modified Weibull extension distribution proposed by Xie et al. [6], see also Al-Zahrani et al. [26]. Dey et al. [27] introduced a new generalization for the generalized exponential distribution by taking the CDF of the generalized exponential as the baseline in (1). They referred to the new distribution by ALT generalized exponential distribution. Dey et al. [27] used the Maclaurin series and binomial expansion to provide a useful mixture representation to the PDF in (2). The expansion of the PDF in (2) can be written as follows fALT (x) =

∞ ∑ k ∑

ωk,j hj+1 (x)

(3)

k=0 j=0

and the corresponding CDF is FALT (x) =

∞ ∑ k ∑ k=0 j=0

ωk,j Hj+1 (x),

(4)

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Fig. 1. Density function of the ALTW distribution with λ = 1 and various values of α and β .

where hθ (x) = θ g(x)Gθ −1 (x) is the exp −G PDF with power parameter θ > 0, Hθ (x) is the corresponding CDF and

ωk,j =

(α − 1)j+1

( )

(j + 1)(α + 1)k+1 log(α )

j

k

.

(5)

We now introduce the notion of ALTW distribution. Let X follows Weibull random variable with PDF and CDF given by β

g(x) = βλ xβ−1 e−λx , x > 0, β, λ > 0 and β

G(x) = 1 − e−λx , x > 0, β, λ > 0, respectively, where λ is a scale parameter and β is a shape parameter, respectively. The CDF of the ALTW distribution is given by

F (x) =

⎧ ⎨ ⎩

β

1−

log[α − (α − 1)(1 − e−λx )]

1−e

−λxβ

if x > 0; α, β, λ > 0, α ̸ = 1

log(α )

(6)

if x > 0; α, β, λ > 0, α = 1

and the corresponding PDF is

f (x) =

(α − 1)λβ xβ−1 e−λx

⎧ ⎪ ⎨ ⎪ ⎩

β

if x > 0; α, β, λ > 0, α ̸ = 1

β

log(α )[α − (α − 1)(1 − e−λx )]

λβ x

β−1 −λxβ

(7)

if x > 0; α, β, λ > 0, α = 1.

e

The corresponding hazard rate function is given by

h(x) =

⎧ ⎪ ⎨

(α − 1)λβ xβ−1 e−λx

β

[α − (α − 1)(1 − e−λxβ )] log[α − (α − 1)(1 − e−λxβ )]

⎪ ⎩ λβ xβ−1

if x > 0; α, β, λ > 0, α ̸ = 1

(8)

if x > 0; α, β, λ > 0, α = 1.

Figs. 1 and 2 show the curves for PDF and hazard rate function, respectively, of the ALTW distribution with λ = 1 and various values of α and β . Fig. 2 shows that the hazard function of the APTW distribution can be decreasing, increasing, bathtub or unimodal and then bathtub shapes. One of the advantages of the APTW distribution over the Weibull distribution is that the latter cannot model phenomenon showing bathtub or an unimodal and then bathtub shape failure rates. Special cases: Let X ∼ ALTW (α, β, λ). 1. If α → 1, then X reduces to the Weibull distribution [13]. 2. If α → 1 and β = 1, then X reduces to the exponential distribution.

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Fig. 2. Hazard rate function of the ALTW distribution with λ = 1 and various values of α and β .

3. If α = 2, β = 1, then X reduces to Logarithmic transformed exponential distribution introduced by Maurya et al. [12]. 4. If α = 2, then X reduces to Logarithmic transformed Weibull distribution. 2.1. Mixture representation Here, we present some representations of the PDF and CDF of the ALTW distribution. Using binomial expansion, a very useful linear representation for the PDF in (7) can be written as follows f (x) =

j ∞ ∑ k ∑ ∑

ωk,j,m gW (x; λ(m + 1), β ),

(9)

k=0 j=0 m=0

where gW (x; λ(m + 1), β ) is the PDF of the Weibull distribution with scale parameter λ(m + 1) and shape parameter β and

ωk,j,m =

( ) ωk,j (−1)m (j + 1) j . m+1 m

Several structural properties can be obtained from Eq. (9) and also properties of the Weibull distribution. By integrating Eq. (9), the CDF of X can be written in the mixture form F (x) =

j ∞ ∑ k ∑ ∑

ωk,j,m GW (x; λ(m + 1), β ),

k=0 j=0 m=0

where GW (x; λ(m + 1), β ) is the CDF of the Weibull distribution with scale parameter λ(m + 1) and shape parameter β . Hereafter, a random variable X that follows the distribution in (7) is denoted by X ∼ ALTW (α, β, λ). 3. Mathematical properties In this section, we give some important statistical and mathematical properties of the ALTW distribution such as quantile, ordinary and incomplete moments, mean deviation about mean and median, probability weighted moments, moment generating function, Rényi and δ -entropies and moments of the residual and reversed residual lives. Established algebraic

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expressions to determine some structural properties of the ALTW distribution can be more efficient than computing them directly by numerical integration from its density function. 3.1. Quantile and random number generation Quantiles are fundamental for estimation (for example, quantile estimators) and simulation. The pth quantile xp of the ALTW distribution is the root of the equation

( 1−p )]1/β α −1 −1 log , xp = Q (p) = λ α−1 [

x > 0, α > 0, β > 0.

(10)

In particular, the first three, Q (1/4), Q (1/2) and Q (3/4), can be obtained by setting p = 0.25, p = 0.5 and p = 0.75 in Eq. (10), respectively. Let U ∼ uniform(0, 1), then Eq. (10) can be used to simulate a random sample of size n from the ALTW distribution as follows

( 1−u )]1/β −1 α i −1 , i = 1, 2, . . . , n. log λ α−1

[ xi =

3.2. Moments The rth ordinary moments of the ALTW distribution is given by

µ′r = E(X r ) =

∫

∞

xr f (x)dx −∞

=

j ∞ ∑ k ∑ ∑

ωk,j,m

=

xr gW (x; λ(m + 1), β )dx, 0

k=0 j=0 m=0 j ∞ ∑ k ∑ ∑

∞

∫

ωk,j,m λ(m + 1)

k=0 j=0 m=0

Γ

(

r +β

)

β

(λ(m + 1))

r +β

.

