
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12624-2
10.1016/j.heliyon.2024.e36593
e36593
Research Article
A novel flexible T-X family for generating new distributions with applications to lifetime data
Salahuddin Najma a
Alamgir b
Azeem Muhammad azeemstats@uom.edu.pk
c⁎
Hussain Sundus a
Ijaz Musarrat d
a Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan
b Department of Statistics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
c Department of Statistics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
d Department of Statistics, Rawalpindi Women University, Rawalpindi, Pakistan
⁎ Corresponding author. azeemstats@uom.edu.pk
22 8 2024
15 9 2024
22 8 2024
10 17 e3659326 3 2024
19 8 2024
19 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
In recent decades, many research studies have been conducted for the development of some new modifications of different baseline distributions to cope with real-world problems. This paper proposes a novel generalization of probability distributions, called modified type- II half-logistic distribution. We have derived the new family of probability distributions using T-X family with input as a type II variant of the half-logistic distribution. For the purpose of demonstration, the Weibull distribution is considered as a sub-model. Various algebraic properties of the suggested distribution have been discussed. For efficient parameter estimation, we have used the maximum likelihood principle. Additionally, two real-world data sets from the literature have been considered to illustrate the practical usefulness and significance of the suggested model.

Keywords

T-X distribution family
Half-logistic distribution
Transmutation map approach
Moments
Maximum likelihood estimation
==== Body
pmc1 Introduction

In the past few years, some well-known distributions have been used by different researchers to develop some novel families of distributions. The addition of some extra parameters to the existing distributions is a common practice for the construction of these new distributions. The new probability distributions may account for skewness in the symmetrical models, attain different hazard rates, and produce better fits than the already existing distributions. These distributions help in fitting various types of data sets in real-world problems. In the existing literature, various generalized classes of distributions have been developed, for instance, generalized Kummer-beta distribution [1], exponentiated half-logistic distribution [2], Marshall–Olkin generalized G-family [3], exponentiated generalized-G family [4], beta-normal distribution [5], skew kurtotic-normal distribution [6], Alzaatreh et al. [7] method, Weibull-G distribution family [8], Type I half-logistic [9], additive Weibull-geometric and Weibull-generated distributions [10,11], exponentiated extended class of distributions [12], Type II half-logistic-G distribution [13], extended half-logistic distribution [14], and the odd Fréchet-G distribution family [15], among others.

Liu et al. [16] investigated the exponential stability for neural networks. For this purpose, stochastic analysis techniques were used to propose sufficient conditions. Du et al. [17] studied a generalized discrete neural network by using the Mawhin's continuation theorem. Abouelmagd et al. [18] found the secular solution around the points of triangular equilibrium in restricted thee-body problems.

Alzaatreh et al. [7] proposed the T-X transformation which generates a versatile family of distributions. This method can be used to produce a number of new flexible distributions. The Cumulative Distribution Function (CDF) for the T-X family may be derived as:(1) G(x;ζ)=∫aω{F(x;ξ)}z(t;ψ)dz,

where z(t;ψ) is the pdf of any baseline distribution of T∈(a,b) for −∞≤a≤b<+∞. Moreover, the limit ω{F(x;ξ)} in Eq. (1) is non-monotonic and differentiable arbitrary CDF function.

Another well-known generator was introduced by Shaw and Buckley [6], known as the quadratic transmutation approach, with the CDF as follows:(2) G(x;λ,ζ)=(1+λ)F(x;ξ)−λF(x;ξ)2,

where λ denotes the transmuted parameter.

Let F(x;ξ) be the CDF of any distribution defined in Eq. (2). Setting λ=1,F(x;ξ)=x in Eq. (2) will yield the uniform transmuted CDF given in Eq. (3) and its pdf is given in Eq. (4).(3) G(x)=2x−x2,x>0,

and(4) g(x)=2−2x,x>0.

By considering the pdf of transmuted uniform as baseline distribution defined in Eq. (4) and the CDF of the type-II variant of the half logistic-G distribution family proposed by Ref. [13], the CDF of the modified type-II family is given in Eq. (5) as:(5) G(x;θ,ζ)=[2F(x;ζ)θ1+F(x;ζ)θ][1+1−F(x;ζ)θ1+F(x;ζ)θ],x>0,ζ,θ>0.

The probability density function (pdf), the hazard rate function (hrf) and the reverse hazard rate (rhrf) functions are defined in Eq. (6), Eq. (7), and Eq. (8), respectively.(6) g(x;θ,ζ)=4θF(x;ζ)θ−1f(x;ζ)[1−F(x;ζ)θ](1+F(x;ζ)θ)3,x>0,ζ,θ>0,

(7) h(x;θ,ζ)=4θf(x;ζ)(F(x,ζ)θ(1−F(x,ζ)θ)(1+F(x,ζ)θ)F(x,ζ),x>0,ζ,θ>0,

(8) h′(x;θ,ζ)=θf(x;ζ)[1−F(x;ζ)θ](1+F(x,ζ)θ)F(x,ζ),x>0,ζ,θ>0,

where θ and ζ denote the shape and scale parameters, respectively.

The main motivation behind the development of the new family introduced with the CDF in Eq. (5) is that by the addition of only a single shape parameter can generalize the existing distributions. The method improves the existing distributions flexibility, and characteristics, and provides better fit than other distributions. Also, it can also model different monotonic, non-monotonic, bathtub, upside bathtub hazard rate shapes.

