On a New Version of Weibull Model: Statistical Properties, Parameter Estimation and Applications

1  Introduction

In life testing experiments and reliability research, the Weibull (W) distribution is a lifetime distribution that is heavily favoured. It is a widespread distribution that can be used to model the product’s lifetime, including electrical insulation, ball bearings, and automotive parts. Applications in biology and medicine also regularly use it. In place of many well-known distributions including the exponential, gamma, and inverse W distributions, the W distribution is widely used. If the probability density function (PDF) of the random variable X is given by

g(x;ρ,δ)=ρδxδ−1e−ρxδ,(1)

then the random variable X is said to have a W distribution with scale parameter ρ>0 and shape parameter δ>0. The cumulative distribution function (CDF) of corresponds to (1) is expressed as follows:

G(x;ρ,δ)=1−e−ρxδ,x>0.(2)

However, because the W distribution lacks a bathtub or unimodal hazard rate function (HRF), it cannot be used to represent the lifetime of some systems. Researchers have recently discovered a number of generalizations of the traditional Weibull distribution to address this drawback, see for example the exponentiated W distribution by Mudholkar et al. [1], extended W distribution by Marshall et al. [2], Kumaraswamy W distribution by Cordeiro et al. [3], W-W distribution by Abouelmagd et al. [4], alpha power W (APW) distribution by Nassar et al. [5] and Alpha logarithmic transformed W distribution by Nassar et al. [6], odd Lomax W distribution by Cordeiro et al. [7], the log-normal modified W distribution by Shakhatreh et al. [8], generalized extended exponential-W distribution by [9] and logarithmic transformed W by Nassar et al. [10].

By combining the logarithmic transformed method proposed by Pappas et al. [11] with the alpha power transformation procedure proposed by Mahdavi et al. [12], Alotaibi et al. [13] introduced a new family of continuous distributions which called the logarithmic transformed alpha power (LTAP) family. This family introduces new two-shape parameters to any existing baseline distribution to add more flexibility to its PDF and HRF. The LTAP family of distributions’ CDF and PDF are provided, respectively, as

F(x;λ,α)={1−log⁡[λ−λ−1α−1(αG(x)−1)]log⁡(λ),λ,α>0,λ,α≠1,G(x),λ=α=1,(3)

and

f(x;λ,α)={(λ−1)log⁡(α)log⁡(λ)g(x)αG(x)[λα−1−(λ−1)αG(x)],λ,α>0,λ,α≠1,g(x),λ=α=1,(4)

where λ and α are shape parameters. Alotaibi et al. [13] investigated the main structure properties of the LTAP family and introduced the so-called LTAP exponential (LTAPEx) distribution as a new extension of the traditional exponential distribution.

In this paper, we offer the LTAPW distribution as a novel four-parameter modification of the W distribution. Numerous lifetime distributions, including the exponential, Rayleigh, W, alpha logarithmic transformed W, and APW distributions, are included in the suggested distribution as special instances. We are encouraged to present the LTAPW distribution because (1) it possesses more than ten lifetime distributions as sub-models. (2) It reveals seven hazard rate shapes, including constant, decreasing, increasing, unimodal, bathtub, unimodal then bathtub, and bathtub then unimodal shapes which causes this distribution to be outstanding when compared with other lifetime models. (3) It is displayed in Section 2 that the PDF of the LTAPW distribution can be presented as a mixture of W distribution and this property is beneficial when deriving its main properties. (4) It can be regarded as a practical model for fitting the skewed data and can also be utilized to model COVID-19, engineering, and blood cancer data sets.

The remainder of this paper is structured as follows. We introduce the LTAPW distribution and go through some of its characteristics in Section 2. The estimation of the unknown parameters is described in Section 3. Section 4 conducts a simulation study to assess the performance of the maximum likelihood estimators (MLEs). In Section 5, three actual data sets are utilized to illustrate how useful the LTAPW distribution is. Some concluding remarks are included in Section 6.