(11)

β

In particular,

µ′1 = E(X ) =

j ∞ ∑ k ∑ ∑

ωk,j,m λ(m + 1)

k=0 j=0 m=0

µ′2 = E(X 2 ) =

j ∞ ∑ k ∑ ∑

Γ

(

β+1 β

)

(λ(m + 1))

ωk,j,m λ(m + 1)

k=0 j=0 m=0

Γ

(

β+2 β

β+1 β

,

)

(λ(m + 1))

β+2 β

and Var(X ) = E(X 2 ) − [E(X )]2 . The central moments µr and cumulants kr of X can be determined from (11) as

µr = E(X − µ)r =

r ∑

(−1)k

k=0

( ) r

k

µ′1r µ′r −k

and kr = µr − ′

) r −1 ( ∑ r −1 k=1

k−1

kr µ′r −k ,

where k1 = µ′1 . Thus k2 = µ′2 − µ′1 , k3 = µ′3 − 3µ′2 µ′1 + 2µ′1 , k4 = µ′4 − 4µ′3 µ′1 − 3µ′2 + 12µ′2 µ′1 − 6µ′1 , etc. The 3/2 skewness γ1 = k3 /k2 and kurtosis γ2 = k4 /k22 can be calculated from the second, third and fourth standardized cumulants. Eq. (10) and the nth raw moment of the ALTW distribution are used to obtain the mean, median, variance, skewness and kurtosis for various values of α and β with λ = 1. These values are presented in Table 1. Table 1 indicates that, for fixed λ and β , the median and the mean are increasing functions of α , while the skewness is decreasing function of α . Also, for fixed λ and β , the variance and the skewness are decreasing functions of β . Table 1 shows that the ALTW distribution can be left skewed, right skewed, furthermore, it can be platykurtic (kurtosis < 3) or leptokurtic (kurtosis > 3). Then the ALTW distribution is a flexible distribution and can be used to model the skewed data. 2

3

2

2

4

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Table 1 Median, mean, variance, skewness and kurtosis of the ALTW model for λ = 1 and various values of α and β .

α

β

Median

Mean

Variance

Skewness

Kurtosis

0.5

0.5 1.5 2.5 5

0.2860 0.6589 0.7785 0.8823

1.5501 0.7924 0.8141 0.8766

15.2005 0.3388 0.1394 0.0456

7.5439 1.2524 0.4969 −0.1336

113.0321 4.9658 2.9967 2.7823

1.5

0.5 1.5 2.5 5

0.6394 0.8615 0.9144 0.9563

2.3311 0.9722 0.9311 0.9421

23.7263 0.4001 0.1474 0.0436

6.1083 0.9694 0.2792 −0.3237

75.1183 4.0938 2.7914 2.9465

2.5

0.5 1.5 2.5 5

0.8991 0.9652 0.9790 0.9894

2.8328 1.0641 0.9871 0.9716

29.6972 0.4342 0.1521 0.0430

5.5045 0.8451 0.1823 −0.4087

61.4907 3.7629 2.7259 3.0367

5

0.5 1.5 2.5 5

1.3791 1.1131 1.0664 1.0327

3.6871 1.1955 1.0632 1.0102

40.7879 0.4866 0.1595 0.0426

4.7655 0.6875 0.0587 −0.5171

46.6867 3.3884 2.6647 3.1657

25

0.5 1.5 2.5 5

3.2104 1.4752 1.2627 1.1237

6.5979 1.5180 1.2353 1.0919

87.4154 0.6378 0.1813 0.0430

3.4246 0.3835 −0.1806 −0.7258

25.1192 2.8020 2.6130 3.4550

3.3. Incomplete moments The nth incomplete moment of the ALTW distribution is given by mn (t) = E [X n |x < t] = Eq. (9)

mn (t) =

j ∞ ∑ k ∑ ∑

ωk,j,m

=

0

xn f (x)dx. We can write from

t

∫

xn gW (x; λ(m + 1), β )dx

0

k=0 j=0 m=0 j ∞ ∑ k ∑ ∑

∫t

ωk,j,m λ(m + 1)

γ

(

n+β

β

, λ(m + 1) t β

(λ(m + 1))

k=0 j=0 m=0

)

n+β

β

∫x

where γ (a, x) denote the lower incomplete gamma function defined by γ (a, x) = 0 t a−1 e−t dt and S(x) = 1 − F (x). The important application of the first incomplete moment is related to Bonferroni and Lorenz curves defined by L(p) = m1 (xp )/µ′1 and B(p) = m1 (xp )/pµ′1 , respectively, where xp can be evaluated numerically from Eq. (10) for a given probability. These curves are very useful in economics, demography, insurance, engineering and medicine. 3.4. Conditional moments For the ALTW distribution, it can be easily seen that the conditional moments, E(X n |X > x), can be written as E(X n |X > x) =

1 S(x)

Jn (x),

where Jn (x) =

j k ∞ ∑ ∑ ∑

ωk,j,m

j ∞ ∑ k ∑ ∑

yn gW (y; λ(m + 1), β )dy x

k=0 j=0 m=0

=

∞

∫

ωk,j,m λ(m + 1) Γ

(

n+β

k=0 j=0 m=0

β

) , λ(m + 1)xβ ,

(12)

∫∞

where Γ (a, x) denote the upper incomplete gamma function defined by Γ (a, x) = x t a−1 e−t dt and S(x) = 1 − F (x). An application of the conditional moments is the mean residual life (MRL). MRL function is the expected remaining life, X − x, given that the item has survived to time x. Thus, in life testing situations, the expected additional lifetime given that a component has survived until time x is called the MRL. The MRL function in terms of the first conditional moment as mX (x) = E(X − x|X > x) =

1 S(x)

J1 (x) − x.

where J1 (x) can be obtained from (12) with n = 1.

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Another application of the conditional moments is the mean deviations about the mean and the median. They are used to measure the dispersion and the spread in a population from the center. If we denote the median by M, then the mean deviations about the mean and the median can be calculated as ∞

∫

δµ =

|x − µ| f (x)dx = 2µF (µ) − 2µ + 2J1 (µ) 0

and ∞

∫

δM =

|x − M | f (x)dx = 2J1 (M) − µ, 0

respectively, where J1 (µ) and J1 (M) can be obtained from (12). Also, F (µ) and F (M) can be easily calculated from (6). 3.5. Generating function First, we obtain the moment generating function of the Weibull distribution ∞

∫

M(t) = E(etx ) =

ety f (x)dx

−∞

= βλ

∞

∫

β

ety yβ−1 e−λy dy

0

= λβ

∞ p ∫ ∑ t p=0

=

p!

β

yβ+p−1 e−λy dy

0

(

∞ ∑ λt p Γ

p+β

)

β

p!

p=0

∞

λ

p+β

.

β

Consider the Wright generalized hypergeometric function defined by p

ψq

(α1 , A1 ), . . . , (αp , Ap ) (β1 , B1 ), . . . , (βq , Bq )

[

] ;z =

∞ ∑

∏p

Γ (αj + Aj p) z p

∏q

Γ (βj + Bj p) p!

j=1

p=0

j=1

.

Therefore the moment generating function of Weibull distribution is

[ M(t) = 1 ψ0

(1, − β −1 )

−

( )1/β ] 1

;

t .

