Weibull Distribution with Linear Shape Function

Featured Application

The proposed model, a generalization of Weibull-type distributions, finds wide application in analyzing data concerning the lifetime of technical and biological components. It may be particularly useful in reliability engineering, for example, for modeling the failure rate of mechanical devices, electronic systems, or critical infrastructure components, as well as in medicine for analyzing patient survival. Thanks to the flexibility of shaping the hazard function, the model enables a more accurate fit to data compared to classical approaches, which can support maintenance planning, risk forecasting, and resource optimization. Potential applications also include the analysis of large datasets from IoT sensors, which opens the possibility of its use in predictive maintenance systems.

Abstract

The paper is intended to put forward a modified Weibull-type lifetime model. Modification consists of replacing the shape parameter of the original Weibull model with the shape function. It is self-evidently a novelty among lifetime models. The model in question will further be named the Weibull-sf model. To present the Weibull-sf, we need appropriate background. The background comes from an extensive review performed on 165 Weibull-type lifetime models we found in the source literature. Performing this review, we focused on two properties of the models: modality of failure density functions, as well as shape of the hazard rate functions. It does not matter that these are strongly interrelated, incidentally. The Weibull-sf lifetime model has the valuable property of flexibility. It may have a bathtub-like hazard rate function and bimodal density function. This is exactly what reliability analysts want to have. Foreseeing the huge numerical problems one will face when trying the maximum-likelihood method, we promote the method of the least absolute values that is a “close relative” to the method of least squares. Examples of fitting the Weibull-sf to real data are given. The cumulative failure functions of bimodal models with a bathtub-like hazard rate function and R codes are given.

1. Introduction

Without any particular exaggeration, one may say that nearly everything in lifetime data analysis revolves around the hazard rate function (HRF). Lifetime models (LTMs) are categorized according to the shapes of their HRFs. Special attention is paid to the LTMs of “flat-bottomed” HRFs that are commonly named bathtub HRFs. Unfortunately, over time, this name has been used to describe any HRF having a minimum and evidently not being flat-bottomed. The true, i.e., flat-bottomed bathtub hazard rate [1], is specific to a non-homogeneous population. This population consists of subpopulations of “weak” and “strong” items.

Categorized roughly, LTMs may fall into monolithic or hybrid categories. The most representative monolithic LTMs to be recalled here seem to be the Weibull (W) [2], Gamma (G) and Gamma Weibull (GW). Their failure density functions (FDFs) are

$$f_W\left(t\right)=S_F\left(t-\tau \right){\displaystyle \frac{b}{a}}{\left({\displaystyle \frac{t-\tau }{a}}\right)}^{b-1}exp\left[-{\left({\displaystyle \frac{t-\tau }{a}}\right)}^b\right]\left(t>0\right),$$

(1)

$$f_G\left(t\right)={\displaystyle \frac{1}{a\Gamma \left(c\right)}}{\left({\displaystyle \frac{t}{a}}\right)}^{c-1}exp\left[-\left({\displaystyle \frac{t}{a}}\right)\right],$$

(2)

$$f_{WG}\left(t\right)={\displaystyle \frac{b}{a\Gamma \left(c\right)}}{\left({\displaystyle \frac{t}{a}}\right)}^{bc-1}exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^b\right],$$

(3)

where $a>0$ is the scale parameter, $b,c>0$ are the shape parameters, $\tau \ge 0$ is the failure free time parameter and $S_F$ is the step function defined as

$$S_F\left(x\right)=1·I\left(x\ge 0\right)+0·I\left(x<0\right).$$

(4)

Please note that LTM (3) came into being by embedding (1) into (2). It makes (3) more flexible than both (1) and (2) owing to the second shape parameter, namely $c$. However, none of the LTMs in question is sufficiently flexible to be applicable to non-homogeneous populations. It is because they cannot be bimodal. The reader may check it on their own.