2  LTAPW Distribution

The primary structural characteristics of the LTAPW distribution are explored in this section. By substituting G(x) of the W distribution given by (2) in the CDF of the LTAP class in (3), one can write the CDF of the LTAPW distribution with scale parameter ρ and shape parameters λ,α and δ as follows:

F(x;θ)=1−log⁡[λ−λ−1α−1(α1−e−ρxδ−1)]log⁡(λ),x>0,λ,α,ρ,δ>0,(5)

where θ=(λ,α,ρ,δ)⊤ is the vector of the unknown parameters, and the associated PDF is given by

f(x;θ)=(λ−1)log⁡(α)log⁡(λ)ρδxδ−1e−ρxδ[1−λ+(λα−1)αe−ρxδ−1].(6)

The reliability function (RF) and HRF of the LTAPW distribution are given, respectively, by

R(x;θ)=log⁡[λ−λ−1α−1(α1−e−ρxδ−1)]log⁡(λ),(7)

and

h(x;θ)=(λ−1)log⁡(α)ρδxδ−1e−ρxδ[1−λ+(λα−1)βe−ρxδ−1]log⁡[λ−λ−1α−1(α1−e−ρxδ−1)].(8)

Numerous sub-models can be obtained directly from the LTAPW distribution. Some important special models are listed in Table 1. Fig. 1 depicts various plots of the PDF of the LTAPW distribution using ρ=1 consistently and by taking into account different values for the shape parameters λ,α and δ. It reveals how the newly added shape parameters provide the PDF of the LTAPW distribution with more powerful flexibility than the traditional W distribution. It can be observed from Fig. 1 that the PDF the LTAPW distribution can be right-skewed, left-skewed, or approximately symmetric. Fig. 2 displays the different shapes for the HRF of the LTAPW distribution with ρ=1. Beside the constant HRF, Fig. 2 demonstrates that the HRF of the LTAPW distribution has six different shapes including decreasing, increasing, unimodal, bathtub, unimodal then bathtub, and bathtub then unimodal shapes. One of the benefits of the LTAPW distribution over the traditional W distribution is that the latter can model only data with constant, increasing and decreasing failure rates.

Figure 1: Different plots of the LTAPW distribution’s PDF

Figure 2: Different plots of the LTAPW distribution’s HRF

Throughout the paper, we will use X∼LTAPW(θ) to refer to that the random variable X follows the LTAPW distribution with PDF given by (6). The LTAPW distribution’s quantile function, linear representation, moments, entropies, and other characteristics are derived in the following subsections.

2.1 Quantile Function

For estimation (for instance, quantile estimators) and simulation, quantiles are essential. For the LTAPW distribution, the qth quantile xq,0<q<1, can be expressed as follows:

xq=QLTAPW(q)=[−1ρlog⁡{1−[log⁡(1+λ(1−λ−q)(α−1)λ−1)log⁡α]}]1δ.(9)

By substituting q=0.5 in (9), the median of the LTAPW distribution can be acquired as follows:

M=[−1ρlog⁡{1−[log⁡(1+(1−λ0.5)(α−1)λ−1)log⁡α]}]1δ.(10)

By putting q=0.25 and 0.75 in (9), the first and third quartiles of the LTAPW distribution can be obtained, respectively. Allowing U to be uniform (0,1), Eq. (9) can be employed to generate a random sample of size n from the LTAPW distribution as follows:

xi=[−1ρlog⁡{1−[log⁡(1+λ(1−λ−ui)(α−1)λ−1)log⁡α]}]1δ,i=1,…,n.(11)

2.2 Mixture Representation

In this subsection, we offer some mixture expressions for the PDF and CDF of the LTAPW distribution. Employing the generalized binomial expansion and the power series given, respectively, by

(1−z)−r=∑k=0∞Γ(k+r)Γ(r)k!zk,−1<z<1,r>0(12)

and

αz=∑m=0∞(log⁡α)mm!zm.(13)

Applying (12) and (13) to the PDF in (6), then the following is a useful linear expression for the PDF of the LTAPW distribution:

f(x;θ)=∑s=0∞∑i=0sϖs,ig(x;(i+1)ρ,δ).(14)

where g(x;(i+1)ρ,δ) is the PDF of the W distribution with scale parameter (i+1)ρ and shape parameter δ, and

ϖs,i=∑k=0∞∑m=0k∑j=0m(−1)m+j+i(j+1)siΓ(k)(log⁡(α)s+1s(1+λ)k+1log⁡(λ)Γ(i+2)Γ(j)Γ(k−m)Γ(m−j)Γ(s−i)(λ−1α−1)m+1.

By integrating (14), the linear expression for the CDF of the LTAPW distribution given by (5), can be obtained as

F(x;θ)=∑s=0∞∑i=0sϖs,iG(x;(i+1)ρ,δ).(15)

where G(x;(i+1)ρ,δ) is the CDF of the W distribution with scale parameter (i+1)ρ and shape parameter δ.