λ

(13)

Combining Eqs. (9) and (13), the moment generating function of the ALTW distribution is given by MX (t) =

j ∞ ∑ k ∑ ∑

[ ωk,j,m 1 ψ0

(1, − β −1 )

−

k=0 j=0 m=0

( ;

1

λ(m + 1)

)1/β ]

t .

3.6. Moments of the residual and reversed residual lives Several functions can be defined from the residual life, for example the hazard rate function, mean residual life function and the left censored mean function. It is well known that these three functions uniquely determined F (x) (see Zoroa et al. [28]). The nth moment of residual life of X , ϕn (t) = E [(X − t)|X > t ] for n = 1, 2, . . . , uniquely determines F (x), we have

ϕn (t) =

∫

1 1 − F (t)

∞

(x − t)n dF (x).

t

For ALTW distribution, we can write

ϕn (t) =

j ∞ k n 1 ∑∑∑∑

R(t)

k=0 j=0 m=0 p=0

ωk,j,m (−1)n−p t n−p

( ) λ(m + 1) Γ n p

(

p+1

β

, λ(m + 1)t β

(λ(m + 1))

p+1

) .

β

The MRL function corresponding to ϕn (t) represent the expected additional life length for a unit that is alive at age x. Guess and Proschan [29] derived an extensive coverage of possible applications of the MRL function in survival analysis, biomedical sciences, life insurance, maintenance and product quality control, economics, social studies and demography.

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In the similar manner, Navarro et al. [30] prove that the nth moment of the reversed residual life, say φn (t) = E [(t − X )n |X ≤ t ] for t > 0 and n = 1, 2, . . . , uniquely determines F (x). We obtain

φn (t) =

1

t

∫

F (t)

(t − x)n dF (x). 0

For ALTW distribution, we can write

ϕn (t) =

j ∞ k n 1 ∑∑∑∑

F (t)

ωk,j,m (−1)p t n−p

( ) λ(m + 1) γ n

k=0 j=0 m=0 p=0

p

(

p+1

β

, λ(m + 1)t β

(λ(m + 1))

p+1

) .

β

The mean reversed residual life (MRRL) function corresponding to φ1 (t) represents the waiting time elapsed for the failure of an item under the condition that this failure had occurred in (0, t). The MRRL function of X can be obtained by setting n = 1 in the above equation. 3.7. Probability weighted moments The probability weighted moments (PWMs) can be used to derive the estimators of the parameters and quantiles of generalized distributions. These moments have low variances and no severe biases, and they compare favorably with estimators obtained by maximum likelihood method. The (r , s)th PWM of X (for r ≥ 1, s ≥ 0 ) can be defined as ∞

∫

ρr ,s = E [X r F s (x)] =

xr F s (x)f (x)dx 0

=

j l+p ∞ ∑ k s ∞ ∑ ∑ ∑ ∑ ∑

ωk,j,m (−1) φp (l)(α − 1) q

l+p

l

k=0 j=0 m=0 l=0 p=0 q=0

× βλ(m + 1)

∞

∫

(s) (l + p) q

β

xr +β−1 e−λ[(m+1)+q]x dx 0

=

j l+p ∞ ∑ k s ∞ ∑ ∑ ∑ ∑ ∑

ωk,j,m (−1) φp (l)(α − 1) q

l+p

k=0 j=0 m=0 l=0 p=0 q=0

λ(m + 1) Γ

(

r +β

l

q

)

β

×

(s) (l + p)

r +β

.

β

[λ((m + 1) + q)] 3.8. Rényi and δ−entropies

Entropy is used to measure the randomness of systems and it is widely used in areas like physics, molecular imaging of tumors and sparse kernel density estimation. If X has the PDF f (·), Rényi entropy [31] can be defined as Hδ (x) =

1 1−δ

∞

(∫ log

)

f δ (x)dx ,

δ > 0,

δ ̸= 1.

0

Some recent applications of the Rényi entropy have been considered such as sparse kernel density estimations [32]; highresolution scalar quantization [33]; estimation of the number of components of a multicomponent non stationary signal [34]; identification of cardiac autonomic neuropathy in diabetes [35]; and signal segmentation in time-frequency plane [36]. Using Eq. (7), we have f δ (x) =

(

λβ (α − 1) log α

)δ

xδ (β−1) e−δλx

β

[α − (α − 1)(1 − e−λxβ )]δ

.

After some algebra, we can write f δ (x) =

∞ ∑ ∞ ∑

β

υi,j xδ(β−1) e−λ(δ+j)x ,

i=0 j=0

where

υi,j =

(

λβ (α − 1) log α

)δ

(−1)j α −(δ+i) δ (i)

( )

j!

j

i

and a(i) = Γ (a + i)/Γ (a) is the rising factorial defined for any real a.

M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

447

Then, the Rényi entropy of X reduces to

⎛ 1

Hδ (x) =

1−δ

log ⎝

∞ ∑ ∞ ∑

∞

∫

υi,j

⎞ xδ (β−1) e

−λ(δ+j)xβ

dx⎠ .

0

i=0 j=0

Finally, it can be expressed as 1

Hδ (x) =

1−δ

⎡ ∞ ∑ ∞ ∑ log ⎣ υi,j i=0 j=0

(

Γ

δ (β+1)+1 β

β{λ(δ + j)}

)

⎤

δ (β+1)+1 β

⎦.

(14)

The δ−entropy, say Iδ (x), is defined by Iδ (x) =

1

δ−1

[

∞

∫

]

f δ (x)dx ,

log 1 −

δ > 0,

δ ̸= 1,

0

and then it follows from Eq. (14). 3.9. Order statistics Moments of order statistics play an important role in quality control testing and reliability to predict the failure of future items based on the times of few early failures. We know that if X(1:n) ≤ · · · ≤ X(n:n) denotes the order statistics of a random sample X1 , . . . , Xn from a continuous population with CDF GX (x) and PDF gX (x) then the PDF of Xj:n is given by n!

gXj:n (x) =

(j − 1)!(n − j)!

gX (x)(GX (x))j−1 (1 − GX (x))n−j ,

for j = 1, . . . , n. The CDF and PDF of the jth order statistic for the ALTW distribution are given by GXj:n (x) =

n ( ) ∑ n

l

l=k

=

n l ∑ ∑

[F (x)]l [1 − F (x)]n−l

(−1)u

( )( ) n

l

l

u

l=k u=0

=

n l ∑ ∑

(−1)

u

[F¯ (x)]n−l+u

( )( )[

β

n

l

log{α − (α − 1)(1 − e−λx )}

l

u

log α

l=k u=0

]n−l+u

and gXj:n (x) =

× =

j−1 ∑

n! (j − 1)!(n − j)!

u

(

(−1)

u=0

λβ (α − 1)x

)[

β

j−1

log{α − (α − 1)(1 − e−λx )}

u

log α

]n−j+u

β−1 −λxβ

e

β

log(α )[α − (α − 1)(1 − e−λx )] +u+k+l ∞ ∑ ∞ n−j∑ ∑ k=0 l=0

υk,l,m gw (x; λ(m + 1), β ),

(15)

m=0

where

υk,l,m =

j−1 ∑

n! (j − 1)!(n − j)!