This manuscript is organized in the following order: In Section 2, the new suggested family of distribution has been presented. Section 3 presents the analytical properties of the new distribution. Parametric estimation has been discussed in Section 4 and the applications and simulation studies have been presented in Section 5. The last section, Section 6, provides the conclusion of the study.

2 Modified type II half logistic Weibull (MTIIHLW) family

A particular form of the modified type II half-logistic distribution family can be considered using Weibull distribution. By definition, the CDF of the Weibull distribution is presented in Eq. (9).(9) F(x;λ,β)=1−e−λxβ,

where λ denotes the scale parameter, with β denoting the shape parameter.

The Weibull distribution has numerous real-world applications in reliability theory and survival analysis. For instance, Costa et al. [19] used Weibull distribution for dielectric breakdown in oxides of electronic devices. Picoli et al. [20] applied Weibull distribution to the data in a basketball championship. Muraleedharan et al. [21] used the Weibull model for significant and maximum wave heights. Sarkar et al. [22] used Weibull for modelling wind speed data in India. Ishaq and Abiodun [23] used the Weibull model for Currency exchange data. Cordeiro et al. [24] used Weibull distribution for modeling the failure of the windshield equipment used in aircrafts.

The main weakness of the Weibull distribution is that it cannot model non-monotonic hazard rates, especially, the bathtub shaped hazard rate. Hence, in order to obtain a flexible Weibull distribution covering every type of hazard rate, a number of new modifications of the Weibull model have been introduced by different researchers. For example, the flexible Weibull distributions [[25], [26], [27], [28]], the Lindley Weibull distribution [29], and the Gompertz Flexible Weibull distribution [30].

Using the generator suggested in Eq. (5), the pdf, CDF, reliability and hazard functions have been presented in Eq. (10) to Eq. (13) as:(10) G(x;θ,λ,β)=[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ],

(11) g(x;λ,β,θ)=4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ),

(12) R(x;λ,β,θ)=1−[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ],

and(13) h(x;λ,β,θ)=4θλβxβ−1(1−e−λxβ)θ(1−(1−e−λxβ)θ)(1+(1−e−λxβ)θ)(eλxβ−1).

Fig. 1, Fig. 2 present the graphical form of the MTIIHLW model for a variety of parameter values. Fig. 1 displays various shapes of the MTIIHLW model. As the shape parameter θ value increases, the MTIIHLW approaches to a rightly skewed and reversed j-shaped distribution. The reliability function has been presented in Eq. (12). Likewise, Fig. 2 shows different monotonic and non-monotonic behaviors of the proposed model for different parameter values, thus accommodating various real-life phenomena with increasing, decreasing, upside bathtub and constant hazard functions.Fig. 1 pdf of MTIIHLW distribution.

Fig. 1

Fig. 2 Hazard function of MTIIHLW distribution.

Fig. 2

3 Mathematical properties of MTIIHLW model

In this section, various important statistical properties of the MTIIHLW model have been derived.

3.1 Moments, moment generating function, and weighted moments

By definition, the moments of the TIIHLW distribution is derived as follows:(14) E(xr)=∫0∞xrg(x;λ,β,θ)dx.

Using Eq. (11) in Eq. (14) and simplifying, we get:E(xr)=∫0∞xr4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)dx,

orE(xr)=4θλβ∫0∞xr+β−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)dx,

orE(xr)=4θλβ∫0∞xr+β−1[1−(1−e−λxβ)θ]((1−e−λxβ)θ+1)−3(1−e−λxβ)θ−1dx.

Now simplifying the above expression and using the general binomial expansion formula as follows:(1+y)−α=∑j=0∞(−α+j−1j)(−1)jyj,

or((1−e−λxβ)θ+1)−3=∑j=0∞(−3+j−1j)(−1)j(1−e−λxβ)θj,

orE(xr)=4θλβ∫0∞xr+β−1[1−(1−e−λxβ)θ](1−e−λxβ)θ−1∑j=0∞(−3+j−1j)(−1)j(1−e−λxβ)θjdx,

orE(xr)=4θλβ∫0∞xr+β−1[1−(1−e−λxβ)θ]∑j=0∞(−3+j−1j)(−1)j(1−e−λxβ)θj+θ−1dx,

orE(xr)=4θλβ[∫0∞xr+β−1∑j=0∞(−3+j−1j)(−1)j(1−e−λxβ)θj+θ−1dx−∫0∞xr+β−1∑j=0∞(−3+j−1j)(−1)j(1−e−λxβ)θj+2θ−1dx].

Since(1−e−λxβ)θj+θ−1=∑i=1∞(θ+θj+ii)(−1)ie−iλxβ,

soE(xr)=4θλβ[∫0∞xr+β−1∑j=0∞(−3+j−1j)(−1)j∑i=1∞(θ+θj+ii)(−1)ie−iλxβdx−∫0∞xr+β−1∑j=0∞(−3+j−1j)(−1)j∑i=1∞(2θ+θj+ii)(−1)ie−iλxβdx].