2.3 Moments

Moments are crucial to statistics and the use of statistics. Moments can be utilized to examine a probability distribution’s tendency, variation, and symmetry, among other essential aspects. The rth moment for the LTAPW distribution can be derives from (14) as follows:

μr′=E(Xr)=∫−∞∞xrf(x)dx=∑s=0∞∑i=0sϖs,i∫0∞xrg(x;(i+1)ρ,δ)dx=∑s=0∞∑i=0sϖs,iΓ(rδ+1)[(i+1)ρ]rδ,r=1,2,3,…,(16)

where Γ(.) is the gamma function. Particularly, the first two moments of the LTAPW distribution can be obtained directly from (16) by setting r=1,2, respectively, as

μ1′=E(X)=∑s=0∞∑i=0sϖs,iΓ(1δ+1)[(i+1)ρ]1δ

and

μ2′=E(X2)=∑s=0∞∑i=0sϖs,iΓ(2δ+1)[(i+1)ρ]2δ.

Incomplete moments (IMs), moment generating functions (MGF), characteristic function (CF), and conditional moments (CMs) are four different forms of moments for the LTAPW distribution which can be expressed according to the following theorem:

Theorem. If X∼ LTAPW, then

1.   The rth IM of X is

ψr(t)=∑s=0∞∑i=0sϖs,iγ(r+1,(i+1)ρtδ)[(i+1)ρ]iδ,(17)

where γ(.,.) is the lower incomplete gamma.

2.   The MGF of X is

MX(t)=∑k=0∞tkk!E(Xk)=∑k=0∞μk′tkk!,

where μk′ is kth moment for the LTAPW distribution and given by (16).

3.   The CF of X is

ΦX(t)=∑k=0∞(it)kk!E(Xk)=∑k=0∞μk′(it)kk!,i2=−1.

4.   The rth CM of X is

ΨX(t)=E(Xr|T>t)=ψr(t)1−F(t),

where ψr(t) is the IMs and given by (17).

2.4 Mean Deviations, Bonferroni and Lorenz Curves

The mean deviations from the mean and median are applications of the IMs. They are employed to evaluate how far a population has spread from its center. The mean deviations from the mean and the median can be derived, respectively, as follows:

Δ(μ)=∫0∞|x−μ|f(x)dx

and

Δ(M)=∫0∞|x−M|f(x)dx,

where μ=μ1′. These measure can be expressed as follow, see Cordeiro et al. [14],

Δ(μ)=2μF(μ)−2ψ1(μ) and Δ(M)=μ−2ψ1(M)

where ψ1(μ) and ψ1(M) are defined from (17) as

ψ1(μ)=∑s=0∞∑i=0sϖs,iγ(2,(i+1)ρμδ)[(i+1)ρ]iδandψ1(M)=∑s=0∞∑i=0sϖs,iγ(2,(i+1)ρMδ)[(i+1)ρ]iδ.

Moreover, Bonferroni and Lorenz curves, denoted by ΛB and ΛL, respectively, are employed in economics, reliability, demography, insurance, and medicine. They can be easily acquired using the IMs follows:

ΛB(τ)=ψ1(q)τμ and ΛL(τ)=ψ1(q)μ,

where q=QLTAPW(τ) obtained from (9) and

ψ1(q)=∑s=0∞∑i=0sϖs,iγ(2,(i+1)ρqδ)[(i+1)ρ]iδ.

2.5 Entropies of LTAPW Distribution

The amount of information that can be expected from a random variable is called entropy. In many fields, including physics, finance, statistics, chemistry, insurance and biological phenomena, estimating entropy is essential. Less information in a sample is directed to retaining higher entropy. For the LTAPW distribution, the Rényi entropy (RE), denoted by Iζ(x) with ζ>0,ζ≠1, can be derived as

Iζ(x)=11−ζlog⁡{∫0∞[f(x)]ζdx}=11−ζlog⁡{(ρδ(λ−1)log⁡(α)log⁡(λ))ζ∫0∞xζ(δ−1)e−ζρxδαζ(1−e−ρxδ)[(λα−1)−(λ−1)α1−e−ρxδ]ζdx}=11−ζlog⁡{(ρδ(λ−1)log⁡(α)log⁡(λ))ζ∫0∞xζ(δ−1)e−ζρxδαζ(1−e−ρxδ)×[(λα−1)−(λ−1)α1−e−ρxδ]−ζdx}.(18)