× φk (n − j + u)

(−1)n−j+u+m

(

j−1

u=0

(α − 1)

u

)(

n−j+u+k+l

)

m

n−j+u+k+l+1

(m + 1) α l+1 (log α )n−j+u

and gw (x; λ(m + 1), β ) denotes the Weibull density function with parameters λ(m + 1) and β . Thus, the density function of the ALTW order statistics is a linear mixture of the Weibull densities. Based on Eq. (15), we can obtain some structural properties of Xj:n from those Weibull properties. The qth moment of Xj:n is given by q

E [Xj:n ] =

+u+k+l ∞ ∑ ∞ n−j∑ ∑ k=0 l=0

q

υk,l,m E [Yλq(m+1) ],

m=0

where Yλ(m+1) ∼ Weibull distribution with parameters λ(m + 1) and β .

(16)

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M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

Some other important measures useful for lifetime models are the L-moments due to Hosking [37]. It is a linear function of expected order statistics defined by

λk =

k−1 1∑ (−1)i k

(

k−1

)

i

i=0

E(Xk−i:k ), k ≥ 1.

Based upon the moments in Eq. (16), we can obtain explicit expressions for L-moments of X as infinite weighted linear combinations of suitable ALTW means. 4. Methods of estimation In this section, we describe five methods of estimation for estimating the parameters α , β and λ of the ALTW distribution. The methods are: maximum likelihood estimation, least squares and weighted least-squares estimation, percentile based estimation and maximum product spacing estimation. Since the moments of ALTW distribution are not in closed form, the method of moment estimation is not considered here. The performance of the five different estimation methods are studied through Monte Carlo simulation. 4.1. Method of maximum likelihood estimation The maximum likelihood estimation (MLE) method is the most frequently used method of parameter estimation [38]. It enjoys many desirable properties including consistency, asymptotic efficiency, invariance property as well as its intuitive appeal. Let x1 , . . . , xn be a random sample of size n from the PDF in (7), then the log-likelihood function without the constant term is given by L(Θ ) = n log

(

(α − 1)λβ

)

log(α )

+ (β − 1)

n ∑

log(xi ) − λ

i=1

n ∑

β

xi −

i=1

n ∑

β

log(1 + (α − 1)e−λxi ).

(17)

i=1

Differentiating (17) partially with respect to α, β and λ and equate the results to zero, the resulting equations are β

n

∑ e−λxi n n ∂ L(Θ ) = − − = 0, ∂α α − 1 α log(α ) ψi i=1

n

n

n

i=1

i=1

i=1

β

β

∑ ∑ β ∑ x log(xi )e−λxi ∂ L(Θ ) n i = + log(xi ) − λ xi log(xi ) + λ(α − 1) =0 ∂β β ψi and n

n

i=1

i=1

β

β

∑ β ∑ x e−λxi n ∂ L(Θ ) i = − xi + (α − 1) = 0, ∂λ λ ψi where β

ψi = 1 + (α − 1)e−λxi . The MLEs of α, β and λ denoted by αˆ , βˆ and λˆ , respectively are obtained by solving the above three equations simultaneously. To check that the global maximum has been attained, a number of starting values have been used. 4.2. Method of ordinary and weighted least-squares Swain et al. [39] proposed the least square (LS) and weighted least square (WLS) estimators to estimate the parameters of the beta distribution. Let x(1) < x(2) < · · · < x(n) be the order statistics of a random sample of size n from the PDF given in (7), then the least square estimators (LSEs) of the unknown parameters α, β and λ of the ALTW distribution can be obtained by minimizing the following function n ∑

⎡⎛ ⎣⎝1 −

i=1

β

−λx(i)

log[1 + (α − 1)e log(α )

⎞ ]⎠

⎤2 −

i n+1

⎦ ,

with respect to unknown parameters α, β and λ or by solving the following nonlinear equations n ∑ i=1

⎡⎛ ⎣⎝1 −

β

−λx(i)

log[1 + (α − 1)e log(α )

⎞ ]⎠

⎤ −

i n+1

⎦ ∆1 (x(i) |α, β, λ) = 0,

M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

n ∑ i=1

n ∑

⎡⎛

⎣⎝1 − log[1 + (α − 1)e log(α ) ⎡⎛ ⎣⎝1 −

log[1 + (α − 1)e

β

−λx(i)

β

−λx(i)

log(α )

i=1

⎞

449

⎤

]⎠

−

⎞ ]⎠

i n+1

⎦ ∆2 (x(i) |α, β, λ) = 0, ⎤

−

i n+1

⎦ ∆3 (x(i) |α, β, λ) = 0

where

∆1 (x(i) |α, β, λ) =

log(ψ(i) )

−

α log(α )2

e

β

−λx(i)

log(α )ψ(i)

β

∆2 (x(i) |α, β, λ) =

λ(α − 1)x(i) log(x(i) )e

,

(18)

β

−λx(i)

(19)

log(α )ψ(i)

and β

∆3 (x(i) |α, β, λ) =

β

−λx(i)

(α − 1)x(i) e

log(α )ψ(i)

.

(20)

The weighted least square estimators (WLSEs) of α, β and λ can be obtained by minimizing the following function n ∑ (n + 1)2 (n + 2)

n−i+1

i=1

⎡⎛ ⎣⎝1 −

log[1 + (α − 1)e

β

−λx(i)

log(α )

⎞ ]⎠

⎤2 −

i n+1

⎦ ,

(21)

with respect to the unknown parameters α, β and λ. Also, the WLSEs can be obtained by solving the following nonlinear equations n ∑ (n + 1)2 (n + 2)

n−i+1

i=1

n ∑ (n + 1)2 (n + 2)

n−i+1

i=1

⎡⎛ ⎣⎝1 − ⎡⎛ ⎣⎝1 −

log[1 + (α − 1)e

β

−λx(i)

log(α ) log[1 + (α − 1)e

β

−λx(i)

log(α )

⎞ ]⎠ ⎞ ]⎠

⎤ −

i n+1

⎦ ∆1 (x(i) |α, β, λ) = 0, ⎤

−

i n+1

⎦ ∆2 (x(i) |α, β, λ) = 0

and n ∑ (n + 1)2 (n + 2)

n−i+1

i=1

⎡⎛ ⎣⎝1 −

log[1 + (α − 1)e log(α )

β

−λx(i)