The prime example of the hybrid LTM is the compound Weibull (CW) proposed by [3]. Its FDF is given by

$$\begin{array}{ccc}\hfill f_{CW}\left(t\right)& =& \omega {\displaystyle \frac{b_1}{a_1}}{\left({\displaystyle \frac{t}{a_1}}\right)}^{b_1-1}exp\left[-{\left({\displaystyle \frac{t}{a_1}}\right)}^{b_1}\right]+\hfill \\ & +& (1-\omega ){\displaystyle \frac{S_F(t-\tau )b_2}{a_2}}{\left({\displaystyle \frac{t-\tau }{a_2}}\right)}^{b_2-1}exp\left[-{\left({\displaystyle \frac{t-\tau }{a_2}}\right)}^{b_2}\right].\hfill \end{array}$$

(5)

where $a_1,a_2,b_1,b_2>0;\tau \ge 0;\omega \in \left(0,1\right)$.

There is no doubt that (5) can be regarded as an extension of (1), however far-reaching.

As mentioned above, monolithic LTMs are unimodal. In contrast, hybrid LTMs may be bimodal. Of course, they can also be unimodal for (1) or (2), to the great surprise of the analyst, which turns out to be inapplicable to a homogeneous population. Struck by the superiority of (5) over (1)–(3), one must not overlook the fact that (5) has twice as many parameters as (3). Therefore, employing (5), one should have at its disposal much more input data than when employing (3). It is essential to guarantee that the (5) and (3) sets of estimates are at the same level of accuracy. We face the problem of equilibrating flexibility and data consumption.

To familiarize ourselves with the problem, let us consider the results of the following simple, but very instructive, Monte Carlo experiment. A set of input data that comprises one hundred samples each of 30 items was drawn from the exponential population. The population scale parameter was set equal to one. Then the (1)–(3) LTMs were sequentially fitted to the dataset. Parameters were estimated with the maximum-likelihood (ML) method.

Table 1. Standard deviation of scale parameter estimate for LTMs.

LTM	a Estimate
Exponential	0.178
Gamma	0.242
Weibull	0.321
Gamma Weibull	0.937

shows standard deviations of scale parameter estimates.

LTM (3) produced scale parameter estimates of standard deviation more than five times greater than LTM (1) did. The explanation is simple, saying freely, “underfeeding” of the scale parameter took place because the shape parameters have “eaten” most of the input data for their estimation purposes.

In general, no one disputes the need for LTMs to be flexible. On the other hand, does the LTM need to have as many as eight parameters? The LTM below, called the Kumaraswamy transmuted exponentiated additive Weibull (KTEAW) [4], satisfies the mentioned criterion. The cumulative failure function (CFF) of the KTEAW is defined as

$$F_{KTEAW}\left(t\right)=1-{\left\{1-{\displaystyle \frac{{\left[1-exp\left(-c{t}^d-a{t}^b\right)\right]}^{eg}}{{\left[1+f-f{\left(1-exp\left(-c{t}^d-a{t}^b\right)\right)}^e\right]}^{-g}}}\right\}}^h,$$

(6)

where $\left(t>0;a,c,d,e,f,g,h>0;b>1\right).$

In general, there are currently two techniques to increase flexibility of LTMs: In the formula of the failure density function, more parameters or the same parameter are embedded in more than one place. The reader is prompted to compare (1) with (2). The Weibull distribution turned out to be a little more flexible than the Gamma distribution in Monte Carlo experiments.

The shape parameter can be called static in the sense that it shapes the LTM identically at each time point. In this paper, we will be able to shape the LTM dynamically owing to the following modification: we replace the shape parameter with the shape function. This is an innovative idea. The subject of modification, of course, will be the Weibull LTM, further named the Weibull distribution with shape function (WDSF). The CFF and FDF take the following forms:

$$F\left(t\right)=1-exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{w\left(t\right)}\right],$$

(7)

$$f\left(t\right)={\displaystyle \frac{w\left(t\right)}{a}}·{\left({\displaystyle \frac{t}{a}}\right)}^{w\left(t\right)-1}·e^{-{\left({\displaystyle \frac{t}{a}}\right)}^{w\left(t\right)}}+w^{\prime }\left(t\right)·ln\left({\displaystyle \frac{t}{a}}\right)·{\left({\displaystyle \frac{t}{a}}\right)}^{w\left(t\right)}·e^{-{\left({\displaystyle \frac{t}{a}}\right)}^{w\left(t\right)}}.$$

(8)

The above FDF is a sum of two components. This is a unique property of the LTM in question. Although born as a monolithic LTM, the WDSF turns out to be a hybrid-like LTM. Please note that WDSF is free of the fraction parameter $\omega$, which is a data guzzler in (5). As it is easy to guess, we, further in this paper, consider the simplest version of WDSF that involves a linear shape function.