Using the transformation z =xβ,z1β=x,dzβ=dx , we haveE(xr)=4θλβ[∑j=0∞∑i=0∞(−3+j−1j)(θ+θj+ii)(−1)j+i∫0∞zr+β−1βe−iλzdzβ−∑j=0∞∑i=0∞(−3+j−1j)(2θ+θj+ii)(−1)j+i∫0∞zr+β−1βe−iλzdzβ],

or(15) μr=4θλΓ(rβ−1β+2)λrβ−1β+2[∑j=0∞∑i=0∞(−3+j−1j)(θ+θj+ii)(−1)j+i−∑j=0∞∑i=0∞(−3+j−1j)(2θ+θj+ii)(−1)j+i].

We can obtain the moments simply by putting r = 1, 2, …., in Eq. (15).

The MGF of the MTIIHLW can be derived as:Mx(t)=∑l=0∞tll!E(X)r.

Using Eq. (15) in Mx(t), we get Eq. (16) as:(16) Mx(t)=θλ∑l=0∞tll!Γ(rβ−1β+2)λrβ−1β+2[∑j=0∞∑i=0∞(−3+j−1j)(θ+θj+ii)(−1)j+i−∑j=0∞∑i=0∞(−3+j−1j)(2θ+θj+ii)(−1)j+i].

Now the weighted moment of MTIIHLW by definition is given as:φr,m=E(XrG(x,ζ)m)=∫−∞∞xrg(x,ζ)(G(x,ζ))mdx,

org(x,λ,β,θ)(G(x,λ,β,θ))m=4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)[[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]]m,

org(x,λ,β,θ)(G(x,λ,β,θ))m=4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)[2(1−e−λxβ)θ1+(1−e−λxβ)θ]m[1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]m,

org(x,ζ)(G(x,ζ))m=4θλβxβ−1[1−(1−e−λxβ)θ]((1−e−λxβ)θ+1)−3(1−e−λxβ)θ−12m(1−e−λxβ)mθ[1+(1−e−λxβ)θ]m[1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]m,

or[1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]m=∑k=0∞(mk)(−1)m[1−(1−e−λxβ)θ1+(1−e−λxβ)θ].

Further simplification leads to:g(x,λ,β,θ)(G(x,λ,β,θ))m=4θλβxβ−1[1−(1−e−λxβ)θ]((1−e−λxβ)θ+1)−3(1−e−λxβ)θ−12m(1−e−λxβ)mθ[1+(1−e−λxβ)θ]m∑k=0∞(mk)(−1)m[1−(1−e−λxβ)θ]k[1+(1−e−λxβ)θ]−k,

org(x,λ,β,θ)(G(x,λ,β,θ))m=2m+2θλβxβ−1∑k=0∞(mk)(−1)m[1−(1−e−λxβ)θ]k+1((1−e−λxβ)θ+1)−3−m−k(1−e−λxβ)mθ+θ−1.

Using the following general binomial expansion in the above expression gives:[1−(1−e−λxβ)θ]k+1=∑j=0∞(k+1j)(−1)j(1−e−λxβ)θj,

or[1+(1−e−λxβ)θ]−3−m−k=∑j=0∞(−3−m−kj)(−1)j(1−e−λxβ)θj.

Further simplification gives:g(x,λ,β,θ)(G(x,λ,β,θ))m=2m+2θλβxβ−1∑k,j=0∞(mk)(k+1j)(−3−m−kj)(−1)m+2j(1−e−λxβ)2θj+mθ+θ−1,

org(x,ζ)(G(x,ζ))m=∑k,j=0∞η(1−e−λxβ)2θj+mθ+θ−1.

Substituting the result in φr,m, we have:φr,m=E(XrG(x,λ,β,θ)m)=∫0∞xr∑k,j=0∞η(1−e−λxβ)2θj+mθ+θ−1dx,

orφr,m=E(XrG(x,λ,β,θ)m)=∑k,j=0∞η∫0∞xr(1−e−λxβ)2θj+mθ+θ−1dx.

Since(1−e−λxβ)2θj+mθ+θ−1=∑i=1∞(2θ+mθ+θ−1i)(−1)ie−iλxβ,

soφr,m=E(XrG(x,λ,β,θ)m)=∑k,j,i=0∞η(2θ+mθ+θ−1i)(−1)i∫0∞xre−iλxβdx,

orφr,m=E(XrG(x,λ,β,θ)m)=∑k,j,i=0∞η(2θ+mθ+θ−1i)(−1)i∫0∞x(r+1)−1e−iλxβdx.

Let =xβ,z1β=x,z1β−1dzβ=dx , thus we have:φr,m=E(XrG(x,λ,β,θ)m)=∑k,j,i=0∞η(2θ+mθ+θ−1i)(−1)i∫0∞z(rβ+1β)−1e−iλzdzβ,

orφr,m=E(XrG(x,λ,β,θ)m)=1β∑k,j,i=0∞η(2θ+mθ+θ−1i)(−1)i∫0∞z(rβ+1β)−1e−iλzdz.

Solving the integral, the weighted moments have been presented in Eq. (17) as:(17) φr,m=E(XrG(x,λ,β,θ)m)=∑k,j,i=0∞η(2θ+mθ+θ−1i)(−1)iΓ(rβ+1β)βλ(rβ+1β).

3.2 Quantile function

Using the suggested MTIIHLW distribution, we can use Q(u)=F−1(u) to get:(18) Q=log[11−−q−21−q+2qθ]1βλ.