⎞ ]⎠

⎤ −

i n+1

⎦ ∆3 (x(i) |α, β, λ) = 0

where ∆1 (x(i) |α, β, λ), ∆2 (x(i) |α, β, λ) and ∆3 (x(i) |α, β, λ) are given by (18)–(20). 4.3. Method of percentile estimation The method of percentile was originally proposed by Kao [40,41]. Let pi = n+i 1 be an estimate of F (x(i) |α, β, λ), then the percentile estimators (PCEs) of α, β and λ can be obtained by minimizing the function n ∑ i=1

⎡

1−pi −1 ⎣x(i) − − log α λ α−1

(

1

{

}) β1

⎤2 ⎦ ,

with respect to α, β and λ. Also, the PCEs of α, β and λ can be obtained by solving the following nonlinear equations n ∑ i=1

n ∑ i=1

1−pi −1 ⎣x(i) − − log α λ α−1

}) β1

⎤

⎡

}) β1

⎤

⎡

(

1

{

1−pi −1 ⎣x(i) − − log α λ α−1

(

1

{

⎦ φ1 (x(i) |α, β, λ) = 0,

⎦ φ2 (x(i) |α, β, λ) = 0

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M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

and n ∑ i=1

⎡

1−pi −1 ⎣x(i) − − log α λ α−1

(

{

1

}) β1

⎤ ⎦ φ3 (x(i) |α, β, λ) = 0,

where

α 1−pi − 1 φ1 (x(i) |α, β, λ) = − log λβ λ α−1 1

φ2 (x(i) |α, β, λ) =

1

(

(

β2

1

{

α 1−pi − 1 − log λ α−1

{

1

}) β1 −1

1 − α −pi (1 + α pi − pi ) (α − 1)(α 1−pi − 1)

}) β1

( log −

1

{

α 1−pi − 1 α−1

λ

log

(

) α 1−pi − 1 . α−1

,

})

and

α 1−pi − 1 φ3 (x(i) |α, β, λ) = 2 − log λ β λ α−1 1

(

1

{

}) β1 −1 log

4.4. Method of maximum product of spacings The method of maximum product of spacings (MPS) as an alternative to MLE was introduced by Cheng and Amin [42,43]. Let Di (α, β, λ) is the uniform spacings of a random sample from the ALTW distribution defined by Di (α, λ) = F x(i) | α, β, λ − F x(i−1) | α, β, λ ,

(

)

(

)

i = 1, 2, . . . , n,

where F (x(0) | α, β, λ) = 0 and F (x(n+1) | α, β, λ) = 1. The MPS estimators (MPSEs) of α, β and λ are obtained by maximizing the following function 1 n+1

n+1 ∑

log Di (α, β, λ),

i=1

with respect to α, β and λ. Alternatively, MPSEs can be obtained by solving the following nonlinear equations 1 n+1

1 n+1

n+1 ∑

1

i=1

Di (α, β, λ)

n+1 ∑

1

i=1

Di (α, β, λ)

n+1 ∑

1

i=1

Di (α, β, λ)

[ ] ∆1 (x(i) |α, β, λ) − ∆1 (x(i−1) |α, β, λ) = 0,

[ ] ∆2 (x(i) |α, β, λ) − ∆2 (x(i−1) |α, β, λ) = 0

and 1 n+1

[ ] ∆3 (x(i) |α, β, λ) − ∆3 (x(i−1) |α, β, λ) = 0,

where ∆1 (x(i) |α, β, λ), ∆2 (x(i) |α, β, λ) and ∆3 (x(i) |α, β, λ) are given by (18)–(20). 5. Simulation study In this section a Monte Carlo simulation study is conducted to compare the performance of the different estimators of the unknown parameters for the ALTW distribution. The performance of the different estimators proposed in the previous section are evaluated in terms of their mean squared errors (MSEs). All the computations in this section are done by Mathcad program Version 14.0. We generate 1000 samples of the ALTW distribution, where n = (20, 50, 100, 200) and by choosing λ = (1, 2), β = (0.5, 2.5) and α = (0.5, 2.5). The average values of estimates and MSEs of MLEs, LSEs, WLSEs, PCEs and MPSEs are obtained and displayed in Tables 2–5. From Tables 2–5, we observe that all the estimates show the property of consistency i.e., the MSEs decrease as sample size increase. Comparing the different methods of estimation, the results show that the PCE produces the best results for estimating the parameters α, β and λ, in terms of MSEs in most of the cases. The ordering of performance of estimators in term of MSEs (from best to worst) for α is PCE, MPSE, LSE, WLSE and MLE. The ordering of performance for β is PCE, WLSE, LSE, MPSE and MLE and PCE, WLSE, MPSE, LSE and MLE for the parameter λ.

M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

451

Table 2 Average values of estimates and the corresponding MSEs (in parentheses) for n = 20. Parameters

MLEs

LSEs

WLSEs

PCEs

MPSEs

λ=1 β = 0.5 α = 0.5

0.738(0.401) 0.818(0.258) 0.583(0.597)

0.757(0.310) 0.665(0.070) 0.417(0.439)

0.803(0.335) 0.638(0.048) 0.476(0.631)

1.124(0.223) 0.468(0.049) 0.649(0.123)

0.972(0.318) 0.748(0.219) 0.477(0.226)

λ=1 β = 0.5 α = 2.5

0.719(0.446) 0.754(0.159) 2.704(1.747)

0.781(0.289) 0.614(0.064) 2.489(0.944)

0.895(0.154) 0.544(0.019) 2.646(1.141)

0.906(0.153) 0.540(0.035) 2.306(0.613)

0.936(0.375) 0.694(0.151) 2.551(0.872)

λ=1 β = 2.5 α = 0.5

0.568(0.569) 3.190(1.589) 0.397(0.648)

0.907(0.205) 2.752(0.571) 0.552(0.657)

1.039(0.098) 2.450(0.235) 0.704(0.532)

1.076(0.064) 2.320(0.285) 0.655(0.044)

0.984(0.277) 3.126(1.331) 0.520(0.526)

λ=1 β = 2.5 α = 2.5

0.783(0.209) 3.099(0.896) 2.328(1.150)

0.884(0.116) 2.672(0.483) 2.607(1.130)

0.827(0.128) 2.754(0.531) 2.125(0.754)

1.003(0.041) 2.361(0.209) 2.367(0.301)

0.951(0.221) 2.999(1.062) 2.408(0.751)

λ=2 β = 0.5 α = 0.5

1.792(1.659) 0.723(0.131) 0.641(0.143)

1.895(1.432) 0.623(0.087) 0.702(0.094)

1.926(0.994) 0.607(0.065) 0.771(0.278)

2.158(0.468) 0.514(0.043) 0.574(0.046)

2.314(1.911) 0.714(0.166) 0.579(0.074)