By the end of this section, let us return to the very origins of the reliability domain. Let us recall two books published by reliability pathfinders of that time. These are [5,6]. A device stops working properly, not because the Finger of Fate points it out and says “fail”. It does it because the physical failure process reached its critical level. In the mentioned books, commonly used LTMs have assigned a particular mathematical failure mechanism or process.

For instance, both books assign the Weibull LTM to the model of the weakest link of the chain of elements forming a very long series reliability structure. This point of view appears to be timeless. Therefore, in this paper we have the courage to extend the Weibull LTM assigning the following failure process: As time flows and the wear-out process advances, the series reliability structure steadily elongates, causing device reliability to decrease. Referential mathematical considerations will be presented in Section 3.

The main goal of our work is to complement the literature on the theory of reliability models by introducing a new distribution with a linear shape function, which is a modification of the Weibull LTM. The additional goal of our paper is to define an estimation method that measures the absolute values of the differences between the empirical and theoretical reliability functions (RFs) (see Section 4).

The rest of the paper is organized as follows. Section 2 is devoted to the review of modified Weibull distributions. The properties of the two-version WDSF such as the CFF, RF, FDF, HRF, hazard rate average function (HRAF), quantile (Q) and pseudo-random number generator (PRNG) are described in Section 3. The estimation methods used are described in Section 4. Illustrative examples of the applicability and flexibility of the WDSF are presented in Section 5. Concluding remarks are provided in Section 7. CFF formulas of bimodal LTMs defined with five to eight parameters and having a bathtub HRF are provided in Appendix A. As the popularity of the R environment has increased significantly recently, main properties of the new distribution have been implemented in R software, version 4.3.1 [7]. Their full codes are in Appendix B.

2. A Review of Modified Weibull Distributions

Generalized Weibull distributions can be constructed in many ways. The first, and in our opinion, the most important way is to define distributions with the Weibull distribution as their special case (including a mixture of two or more Weibull variables). Other ways are, i.e., adding a constant to the hazard rate of the Weibull model; transformations (linear, inverse or log) of the Weibull random variable; transformations of the CFF or survival function of the Weibull models in such a way that the new model remains a CFF or survival function. More details can be found in [8].

By reviewing the statistical literature, we found 165 generalized Weibull distributions with 2–8 parameters. Among them, four distributions have a domain different from $t>0$. They are the reflected Weibull distribution [9] defined for $t<0$ as well as the Log-Weibull [10,11], modified odd Weibull normal [12] and extended odd Weibull normal [13] distributions defined for $t\in R$.

In the rest of this section, we will focus on such generalized Weibull distributions for which the Swedish research’s distribution is their special case. In this case, the large family of generalized Weibull distributions reduces to 71 distributions with three to eight parameters, named by the authors as modified Weibull distributions.

Modified Weibull distributions are divided into six groups, according to the number of their parameters. The list of these models is presented below. Information on hazard rate function shapes is provided in the superscript (¹unimodal, ²increasing, ³decreasing, ⁴bathub). Pseudo-bimodal lifetime models with bathtub hazard rate function are in underline (22 models). Bimodal lifetime models with bathtub hazard rate function are in bold (seven models) and their CFFs are presented in Appendix A.

Group I includes 19 models with three parameters. These are the following: generalized Gamma or Gamma Weibull^1–4 [14], generalization of Gamma Weibull^1–4 [15], exponentiated Weibull^1–4 [16], generalized Weibull^1–4 [17], exponentiated Weibull^1–4 [18], power generalized Weibull^1–4 [19], modified Weibull extension^2–4 [20], modified Weibull^2–4 [21], Marshall–Olkin Extended Weibull^1–4 [22], extended Weibull type^1,4 [23], generalized power Weibull^2–4 [24], Extended Weibull^1–4 [25], Sarhan and Zaindin’s Modified Weibull^2,3 [26], Weibull Geometric^1–4 [27], transmuted Weibull^1–4 [28], complementary Weibull geometric^1–4 [29], Alpha power Weibull^1–4 [30], MIT Weibull^1–4 [31] and Semi-Modified Alpha Power Weibull Distribution [32].