The quantile function can be used for generation of random samples from the suggested distribution. For median, we can use q=12 in Eq. (18), and for obtaining quartiles, we can use q=14,34.

3.3 Skewness and Kurtosis

The skewness and kurtosis of a distribution are important measures in statistical analysis. For this reason, Bowley's co-efficient skewness and Moor's Kurtosis measure have been presented in Eq. (19) and Eq. (20) as:(19) SGalton=Q(34)+Q(14)−2Q(24)Q(34)−Q(14),

and(20) KMoors=Q(78)+Q(38)−Q(58)−Q(18)Q(34)−Q(14).

Using the above formulas, the empirical values for the skewness and kurtosis measures under the suggested MTIIHLW distribution are presented in Table 1 for various choices of parameter values.Table 1 Skewness and Kurtosis measures.

Table 1λ	β	θ	Skewness	Kurtosis	
1	0.5	2	0.4142582	1.64514	
0.5	0.5	1	0.6238205	2.268221	
2	2	5	0.04747	1.2567	
2	3	5	0.02262	1.2541	
2	3	5.5	0.02332	1.2547	
2	3	6	0.02401	1.2552	
2	3	7	0.02531	1.2560	

Table 1 indicates that for fixed values of λ=2,β=3, the Skewness and Kurtosis measures are increasing functions of θ. Likewise, for λ=2,θ=5, and β=2,θ=1, the skewness and Kurtosis can be observed as decreasing functions of the parameters β and λ, respectively.

3.4 Entropy measure

In the reliability analysis, the entropy measure plays a pivotal role. It quantifies the amount of variation as well as the extent of uncertainty in a particular data set. A small value of entropy shows a low level of uncertainty in a data set. We have computed the level of uncertainty in a random variable X following the suggested MTIIHLW distribution, using the Renyi [31] and Havrda & Charvat [32] entropies. The entropies can be derived as given in Eq. (21) and Eq. (22):(21) REx=1(ρ−1)log[(αβλ)ρ(1−exp(−α))ρ∑n=0∞(−1)n(ρ(β−1)k)1λ(ρ+n)],

and(22) Hx(q)=1q−1{1−(αβλ)q(1−exp(−α))q∑n=0∞(−1)n(q(β−1)k)1λ(q+n)}.

Proof The Renyi and q-entropy can be derived as:REx(ρ)=1ρ−1log{∫−∞∞gρ(x)},Hx(q)=1q−1{1−∫−∞∞gq(x)}.

The Renyi entropy using Eq. (10), for ρ>1 , is as follows:REx(ρ)=1ρ−1log{∫0∞[4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)]dxρ},

orREx(ρ)=1ρ−1log{∫0∞(4θλβ)ρxρ(β−1)[1−(1−e−λxβ)θ]ρ((1−e−λxβ)θ+1)−3ρ(1−e−λxβ)ρ(θ−1)}.

Using the following identities:[1−(1−e−λxβ)θ]ρ=∑j=0∞(ρj)(−1)j(1−e−λxβ)θj,

and(1+(1−e−λxβ)θ)−3ρ=∑j=0∞(−3ρj)(−1)j(1−e−λxβ)θj,

we get:REx(ρ)=1ρ−1log[(4θλβ)ρ∑j=0∞(ρj)(−3ρj)(−1)2j∫0∞xρ(β−1)(1−e−λxβ)2θj+ρ(θ−1)dx].

Using (1−e−λxβ)2θ+ρ(θ−1)=∑i=1∞(2θ+ρ(θ−1)i)(−1)ie−iλxβ to have:REx(ρ)=1ρ−1log[(4θλβ)ρ∑j=0∞∑i=0∞(ρj)(−3ρj)(2θ+ρ(θ−1)i)(−1)2j+i∫0∞xρ(β−1)e−iλxβdx],

orREx(ρ)=1ρ−1log[(4θλβ)ρ∑j=0∞∑i=0∞(ρj)(−3ρj)(2θ+ρ(θ−1)i)(−1)2j+i∫0∞zρ(β−1)+1β−1e−iλzdzβ].

On solving the above expression, the Renyi entropy becomes:REx(ρ)=1ρ−1log[(4θλβ)ρβ∑j=0∞∑i=0∞(ρj)(−3ρj)(2θ+ρ(θ−1)i)(−1)2j+iΓ(ρ(β−1)+1β)(λi)(ρ(β−1)+1β)].

The q-entropy or β-entropy can be obtained as:Hx(q)=1q−1{1−∫0∞fq(x)dx}.

Consider the integral:∫0∞fq(x)dx=∫0∞[4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ)]qdx,

or∫0∞fq(x)dx=∫0∞(4θλβ)qxq(β−1)[1−(1−e−λxβ)θ]q(1+(1−e−λxβ)θ)−3q(1−e−λxβ)q(θ−1)dx.

since[1−(1−e−λxβ)θ]q=∑j=0∞(qj)(−1)j(1−e−λxβ)θj,

and(1+(1−e−λxβ)θ)−3q=∑j=0∞(−3qj)(−1)j(1−e−λxβ)θj,

so∫0∞fq(x)dx=∫0∞(4θλβ)qxq(β−1)∑j=0∞∑j=0∞(−3qj)(qj)(−1)2j(1−e−λxβ)2θj+q(θ−1)dx,

or∫0∞fq(x)dx=(4θλβ)q∑j=0∞∑j=0∞∑i=0∞(2θ+q(θ−1)i)(−3qj)(qj)(−1)2j+i∫0∞zq(β−1)+1β−1e−iλzdzβdx,

or∫0∞fq(x)dx=(4θλβ)qβ∑j=0∞∑j=0∞∑i=0∞(2θ+q(θ−1)i)(−3qj)(qj)(−1)2j+iΓ(q(β−1)+1β)(λi)(q(β−1)+1β).