λ=2 β = 0.5 α = 2.5

1.492(1.321) 0.755(0.148) 2.211(1.719)

1.619(0.915) 0.615(0.063) 2.535(1.450)

1.592(0.830) 0.630(0.056) 2.349(1.424)

2.005(0.141) 0.503(0.028) 2.409(1.150)

1.882(1.002) 0.709(0.155) 2.111(1.305)

λ=2 β = 2.5 α = 0.5

1.754(1.081) 3.734(2.431) 0.840(0.250)

1.860(1.516) 2.655(0.364) 0.466(0.089)

1.858(0.785) 2.640(0.377) 0.687(0.189)

1.862(0.326) 2.382(0.224) 0.513(0.091)

2.205(0.554) 2.771(0.452) 0.411(0.032)

λ=2 β = 2.5 α = 2.5

1.484(0.911) 3.334(1.519) 2.133(1.564)

1.725(0.671) 2.796(0.813) 2.407(1.207)

1.819(0.526) 2.733(0.465) 2.390(1.234)

2.017(0.204) 2.390(0.222) 2.598(0.812)

2.227(0.727) 2.776(0.527) 2.591(0.658)

Table 3 Average values of estimates and the corresponding MSEs (in parentheses) for n = 50. Parameters

MLEs

LSEs

WLSEs

PCEs

MPSEs

λ=1 β = 0.5 α = 0.5

0.944(0.199) 0.588(0.047) 0.674(0.488)

0.833(0.136) 0.570(0.025) 0.456(0.337)

0.870(0.188) 0.580(0.025) 0.575(0.596)

0.995(0.192) 0.506(0.038) 0.446(0.093)

1.002(0.100) 0.551(0.021) 0.488(0.167)

λ=1 β = 0.5 α = 2.5

0.743(0.271) 0.632(0.050) 2.328(0.950)

0.791(0.215) 0.586(0.032) 2.596(0.821)

0.907(0.086) 0.522(0.005) 2.490(0.412)

1.071(0.191) 0.480(0.026) 2.708(0.248)

0.900(0.189) 0.586(0.037) 2.535(0.513)

λ=1 β = 2.5 α = 0.5

0.768(0.257) 2.752(0.450) 0.429(0.461)

0.898(0.087) 2.706(0.228) 0.448(0.198)

1.068(0.051) 2.465(0.067) 0.688(0.192)

0.923(0.057) 2.658(0.142) 0.529(0.036)

0.899(0.144) 2.909(0.493) 0.437(0.485)

λ=1 β = 2.5 α = 2.5

0.923(0.102) 2.691(0.181) 2.617(0.763)

0.948(0.052) 2.560(0.116) 2.526(0.359)

0.934(0.057) 2.592(0.114) 2.469(0.257)

1.020(0.020) 2.415(0.072) 2.647(0.175)

0.931(0.149) 2.788(0.417) 2.536(0.620)

λ=2 β = 0.5 α = 0.5

1.672(0.644) 0.630(0.044) 0.481(0.064)

1.741(0.563) 0.593(0.038) 0.615(0.087)

1.740(0.433) 0.580(0.026) 0.523(0.064)

2.019(0.071) 0.451(0.021) 0.546(0.015)

1.979(0.483) 0.594(0.040) 0.598(0.048)

λ=2 β = 0.5 α = 2.5

1.546(0.825) 0.647(0.062) 2.349(1.377)

1.604(0.611) 0.590(0.033) 2.405(0.966)

1.554(0.640) 0.602(0.029) 2.355(1.216)

2.019(0.076) 0.456(0.018) 2.401(0.110)

1.776(0.551) 0.604(0.043) 2.412(1.241)

λ=2 β = 2.5 α = 0.5

1.760(0.550) 3.038(0.781) 0.551(0.128)

1.866(0.247) 2.710(0.301) 0.478(0.032)

1.884(0.255) 2.700(0.166) 0.496(0.068)

2.028(0.107) 2.404(0.060) 0.576(0.017)

2.082(0.228) 2.688(0.177) 0.442(0.031)

λ=2 β = 2.5 α = 2.5

1.723(0.497) 2.900(0.398) 2.643(1.129)

1.817(0.263) 2.610(0.149) 2.432(0.524)

1.855(0.198) 2.600(0.109) 2.534(0.968)

2.005(0.064) 2.429(0.071) 2.610(0.171)

2.021(0.212) 2.656(0.138) 2.458(0.538)

6. Data analysis In this section, we compare the fits of the ALTW distribution with some other competitive models given in Table 6 by means of two real data sets to illustrate the potentiality of the ALTW model. We estimate the unknown parameters of the distributions by the maximum likelihood method. For model comparison, we consider four well-known statistics: the maximized log-likelihood, Anderson–Darling (A∗ ), Cramér–Von Mises (W ∗ ) and Kolmogorov–Smirnov (KS) (with its

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M. Nassar et al. / Journal of Computational and Applied Mathematics 336 (2018) 439–457

Table 4 Average values of estimates and the corresponding MSEs (in parentheses) for n = 100. Parameters

MLEs

LSEs

WLSEs

PCEs

MPSEs

λ=1 β = 0.5 α = 0.5

0.929(0.142) 0.559(0.019) 0.569(0.265)

0.830(0.135) 0.573(0.018) 0.419(0.241)

0.871(0.169) 0.577(0.021) 0.534(0.350)

1.091(0.191) 0.477(0.022) 0.491(0.056)

0.964(0.075) 0.545(0.012) 0.466(0.105)

λ=1 β = 0.5 α = 2.5

0.811(0.162) 0.573(0.018) 2.348(0.837)

0.820(0.163) 0.558(0.013) 2.528(0.819)

0.929(0.064) 0.518(0.003) 2.519(0.335)

1.037(0.148) 0.493(0.021) 2.514(0.151)

0.907(0.108) 0.546(0.010) 2.443(0.509)

λ=1 β = 2.5 α = 0.5

0.803(0.149) 2.816(0.353) 0.419(0.337)

0.881(0.068) 2.626(0.184) 0.440(0.082)

1.081(0.040) 2.378(0.066) 0.756(0.175)

0.991(0.008) 2.266(0.131) 0.507(0.001)

0.900(0.100) 2.689(0.202) 0.468(0.305)

λ=1 β = 2.5 α = 2.5

0.950(0.043) 2.591(0.094) 2.505(0.242)

0.964(0.033) 2.539(0.108) 2.608(0.221)

0.959(0.034) 2.569(0.082) 2.600(0.185)

1.007(0.007) 2.441(0.039) 2.583(0.052)

0.943(0.084) 2.653(0.193) 2.573(0.501)

λ=2 β = 0.5 α = 0.5

1.751(0.395) 0.576(0.017) 0.472(0.047)

1.736(0.387) 0.566(0.015) 0.479(0.042)