Group II includes 18 models with four parameters. These are the following: four-parameter generalized Gamma^1–4 [33], additive Weibull^2–4 [34], Generalized Modified Weibull^1,2,4 [35], Kumaraswamy Weibull^1–4 [36], Exponentiated Generalized Gamma^1,4 [37], exponentiated modified Weibull^1–4 [38], Transmuted modified Weibull² [39], Exponentiated transmuted Weibull^2–4 [40], Generalized Weibull-Exponential^1–3 [41], Weibull Lomax^2,3 [42], generalized power generalized Weibull^1–4 [43], Generalization of Generalized Gamma^1–4 [44], Weibull Lomax^2–4 [45], additive Chen-Weibull^2,4 [46], Poisson modified Weibull^1–4 [47], modified power generalized Weibull^1–4 [48], Generalized New Extended Weibull [49] and new modified exponentiated Weibull (NMEW)^2,4 [50].

Group III includes 27 models with five parameters. These are the following: Beta modified Weibull^1–4 [51], Kumaraswamy generalized Gamma^1–4 [52], beta generalized Weibull^1–4 [53], Transmuted Exponentiated Modified Weibull^2,3 [54], Beta Generalized Gamma^1–4 [55], transmuted additive Weibull^2,4 [56], Exponentiated Kumaraswamy Weibull^1–4 [57], new modified Weibull^2–4 [58], Kumaraswamy modified Weibull^1–4 [59], beta transmuted Weibull^1–3 [60], McDonald Weibull^1–4 (McW) [61], Exponentiated Transmuted Modified Weibull^1,2,4 [62], exponentiated generalized modified Weibull^2,4 [63], Gamma generalized modified Weibull [64], generalized modified Weibull geometric⁴ [65], additive modified Weibull^2–4 [66], transmuted exponentiated Weibull geometric^1–4 [67], transmuted new generalized Weibull^2–3 [68], Burr XII modified Weibull^1–4 [69], log-logistic modified Weibull^1,2,4 [70], Kumaraswamy alpha power Weibull^1–4 [71], exponentiated additive Weibull^1–4 (EAW) [72,73], generalized extended exponential Weibull^1–4 [74], generalized Weibull generalized exponential^1–2 [75], new generalized modified Weibull^1–4 [76] and improved modified Weibull^3,4 [77].

Group IV includes seven models with six parameters. These are the following: the mixture of two Weibull⁴ [3], McDonald modified Weibull^1–4 (McMW) [78], McDonald extended Weibull^1–4 (McEW) [79], Additive Weibull log logistic^1–4 [80], underbarBeta exponentiated modified Weibull^1–4 [81], exponentiated power generalized Weibull binomial^1–4 [82], and McDonald Generalized Power Weibull^1–4 (McGPW) [83].

Concluding the review of modified Weibull distributions, we would like to mention two distributions with seven and eight parameters. The Kumaraswamy transmuted exponentiated modified Weibull^1–4 (KTEMW) with 54 special cases forms group V [84]. The Kumaraswamy transmuted exponentiated additive Weibull^1–4 (KTEAW) [4] with 79 special cases (including KTEMW) forms group VI.

3. Properties of Weibull Distribution with Shape Function

The main goal of the paper is to present the two-version Weibull distribution with the shape function previously denoted as the WDSF. This section describes its properties such as the CFF, RF, FDF, HRF, hazard rate average function (HRAF), quantile (Q), and pseudo-random number generator (PRNG). Both versions of our model will be used in Section 5.