Using series presented in (12) and (13), the RE in (18) can be written as follows:

Iζ(x)=11−ζlog⁡{(δρlog⁡(λ))ζ∑t=0∞∑m=0tϖt,m∫0∞xζ(δ−1)e−ρ(m+ξ)xδdx}=11−ζlog⁡{(δρlog⁡(λ))ζ∑t=0∞∑m=0tϖt,mΓ(ζ(δ−1)δ+1)δ[ρ(m+ζ)]ζ(δ−1)δ+1}=ζ1−ζlog⁡{δρlog⁡(λ)}+11−ζlog⁡{∑t=0∞∑m=0tϖt,mΓ(ζ(δ−1)δ+1)δ[ρ(m+ζ)]ζ(δ−1)δ+1},

where

ϖt,m=∑k=0∞(−1)mCk(tm)(j+ζ)t[log⁡(α)]n+ζt!.

and

Ck=∑i=0k∑j=0i(−1)i+j(ki)(ij)Γ(k+ζ)Γ(ζ)k!(λ−1(α−1)(λ+1))k+ζ

2.6 Order Statistics

Suppose that X(1)≤X(2)≤⋯≤X(n) indicate the order statistics (OS) acquired from a sample of size n selected from a population that follows the LTAPW distribution with CDF and PDF given by (5) and (6), respectively. Then, one can get the PDF of the ith OS X(i), as follows:

fX(i)(x)=f(x)∑k=0n−idkFk+i−1(x)(19)

where dk=(−1)kB(i,n−i+1)(n−ik) and B(.,.) is the beta function. Using (5), (6) and after some simplifications, we can write

f(x)Fk+i−1(x)=∑j=0k+i−1Tjxδ−1e−ρxδ[1+λα−11−λαe−ρxδ−1][log⁡(λ−λ−1α−1(α1−e−ρxδ−1))]j,(20)

where

Tj=ρδlog⁡(α)(−1log⁡(λ))j+1(k+i−1j)(21)

In this case, the PDF of the ith OS for the LTAPW distribution can be expressed as

fX(i)(x)=∑k=0n−i∑j=0k+i−1Tk,jxδ−1e−ρxδ[1+λα−11−λαe−ρxδ−1][log⁡(λ−λ−1α−1(α1−e−ρxδ−1))]j,(22)

Tk,j=dkTj.

On the other hand, the CDF of the ith order statistic for the LTAPW distribution corresponds to (22) is given by the following formula:

FX(i)(x)=∑k=in(nk)[F(x)]k[1−F(x)]n−k=∑k=in∑j=0k(−1)jΓ(n+1)Γ(j+1)Γ(n−k+1)Γ(k−j+1)[log⁡(λ−λ−1α−1(α1−e−ρxδ−1))log⁡(λ)]n+j−k.

2.7 Moments of Residual Life

The residual life (RL) function in reliability and life testing refers to the extra lifetime presumed that a component has stayed until time t. The rth moment of the RL of the random variable X is defined as follows:

Rr(t)=E((X−t)r|X>t)=11−F(t)∫t∞(x−t)rf(x)dx,r≥1

Based on (5), (14) and applying the binomial expansion of (x−t)n, we have

Rr(t)=11−F(t)∑k=0r(−t)r−k(rk)∫t∞xrf(x)dx=11−F(t)∑k=0r∑s=0∞∑i=0sϖs,i(−t)r−k(rk)∫t∞xrg(x;(i+1)ρ,δ)dx

where g(x;(i+1)ρ,δ) is the PDF of the W distribution with scale parameter (i+1)ρ and shape parameter δ. Then,

Rr(t)=log⁡(λ)log⁡(λ−λ−1α−1(α1−e−ρtδ−1))∑k=0r∑s=0∞∑i=0sϖs,i(−t)r−k(rk)Γ(rδ+1,(i+1)ρtδ)[(r+1)ρ]rδ,

where Γ(a,y)=∫y∞za−1e−zdz is the complementary incomplete gamma function. On the other hand, one can get the rth moment of the reversed RL by utilizing the following formula:

mr(t)=E((t−X)r|X≤t)=1F(t)∫0t(t−x)rf(x)dx,r≥1.