Using binomial expansion, the q-entropy takes the form:Hx(q)=1q−1{1−(4θλβ)qβ∑j=0∞∑j=0∞∑i=0∞(2θ+q(θ−1)i)(−3qj)(qj)(−1)2j+iΓ(q(β−1)+1β)(λi)(q(β−1)+1β)}.

3.5 Mean residual life function

For any component whose additional expected lifetime has to be approximated that it has survived more than time t (fixed), the function of mean residual life i.e., μ(t), is used. The μ(t) function is presented in Eq. (23) and Eq. (24) as:(23) μ(t)=1P(X>t)∫t∞P(X>x)dx,t≥0,

or(24) μ(t)=1S(t)(E(t)−∫0txf(x)dx)−t,t≥0.

Taking∫0txf(x)dx=∫0t4θλβxβ[1−(1−e−λxβ)θ]((1−e−λxβ)θ+1)−3(1−e−λxβ)θ−1dx,

or∫0txf(x)dx=4θλβ[∫0txβ∑j=0∞(−3+j−1j)(−1)j∑i=1∞(θ+θj+ii)(−1)ie−iλxβdx−∫0txβ∑j=0∞(−3+j−1j)(−1)j∑i=1∞(2θ+θj+ii)(−1)ie−iλxβdx],

∫0txf(x)dx=4θ(Γ(β+1β,tβλ)−Γ(β+1β))λ−1β∑j=0∞(−3+j−1j)(−1)j[∑i=0∞(2θ+θj+ii)(−1)i−∑i=0∞(θ+θj+ii)(−1)i],

and(25) S(t)=1−[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ].

Using Eq. (25) in the expression for μ(t), we get MRF as presented in Eq. (26).(26) μ(t)=4θ∑j=0∞(−3+j−1j)(−1)j[∑i=0∞(2θ+θj+ii)(−1)i−∑i=0∞(θ+θj+ii)(−1)i]1−[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ](1λ−(Γ(β+1β,tβλ)−Γ(β+1β))λ−1β)−t

3.6 Order statistics

Let a random sample x_ of k units be taken from a population having the MTIIHLW distribution. Let X1:k<X2:k<X3:k<...<Xk:k denote the order statistics. The distribution of the rth order statistics can be obtained as:gr:k(x)=k!(r−1)!(k−r)!Gr−1(x)[1−G(x)]k−rg(x),

or(27) gr:k(x)=k!(r−1)!(k−r)!∑m=0k−r(−1)m(k−rm)Gm+r−1(x)g(x).

Using Eq. (10) and Eq. (11) in Eq. (27), we havegr:k(x)=k!(r−1)!(k−r)!∑m=0k−r(−1)m(k−rm)[[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]]m+r−14θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ).

Consequently, the pdf of the rth order statistics is presented in Eq. (28) as:(28) gr:k(x)=k!4θλβxβ−1(r−1)!(k−r)!∑m=0k−r(−1)m(k−rm)[[2(1−e−λxβ)θ1+(1−e−λxβ)θ][1+1−(1−e−λxβ)θ1+(1−e−λxβ)θ]]r+m−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ).

4 Parameter estimation methods

This section presents the various estimation methods to estimate the parameters of our proposed MTIIHLW distribution.

4.1 Maximum likelihood estimation (MLE)

Let us consider a sample X=(x1,x2,...xn) taken from the MTIIHLW (α,β,θ) distribution. The likelihood function can be expressed as:(29) l(x;α,β,λ)=∏i=1n4θλβxβ−1[1−(1−e−λxβ)θ](1−e−λxβ)θ((1−e−λxβ)θ+1)3(1−e−λxβ).

For the log likelihood function, taking logarithm on both sides of Eq. (29) to have:(30) logl(x;λ,β,θ)=∑i=1nlog(4)+∑i=1nlog(θ)+∑i=1nlog(β)+∑i=1nlog(λ)+(β−1)∑i=1nlog(x)+∑i=1nlog[1−(1−e−λxβ)θ]+(θ−1)∑i=1nlog(1−e−λxβ)−3∑i=1nlog[1+(1−e−λxβ)θ]

For MLE's of the parameters, differentiating Eq. (30) with respect to λ,β,θ, and using logl(x;λ,β,θ)∂λ=0 , logl(x;λ,β,θ)∂β=0 , logl(x;λ,β,θ)∂θ=0. We get Eq. (31) and Eq. (32) as follows:(31) ∂logl(x;λ,β,θ)∂λ=nλ+(θ−1)∑i=1nxβe−λxβeλxβ−1+∑i=1nθxβ(1−e−λxβ)θe−λxβ((1−e−λxβ)θ−1)(1−e−λxβ)θ−3∑i=1nθxβ(1−e−λxβ)θe−λxβ((1−e−λxβ)θ+1)(1−e−λxβ)θ=0,

and(32) ∂logl(x;λ,β,θ)∂β=nβ−∑i=1nlog(x)−∑i=1nθxβlog(x)λ(1−e−λxβ)θ((1−e−λxβ)θ−1)(1−eλxβ)θ+(θ−1)∑i=1∞xβlog(x)λ(1−e−λxβ)−∑i=1∞θxβlog(x)λ(1−e−λxβ)θ((1−e−λxβ)θ+1)(1−eλxβ)θ=0.