1.734(0.336) 0.573(0.018) 0.415(0.024)

2.026(0.043) 0.456(0.012) 0.539(0.010)

1.880(0.285) 0.562(0.014) 0.426(0.016)

λ=2 β = 0.5 α = 2.5

1.645(0.551) 0.588(0.022) 2.412(1.138)

1.658(0.444) 0.562(0.013) 2.464(0.779)

1.645(0.489) 0.574(0.015) 2.443(1.168)

2.033(0.042) 0.462(0.010) 2.481(0.071)

1.781(0.379) 0.562(0.014) 2.465(0.753)

λ=2 β = 2.5 α = 0.5

1.851(0.319) 2.809(0.302) 0.472(0.033)

1.904(0.181) 2.669(0.142) 0.438(0.011)

1.927(0.167) 2.698(0.124) 0.451(0.031)

1.988(0.061) 2.513(0.029) 0.474(0.003)

2.067(0.131) 2.613(0.101) 0.497(0.025)

λ=2 β = 2.5 α = 2.5

1.856(0.196) 2.659(0.123) 2.487(0.692)

1.920(0.090) 2.582(0.106) 2.475(0.159)

1.881(0.119) 2.629(0.104) 2.448(0.383)

1.989(0.034) 2.486(0.034) 2.540(0.036)

2.010(0.078) 2.581(0.074) 2.493(0.218)

Table 5 Average values of estimates and the corresponding MSEs (in parentheses) for n = 200. Parameters

MLEs

LSEs

WLSEs

PCEs

MPSEs

λ=1 β = 0.5 α = 0.5

0.904(0.095) 0.545(0.009) 0.506(0.216)

0.856(0.096) 0.554(0.008) 0.422(0.214)

0.921(0.053) 0.530(0.004) 0.466(0.124)

1.092(0.123) 0.475(0.015) 0.536(0.042)

0.943(0.050) 0.533(0.005) 0.457(0.079)

λ=1 β = 0.5 α = 2.5

0.931(0.069) 0.526(0.005) 2.606(0.437)

0.896(0.102) 0.535(0.008) 2.535(0.512)

0.955(0.040) 0.518(0.003) 2.551(0.185)

1.074(0.134) 0.479(0.012) 2.423(0.122)

0.962(0.053) 0.520(0.003) 2.574(0.363)

λ=1 β = 2.5 α = 0.5

0.880(0.085) 2.724(0.190) 0.430(0.184)

0.909(0.059) 2.636(0.101) 0.425(0.075)

1.040(0.031) 2.468(0.041) 0.616(0.099)

1.011(0.006) 2.458(0.024) 0.549(0.001)

0.918(0.058) 2.665(0.108) 0.431(0.188)

λ=1 β = 2.5 α = 2.5

0.975(0.028) 2.549(0.037) 2.479(0.235)

0.962(0.026) 2.552(0.052) 2.409(0.161)

0.989(0.020) 2.545(0.038) 2.583(0.112)

1.022(0.006) 2.441(0.019) 2.549(0.039)

0.957(0.054) 2.592(0.074) 2.545(0.342)

λ=2 β = 0.5 α = 0.5

1.931(0.195) 0.528(0.005) 0.570(0.034)

1.915(0.188) 0.531(0.009) 0.595(0.041)

1.852(0.162) 0.551(0.012) 0.423(0.013)

2.042(0.032) 0.462(0.010) 0.482(0.009)

1.982(0.149) 0.528(0.006) 0.534(0.011)

λ=2 β = 0.5 α = 2.5

1.855(0.229) 0.535(0.007) 2.480(0.394)

1.827(0.241) 0.534(0.008) 2.546(0.284)

1.790(0.270) 0.550(0.010) 2.465(0.674)

2.074(0.037) 0.464(0.008) 2.514(0.059)

1.907(0.208) 0.531(0.006) 2.571(0.206)

λ=2 β = 2.5 α = 0.5

1.864(0.177) 2.680(0.148) 0.481(0.031)

1.914(0.107) 2.609(0.088) 0.457(0.006)

1.928(0.083) 2.586(0.052) 0.466(0.007)

1.960(0.025) 2.487(0.013) 0.479(0.001)

2.051(0.101) 2.583(0.089) 5.021(0.019)

λ=2 β = 2.5 α = 2.5

1.938(0.097) 2.571(0.046) 2.450(0.217)

1.939(0.085) 2.555(0.061) 2.512(0.152)

1.948(0.100) 2.578(0.068) 2.544(0.171)

1.984(0.015) 2.492(0.020) 2.416(0.030)

2.010(0.053) 2.532(0.032) 2.471(0.103)

p-value). The smaller these statistics are, the better the fit is. The first data set contains 40 observations and refers to timeto-failure (103h ) of turbocharger of one type of engine. The second data set represents the survival times, in weeks, of 33 patients suffering from acute Myelogeneous Leukaemia. These data have been studied by Feigl and Zelen [44]. The PDFs of these models (for x > 0) are given by BW : f (x) =

] βα β β−1 −b(αx)β [ β a−1 x e 1 − e−(α x) , a, b, λ > 0. B (a, b)

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453

Table 6 The list of competitive models. Distribution

Abbreviation

Author(s)

Beta Weibull Marshall–Olkin Weibull Transmuted geometric Weibull Transmuted complementary Weibull geometric Transmuted exponentiated generalized Weibull Weibull

BW MOW TGW TCWG TEGW W

Lee et al. [7] Ghitany et al. [45] Nofal et al. [46] Afify et al. [47] Yousof et al. [48] Weibull [13]

Fig. 3. Fitted density and estimated survival function for data I.

β

MOW:f (x) = λβα β xβ−1 e−(α x)

β β β−1 −(α x)

TGW:f (x) = θ βα x

e

[

β

1 − (1 − λ) e−(α x)

⎧ ⎪ ⎨

1+λ−

⎪ ⎩ [

(

× 1 + (θ − 1) 1 − e

β

−(α x)

2λ θ

]−2

(

, α, β, λ > 0. β

1 − e−(α x)

(

⎫ ⎪ ⎬

) β

1 + (θ − 1) 1 − e−(α x)

)]−2

) ⎪ ⎭

, α, β, θ > 0 and |λ| ≤ 1.

[ ] β −3 σ + (1 − σ ) e−(αx) [ ] β × σ (1 − λ) − (σ − σ λ − λ − 1) e−(αx) , α, β, σ > 0 and |λ| ≤ 1. β

TCWG:f (x) = σ βα(α x)β−1 e−(α x)

TEGW:f (x) = abβ xβ−1 e−ax

[

β

(

(

β

1 − e−ax

× 1 + λ − 2λ 1 − e

−axβ

)b−1

)b ]

, β, a, b > 0 and |λ| ≤ 1.