Using (5) and (14), it follows:

mr(t)=[1−log⁡[λ−λ−1α−1(α1−e−ρtδ−1)]log⁡(λ)]−1∑k=0r∑s=0∞∑i=0sϖs,itk(i+1)ρδ×∫0txr+δ−k−1e−(i+1)ρxδdx=[1−log⁡[λ−λ−1α−1(α1−e−ρtδ−1)]log⁡(λ)]−1∑k=0r∑s=0∞∑i=0sϖs,itkγ(r−k+δδ,(i+1)ρtδ)[(i+1)ρ]r−kδ,

where γ(a,b) is the lower incomplete gamma function.

3  Maximum Likelihood Estimation

The MLEs of the different unknown parameters and the associated approximate confidence intervals (ACIs) using the asymptotic traits of the MLEs are discussed in this section. Let x1,x2,…,xn is an observed sample of size n selected from a population that follows the LTAPW distribution with PDF given by (6). Then, we can formulate the likelihood function as follows:

L(θ)=(ρδ(λ−1)log⁡(α)log⁡(λ))n∏i=1n[xiδ−1e−ρxiδ(1−λ)+(λα−1)αe−ρxiδ−1].(23)

The log-likelihood function of (23) is expressed as follows:

ℓ(θ)=nlog⁡(ρ)+nlog⁡(δ)+nlog⁡(λ−1)+nlog⁡(log⁡(α))−nlog⁡(log⁡(λ))+(δ−1)∑i=0nlog⁡(xi)−ρ∑i=0nxiδ−∑i=0nlog⁡[(αλ−1)αe−ρxiδ−1+(1−λ)].(24)

To get the MLEs of λ,α,ρ and δ denoted by λ^,α^,ρ^ and δ^, one can maximize the log-likelihood function in (24) with regard to λ,α,ρ and δ, or equivalently by solving the next normal equations simultaneously

∂ℓ∂λ=n(λ−1)−nλlog⁡(λ)−∑i=0nαe−ρxiδ−1(αλ−1)αe−ρxiδ−1−λ+1=0,(25)

∂ℓ∂α=nαlog⁡(α)−∑i=0n(αλ−1)(e−ρxiδ−1)αe−ρxiδ−2+λαe−ρxiδ−1(αλ−1)αe−ρxiδ−1−λ+1=0,(26)

∂ℓ∂ρ=nρ−∑i=0nxiδ+∑i=0n(αλ−1)log⁡(α)xiδe−ρxiδαe−ρxiδ−1(αλ−1)αe−ρxiδ−1−λ+1=0(27)

and

∂ℓ∂δ=nδ+∑i=0nlog⁡(xi)−ρ∑i=0nxiδlog⁡(xi)+∑i=0nρ(αλ−1)log⁡(α)xiδlog⁡(xi)e−ρxiδαe−ρxiδ−1(αλ−1)αe−ρxiδ−1−λ+1=0.(28)

It is clear that λ^,α^,ρ^ and δ^ cannot be acquired in closed forms. Therefore, some numerical methods can be implemented to solve the non-linear system of equations in (25)–(28). Then ACIs of λ,α,ρ and δ are then computed using the asymptotic properties of the MLEs. It is known that θ^∼N(θ,I−1(θ)), where θ^=(λ^,α^,ρ^,δ^)⊤ and I−1(θ) is the asymptotic variance-covarianc matrix. Here, we use the approximate asymptotic variance-covariance matrix of the MLEs denoted by I0−1(θ^) because the obtaining of I0−1(θ) is very complicated. In this case, we can write I0−1(θ^) as follows:

I0−1(θ^)=[−Iλλ−Iλα−Iλρ−Iλδ−Iαλ−Iαα−Iαρ−Iαδ−Iρλ−Iρα−Iρρ−Iρδ−Iδλ−Iδα−Iδρ−Iδδ](λ,α,ρ,δ)=(λ^,α^,ρ^,δ^)−1=[v^(λ^)c^(λ^,α^)c^(λ^,ρ^)c^(λ^,δ^)c^(α^,λ^)v^(α^)c^(α^,ρ^)c^(α^,δ^)c^(ρ^,λ^)c^(ρ^,α^)v^(ρ^)c^(ρ^,δ^)c^(δ^,λ^)c^(δ^,α^)c^(δ^,ρ^)v^(δ^)],(29)