Further simplification yields:∂logl(x;λ,β,θ)∂θ=nθ−∑i=1n(1−e−λxβ)θln(1−e−λxβ)[1−(1−e−λxβ)θ]+∑i=1nlog(1−e−λxβ)−3∑i=1n(1−e−λxβ)θln(1−e−λxβ)[1+(1−e−λxβ)θ],

or∂logl(x;α,β,λ)∂λ2=−nλ2−(θ−1)∑i=1nx2βeλxβ(eλxβ−1)2−2∑i=1nθx2β(1−e−λxβ)θ((1−e−λxβ)θeλxβ+θ)((1−e−λxβ)θ−1)2(eλxβ−1)2,

or∂log2l(x;λ,β,θ)∂β2=−nβ2−2∑i=1nθxβlog2(x)λ(1−e−λxβ)θ(((λxβ−1)(1−e−λxβ)θ−λxβ+1)eλxβ+(1−e−λxβ)θ+θxβλ−1)((1−e−λxβ)θ−1)2(eλxβ−1)2,

or∂log2l(x;λ,β,θ)∂θ2=−nθ2−∑i=1n(1−e−λxβ)θln2(1−e−λxβ)[(1−e−λxβ)θ−1]2−3∑i=1n(1−e−λxβ)θln2(1−e−λxβ)[(1−e−λxβ)θ+1]2,

or∂log2l(x;λ,β,θ)∂λ∂β=−∑i=1n(θ−1)xβlog(x)((xβλ−1)eλxβ+1)(eλxβ−1)2+2∑i=1nθxβlog(x)λ(1−e−λxβ)θ(((λxβ−1)(1−e−λxβ)θ−λxβ+1)eλxβ+(1−e−λxβ)θ+θxβλ−1)((1−e−λxβ)θ−1)2(eλxβ−1)2,

or∂log2l(x;λ,β,θ)∂λ∂θ=∑i=1nxβeλxβ−1+2∑i=1nxβ(1−e−λxβ)θ(log(1−e−λxβ)θ−(1−e−λxβ)θ+1)((1−e−λxβ)θ−1)2(eλxβ−1),

or(33) ∂log2l(x;λ,β,θ)∂β∂θ=2∑i=1nxβlog(x)λ(1−e−λxβ)θ(log(1−e−λxβ)θ−(1−e−λxβ)θ+1)((1−e−λxβ)θ−1)2(eλxβ−1)+∑i=1∞xβlog(x)λ(eλxβ−1).

Using Eq. (33), large sample confidence intervals can be obtained for the parameters λ,β,θ assuming that the MLE's are approximately normally distributed with mean (λ,β,θ). The inverse Fisher information matrix (FI−1) is presented in Eq. (34) as:(34) FI−1=−[∂2logl∂θ2∂2logl∂θ∂β∂2logl∂θ∂λ∂2logl∂θ∂β∂logl∂β2∂2logl∂λ∂β∂2logl∂θ∂λ∂2logl∂λ∂β∂2logl∂λ2]−1,

such that(35) FI=[var(θˆ)cov(θˆ,βˆ)cov(θˆ,λˆ)cov(θˆ,βˆ)var(βˆ)cov(λˆ,βˆ)cov(θˆ,λˆ)cov(λˆ,βˆ)var(λˆ)].

Using Eq. (35), we can obtain the asymptotic (1−ζ)100% confidence intervals for the parameters as:θˆ±Zζ/2var(θˆ),λˆ±Zζ/2var(λˆ),βˆ±Zζ/2var(βˆ),

where Zζ2 denotes the (ζ2)th percentile of the standardized normal distribution.

5 Applications of the MTIIHLW distribution

This section presents the simulations for parameters and applications of MTIIHLW model using real life data. The sub model MTIIHLW of the advised family of distributions is compared with other models including Kumaraswamy Weibull (KW), Alpha Power Weibull (APW), Weibull Exponential (WE), Marshal-Olkin Exponentiated Weibull (MOEW) and Weibull (W) distributions.

5.1 Simulation study

To show the large-sample pattern of the maximum likelihood estimates using the suggested MTIIHLW distribution, Monte Carlo simulations have been performed. Means, biases and mean-squared errors (MSEs) of the MLE's for different sample sizes have been calculated. For obtaining the results, the procedure is repeated N = 10,000 times for n = 50, 300, and 1000. The obtained results of means, biases and MSEs from simulation are presented in Table-2. The general bias and MSE for an estimator are given in Eq. (36) and Eq. (37) as:(36) MSE(θˆ)=∑i=1N(θˆ−θ)2N,

and(37) Bias(θˆ)=∑i=1N(θˆi−θ)N.