The numerical values of goodness-of-fit measures, the MLEs and their corresponding standard errors (in parentheses) of the fitted models are listed in Tables 7 and 8, respectively. In Tables 7 and 8, we compare the fits of the ALTW model with the BW, MOW, TGW, TCWG, TEGW and W distributions. The ALTW distribution has the lowest values for all goodness-of-fit statistics among all fitted models. The fitted PDF and the estimated survival function of the ALTW distribution are displayed in Fig. 3 for data set I and Fig. 4 for data set II, respectively. The results in Tables 7 and 8 show that the ALTW distribution provides a good fit to the two data sets. Now we use the different methods of estimation considered in Section 4 to estimate the unknown parameters. The estimates of the unknown parameters using the five methods of estimation and the values of −ˆ ℓ, K–S (and the corresponding p-value) are displayed in Tables 9 and 10 for data sets I and II, respectively. From Tables 9 and 10 and based on the K-S statistics, we recommend to use the MPS method to estimate the parameters of the ALTW distribution for data set I though

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Table 7 Goodness-of-t statistics, MLEs and standard errors (SE) (in parentheses) for data set I. Model

−ℓˆ

W∗

A∗

K-S

p-value

Estimate (SE)

ALTW

78.609

0.0154

0.1221

0.0524

0.9999

BW

79.038

0.0210

0.1696

0.0878

0.9174

α β λ α β

20001.42(1059.616) 2.6541(0.4099) 0.0326(0.0287) 0.0757(0.0305) 11.2421(3.8504) 0.2401(0.1029) 115.4396(489.0293)

a b MOW

81.320

0.0496

0.3766

0.0918

0.8889

TGW

81.320

0.0496

0.3766

0.0918

0.8889

TCWG

81.320

0.0496

0.3766

0.0918

0.8889

TEGW

82.821

0.0853

0.6238

0.1104

0.7139

α β λ α β λ θ α β λ σ β

0.1886(0.0468) 2.7879(0.8736) 4.8576(5.6685) 0.1886(0.0468) 2.7879(0.8732) −6.06 · 10−5 (0.6475) 0.2059(0.2746) 0.2059(0.2747) 2.7881(0.8733) −8.97 · 10−5 (0.6475) 0.1886(0.0468) 2.8913(0.2361) 0.0054(0.0028) 1.1014(0.3513) −0.5575(0.3371)

a b W

82.583

0.0788

0.5906

0.1077

0.7423

λ β λ

3.8725(0.3250) 0.0006(0.0004)

Table 8 Goodness-of-t statistics, MLEs and standard errors (SE) (in parentheses) for data set II. Model

−ℓˆ

W∗

A∗

K-S

p-value

Estimate (SE)

ALTW

152.168

0.0783

0.5173

0.1080

0.8358

BW

153.557

0.0949

0.6451

0.1363

0.5720

α β λ α β a b

MOW

153.419

0.0940

0.6369

0.1343

0.5909

TGW

153.419

0.0940

0.6369

0.1343

0.5907

TCWG

153.553

0.0945

0.6475

0.1362

0.5729

TEGW

153.550

0.0947

0.6454

0.1361

0.5743

α β λ α β λ θ α β λ σ β a b

W

153.587

0.0948

0.6508

0.1366

0.5688

λ β λ

0.0134(0.0124) 1.2716(0.1231) 0.0031(0.0018) 0.0316(0.2047) 0.6305(1.0312) 1.4252(3.9235) 1.2674(7.8609) 0.0181(0.0132) 0.8807(0.1994) 0.5139(0.5656) 0.0181(0.0132) 0.8813(0.1998) −0.0005(0.5838) 1.9523(2.4402) 0.9990(1.0249) 0.7964(0.1876) 0.1404(0.5701) 0.0256(0.0182) 0.7047(1.0383) 0.0907(0.5361) 1.2137(2.9173) 0.1021(0.6775) 0.7764(0.1073) 0.0628(0.0298)

all the methods of estimation performed well and the LSE for data set II. The relative histogram with the PDFs for various methods are displayed in Fig. 5 for both data sets. These Figures support the results in Tables 9 and 10. 7. Concluding remarks In this paper, we proposed a new three-parameter distribution, so-called the alpha logarithmic transformed Weibull (ALTW) distribution. The proposed ALTW distribution has two shape parameters and one scale parameter. It includes as special sub-models: the exponential, the Weibull, the logarithmic transformed exponential and logarithmic transformed Weibull distributions. The ALTW density function can take various forms depending on its shape parameters. Moreover, the

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Fig. 4. Fitted density and estimated survival function for data II.

Fig. 5. The relative histogram with the fitted density of the ALTW distribution for various methods for data I (left panel) and data II (right panel).

Table 9 The parameter estimates under various methods and the goodness of fit statistics for Data I. Method

αˆ

βˆ

λˆ

−ˆ ℓ

K-S

p-value

MLE LSE WLSE PCE MPSE

20001.420 20220.975 20844.121 20941.363 20875.782

2.6541 2.3904 2.4899 2.3493 2.5496

0.0326 0.0547 0.0451 0.0601 0.0411

78.609 79.063 78.785 79.111 78.636

0.0524 0.0466 0.0462 0.0462 0.0441

0.999899 0.999994 0.999995 0.999995 0.999999

Table 10 The parameter estimates under various methods and the goodness of fit statistics for Data II. Method

αˆ

βˆ

λˆ

−ˆ ℓ

K-S

p-value

MLE LSE WLSE PCE MPSE

0.0134 0.0096 0.0039 0.0011 0.0155

1.2716 1.1132 1.3768 1.3579 1.2860

0.0031 0.0036 0.0011 0.0012 0.0029

152.168 153.621 152.137 153.751 152.160

0.1080 0.0982 0.1118 0.1404 0.1266

0.8358 0.9081 0.8035 0.5335 0.6656

ALTW distribution failure rate function can have the following four forms depending on its shape parameters: (i) increasing; (ii) decreasing (iii) bathtub and (iv) upside down bathtub shaped. Therefore, it can be used quite effectively in analyzing lifetime and survival data. The model parameters are estimated by five methods of estimation, namely, maximum likelihood,

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least squares and weighted least squares, percentile and maximum product of spacings. The simulation results show that percentile estimators (PCE) is the best performing estimator in terms of MSE. The real data applications show that the maximum product of spacing estimator for data set I and LSE for data set II give the best performing estimators for the ALTW distribution. Additionally, the new ALTW model can be used as an alternative to the generalized form of the Weibull distributions and is expected that in some situations it might work better (in terms of model fitting) than the models stated, although it cannot be always guaranteed. In this paper, the ALTW distribution shows its ability to model survival and failure data sets. We hope that the ALTW distribution attract wider sets of applications such as Survival and reliability analysis. 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