where

Iλλ=−n(λ−1)2+nλ2log2⁡(λ)+nλ2log⁡(λ)+∑i=0n(αe−ρxiδ−1)2((αλ−1)αe−ρxiδ−1−λ+1)2,

Iαα=−nα2log2⁡(α)−nα2log⁡(α)+∑i=0n((αλ−1)(e−ρxiδ−1)αe−ρxiδ−2+λαe−ρxiδ−1)2((αλ−1)αe−ρxiδ−1−λ+1)2−∑i=0n(αλ−1)(e−ρxiδ−2)(e−ρxiδ−1)αe−ρxiδ−3+2λ(e−ρxiδ−1)αe−ρxiδ−2(αλ−1)αe−ρxiδ−1−λ+1,

Iρρ=−nρ2−∑i=0n(αλ−1)log⁡(α)xi2δe−2ρxiδαe−ρxiδ(log⁡(α)+eρxiδ)−αλ+α+(αλ−1)αe−ρxiδ+∑i=0n(αλ−1)2log2⁡(α)xi2δe−2ρxiδα2e−ρxiδ−2((αλ−1)αe−ρxiδ−1−λ+1)2,

Iδδ=−nδ2−ρ∑i=0nxiδlog2⁡(xi)+∑i=0nρ2(αλ−1)2log2⁡(α)xi2δlog2⁡(xi)e−2ρxiδα2e−ρxiδ−2((αλ−1)αe−ρxiδ−1−λ+1)2−∑i=0nρ(αλ−1)log⁡(α)xiδlog2⁡(xi)e−2ρxiδαe−ρxiδ(ρxiδ(log⁡(α)+eρxiδ)−eρxiδ)−αλ+α+(αλ−1)αe−ρxiδ,

Iαλ=∑i=0n(αe−ρxiδ−1)((αλ−1)(e−ρxiδ−1)αe−ρxiδ−2+λαe−ρxiδ−1)((αλ−1)αe−ρxiδ−1−λ+1)2−∑i=0n(e−ρxiδ−1)αe−ρxiδ−1+αe−ρxiδ−1(αλ−1)αe−ρxiδ−1−λ+1,

Iρλ=∑i=0n(log⁡(α)xiδe−ρxiδαe−ρxiδ(αλ−1)αe−ρxiδ−1−λ+1−(αλ−1)log⁡(α)xiδe−ρxiδαe−ρxiδ−1(αe−ρxiδ−1)((αλ−1)αe−ρxiδ−1−λ+1)2),

Iδλ=−∑i=0nρ(αλ−1)log⁡(α)xiδlog⁡(xi)e−ρxiδαe−ρxiδ−1(αe−ρxiδ−1)((αλ−1)αe−ρxiδ−1−λ+1)2+∑i=0nρlog⁡(α)xiδlog⁡(xi)e−ρxiδαe−ρxiδ(αλ−1)αe−ρxiδ−1−λ+1,

Iρα=−∑i=0nxiδe−2ρxiδ((1−αλ)log⁡(α)−(αλ+log⁡(α)−1)eρxiδ)α(α(λ−(λ−1)α−e−ρxiδ)−1)−∑i=0n(αλ−1)log⁡(α)xiδe−2ρxiδα2e−ρxiδ−1(αλ+eρxiδ−1)(−αλ+α+(αλ−1)αe−ρxiδ)2,

Iδα=∑i=0nρxiδlog⁡(xi)e−2ρxiδ((αλ−1)log⁡(α)+(αλ+log⁡(α)−1)eρxiδ)α(α(λ−(λ−1)α−e−ρxiδ)−1)−∑i=0nρ(αλ−1)log⁡(α)xiδlog⁡(xi)e−2ρxiδα2e−ρxiδ−1(αλ+eρxiδ−1)(−αλ+α+(αλ−1)αe−ρxiδ)2,

and

Iρδ=−∑i=0n(αλ−1)log⁡(α)xiδlog⁡(xi)e−2ρxiδαe−ρxiδ(ρxiδ(log⁡(α)+eρxiδ)−eρxiδ)−αλ+α+(αλ−1)αe−ρxiδ+∑i=0nρ(αλ−1)2log2⁡(α)xi2δlog⁡(xi)e−2ρxiδα2e−ρxiδ−2((αλ−1)αe−ρxiδ−1−λ+1)2−∑i=0nxiδlog⁡(xi).