Table 2 Average values, biases, and mean squared (MSEs) of MLE's from simulation of MTIIHLW.

Table 2Parameters	n	Mean λˆ	Mean βˆ	Mean θˆ	MSE λˆ	MSE βˆ	Mean θˆ	Bias λˆ	Bias βˆ	Biasθˆ	
λ=2,
β=1.5,
θ=2	50	2.2807	2.6935	2.8065	1.2674	9.2858	10.4965	0.2808	1.1935	0.8065	
300	2.0665	1.5248	2.2460	0.09336	0.12705	0.85067	0.06659	0.02482	0.24606	
1000	2.0038	1.5247	2.0292	0.02965	0.04093	0.18302	0.00372	0.02478	0.029225	
λ=2,
β=2,
θ=3	50	1.9931	3.4779	4.0383	0.7387	13.0558	21.9416	−0.0069	1.4779	1.0383	
300	1.9919	2.1671	3.18653	0.13091	0.4414	2.0080	−0.0081	0.16713	0.1865	
1000	2.03631	2.00332	3.1470	0.03786	0.06136	0.51324	0.03631	0.00333	0.14702	
λ=3,
β=4,
θ=5	50	3.0940	5.4970	6.0349	0.46318	11.9436	25.8293	0.09401	1.4970	1.0349	
300	3.0993	4.0590	5.9959	0.14016	1.1778	8.2993	0.09932	0.059027	0.9959	
1000	3.0407	4.0073	5.3445	0.05454	0.30052	2.3846	0.04074	0.007388	0.3445	
λ=1,
β=2.5,
θ=6	50	1.1864	2.7017	9.1027	0.3974	1.6221	46.6300	0.18646	0.20176	3.1027	
300	1.0350	2.6099	6.7096	0.11137	0.53311	7.9910	0.03495	0.10997	0.7096	
1000	1.07153	2.4293	6.7880	0.05294	0.09267	4.16713	0.07153	−0.07069	0.78803	

From Table 2, it can be noted that the maximum likelihood method outperforms the other estimation methods under the MTIIHLW distribution. Likewise, it is clear that if the sample size is increased, the mean and bias of the ML estimators decrease as expected. The simulation study reveals that the MLE's under the MTIIHLW approach to the true value as n increases i.e., n ≥ 300.

5.2 Real life data applications

We use two real life data sets for analyzing the performance of the suggested model. For this purpose, the commonly used measures are Cramer-von mises (denoted by W), Bayesian information criteria (BIC), the Anderson's darling (A), Hannan and quin information criteria (HQIC), Akaike information criteria (AIC), and the Consistent AIC (abbreviated as CAIC), etc. For analytical form, we can express the different measures in Eq. (38) to Eq. (43) as:(38) A=−n−1n∑i=1n(2i−1)[logF(Xi)+log(1−F(Xn−i+1))],

(39) W=∑i=1n[F(Xi)−2i−12n]+112n,

(40) AICc=AIC+2p(p+1)n−p−1,

(41) CAIC=−2L+P{log(n)+1},

(42) BIC=Plog(n)−2l,

and(43) HQIC=−2L+2Plog{log(n)}.

Generally, a model with smallest values of the above measures is considered the best model.

Data Set 1: Airborne communication transceiver repairs times in hours.

The data set considered here was originally used by Alven [33] which represents the airborne communication transceiver repairs times in hours. The data set is as follows:

0.2,0.3,0.5,0.5,0.5,0.6,0.6,0.7,0.7,0.7,0.8,0.8,1,1,1,1,1.1,1.3,1.5,1.5,1.5,1.5,2.0,2.0,2.2,2.5,2.7,3,3,3.3,3.3,4.0,4.0,4.5,4.7,5.0,5.4,5.4,7.0,7.5,8.8,9.0,10.3,22,24.5.

Data Set 2: Remission time (measured in months) for 128 patients.

The 2nd data was considered from Lee and Wang [34], which represents the patients of bladder cancer, whose remission times are measured in months. The data is as follows:

0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

6 Discussion and conclusion

The usual two-parameter Weibull distribution suffers from a serious issue which limits its implementation in real-world problems. The disadvantage is that it cannot model non-monotonic hazard rates, especially, the bathtub shaped hazard rate. To address this issue, we introduced a novel probability distribution using the Alzaatreh T-X transformation. For the derivation of the suggested distribution, the Weibull probability distribution was used as a baseline-model. The new distribution, named as Modified Type-II half logistic Weibull distribution (MTIIHLW), was observed to perform better the existing distributions. Various mathematical properties, for instance, moments, mean residual life function, entropy measures, quantile function, and order statistics under the suggested distribution were theoretically derived.

Fig. 3 displays the TT-Transform plot for the airborne data, showing non-monotonic (bath-tub hazard) shape. Table 3, Table 4 show the goodness-of-fit values, Maximum likelihood estimates, K-S values and the p-values using the Airborne data. It has been observed that the values of Hannan and quin information criteria (HQIC), AIC, CAIC, BIC and HQIC for MTIIHLW are smaller with a highest p-value showing the superiority of the proposed model in comparison with the other considered variants of Weibull distribution. It can also be seen clearly in Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8 that MTIIHLW model presents the best fit among all models considered for comparison.Fig. 3 TTT plot of airborne communication transceiver repairs times.

Fig. 3

Table 3 Results of various measures using data set 1.

Table 3Model	W	A	AIC	CAIC	BIC	HQIC	
MTIIHLW	0.050919	0.31200	204.3845	204.9699	209.8045	206.405	
KW	0.04929	0.31874	206.3675	207.3675	213.5942	209.0615	
APW	0.12232	0.82959	212.4943	213.0797	217.9143	214.5148	
WE	0.14947	1.06819	215.9137	216.4991	221.3337	217.9343	
MOEW	0.06594	0.44774	206.3904	206.9758	211.8104	208.411	
W	0.11984	0.83876	210.292	210.5778	213.9054	211.6391	

Table 4 Maximum likelihood estimates and K-S values using data set 1.

Table 4Model	MLE's	log (likelihood)	K-S Value	P-Value	
MTIIHLW	1.87293, 0.25451,15.23104	99.19225	0.08482	0.9075	
KW	1.95012, 6.9556, 0.7463,0.4485	99.18375	0.0903	0.8558	
APW	12.43277, 0.83655, 0.64814	103.2472	0.11617	0.5782	
WE	0.04014, 0.776891, 4.32787	103.146	0.14446	0.3047	
MOEW	0.037837, 1.35542, 0.117721	100.1952	0.12361	0.4975	
W	0.32555, 0.9048	103.146	0.11516	0.5894	

Fig. 4 Comparison of the theoretical densities using the data set 1.

Fig. 4

Fig. 5 Comparison of the CDFs using the data set 1.

Fig. 5

Fig. 6 Theoretical density plot of MTIIHLW model for data set 1.

Fig. 6

Fig. 7 PP plot of MTIIHLW model for data set 1.

Fig. 7

Fig. 8 TTT plot for bladder cancer patients.

Fig. 8

Fig. 6 shows the TT-Transform plot for bladder cancer patients, which has a non-monotonic pattern for the hazard rate function. Table 5, Table 6 display the different measures, including AIC, BIC, CAIC, HQIC, MLE's, K-S values and the p-value. The values for MTIIHLW model are smaller with a large P-value, indicating a better fit of the proposed model in comparison with the Weibull sub-models being considered in comparison. Fig. 9, Fig. 10, Fig. 11, Fig. 12 illustrate the values of CDF and pdf of our proposed MTIIHLW model and the available models. The figures display the best fit for the MTIIHLW as discussed numerically with respect to the goodness fit.Table 5 Values of different measures using data set 2.

Table 5Models	W	A	AIC	CAIC	BIC	HQIC	
MTIIHLW	0.03113	0.2100	826.4836	826.6771	835.0397	827.96	
KW	0.04064	0.2702	829.1969	829.5221	840.605	833.8321	
APW	0.12489	0.7394	832.9942	833.1877	841.5503	836.4706	
WE	0.25329	1.5114	849.5531	849.7467	858.1092	853.0295	
MOEW	0.16731	0.98004	836.0764	836.2699	844.6325	839.5528	
W	0.13082	0.783235	832.1747	832.2707	837.8788	834.4923	

Table 6 MLE's, K-S values for data set 2.

Table 6Model	MLE's	-log (likelihood)	K-S Value	P-Value	
MTIIHLW	0.2288,0.6357, -2.6569	410.2418	0.04024	0.9783	
KW	0.55889, 4.71922, 3.13962, 0.42849	410.5984	0.04172	0.9751	
APW	8.67567, 0.2894, 0.7977	413.4971	0.0680	0.5941	
WE	0.02257, 0.8452, 2.9267	421.7766	0.11524	0.0668	
MOEW	0.6413, 0.6406, 7.4457	415.0382	0.079614	0.3917	
W	0.09438, 1.04576	414.0874	0.06972	0.5623	

Fig. 9 Comparison of the theoretical densities using data set 2.

Fig. 9

Fig. 10 Comparison of the theoretical CDF's using data set 2.

Fig. 10

Fig. 11 Theoretical density plot of MTIIHLW Model for data set 2.

Fig. 11

Fig. 12 PP plot of MTIIHLW model for data set 2.

Fig. 12

Parameters were estimated by using the maximum likelihood principle. For this purpose, we have considered two different data sets from the literature. The data sets (i.e. the Airborne communication transceiver data, and the bladder cancer patient data) were used which are non-monotonic. The TTT plot clearly shows non-monotonic (bath-tub) hazard shapes for both data sets being considered. As a practical application, the data sets were used from the literature, which showed the best fit for the new proposed model with a smaller K-S value, Goodness-of fit criterion, and P-value as compared to the sub-models being considered. The simulation study conducted for the estimation of the parameters shows the improvement over the already available models. The findings suggest that the parameters of the suggested model are consistent. We conclude that the proposed model is suitable for practical problems.

Funding for the study

The authors received no funding for this study.

Data availability statement

All relevant data is available within the manuscript.

CRediT authorship contribution statement

Najma Salahuddin: Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Alamgir: Writing – review & editing, Validation, Supervision, Methodology, Formal analysis, Conceptualization. Muhammad Azeem: Writing – review & editing, Visualization, Validation, Investigation, Data curation. Sundus Hussain: Writing – review & editing, Visualization, Software, Methodology. Musarrat Ijaz: Writing – review & editing, Validation, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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