We assume that the device of our interest has a long series reliability structure. It makes the real lifetime distribution close to the Weibull distribution. The formula below is the Weibull LTM with the elongation coefficient of the device’s reliability structure.

$$R\left(t\right)=e^{-{\left({\displaystyle \frac{t}{a}}\right)}^{b}N\left(t\right)}$$

We impose two weak conditions on $N\left(t\right)$ with $dN\left(t\right)/dt\ge 0;t\in R^{+}$ and $N\left(0\right)=1$. Owing to such weak conditions, a wide scope opens itself for defining $N\left(t\right)$ functions not only of reasonable forms but also of monstrous ones. One reasonable form is $N\left(t\right)={\left(t/a\right)}^{ct}$ because it introduces only one parameter into the Weibull LTM and therefore equilibrates flexibility and data consumption. Thus, we get

$$R\left(t\right)=e^{-{\left({\displaystyle \frac{t}{a}}\right)}^b·{\left({\displaystyle \frac{t}{a}}\right)}^{ct}}=e^{-{\left({\displaystyle \frac{t}{a}}\right)}^{b+c·t}}.$$

Definition 1.

Let $w\left(t\right)$ from (7) or (8) be a linear shape function with two parameters given by $w\left(t;b,c\right)=b+ct$; then the CFF of the Weibull distribution with linear shape function in the first version (${\mathit{WDSF}}^I$) is defined as

$$F^I\left(t;\mathit{\vartheta }\right)=1-exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{b+ct}\right]\left(t>0\right),$$

(9)

where $\mathit{\vartheta }=\left(a,b,c\right)$, $a>0$ is the scale parameter, $b>0,c\ge 0$ are the shape parameters and ${\displaystyle \frac{b}{ac}}\ge \underset{z}{max}\left\{-z\left[1+ln\left(z\right)\right]\right\}=0.135335$. If $c=0$, then we get the Weibull distribution.

Definition 2.

Let $w\left(t\right)$ from (7) or (8) be a linear shape function with three parameters given by $w\left(t;b,c,\tau \right)=b+c{\left(t-\tau \right)S}_F\left(t-\tau \right)$ then the CFF of the Weibull distribution with linear shape function in the second version (WDSF^II) is defined as

$$F^{II}\left(t;\mathit{\theta }\right)=1-exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{b+c{\left(t-\tau \right)S}_F\left(t-\tau \right)}\right],$$

(10)

where $\mathit{\theta }=\left(a,b,c,\tau \right)$, $\tau \ge 0$ is the failure free time parameter, $S_F$ is the step function (4) and ${\displaystyle \frac{b-c\tau }{ac}}\ge \underset{z}{max}\left\{-z\left[1+ln\left(z\right)\right]\right\}=0.135335$. If $\tau =0$, then we get the first version of our proposal.

Proposition 1.

The RFs of the WDSF^I and WDSF^II are defined, respectively, as

$$R^I\left(t;\mathit{\vartheta }\right)=exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{b+ct}\right],$$

(11)

$$R^{II}\left(t;\mathit{\theta }\right)=exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{b+c\left(t-\tau \right)S_F\left(t-\tau \right)}\right].$$

(12)

Proof of Proposition 1.

The proof based on the RF definition is trivial. □

It is interesting to compare the ${\mathrm{WDSF}}^I$ and Weibull models on the Weibull probability paper. Taking the logarithm of both sides (11) twice, we get

$$ln\left\{-ln\left[R^I\left(t;\vartheta \right)\right]\right\}=\left(b+ct\right)\left[ln\left(t\right)-ln\left(a\right)\right].$$

Let $ln\left(t\right)=x,y=ln\left\{-ln\left[R^I\left(t;\vartheta \right)\right]\right\}$ then

$$y\left(x;\vartheta \right)=\left(b+c{e}^x\right)\left[x-ln\left(a\right)\right].$$

Figure 1 presents the Weibull probability paper with transformed reliability functions plotted on it. The figure is intended to visualize similarity and dissimilarity between the WDSF^I and Weibull models. One may easily notice the following:

• There may exist even relatively long-time intervals inside in which the $y\left(x;\vartheta \right)$ functions are indistinguishable. It lasts as long as b is much greater as $ct$.
• Both reliability functions cross a horizontal line $y\left(x;\vartheta \right)=0$ in the same point.
• As time passes, $y\left(x;\vartheta \right)$ functions slowly but steadily part. This is the region of the second mode of the WDSF^I lifetime model.

Proposition 2.

The FDFs of the WDSF^I and WDSF^II are respectively given by

$$f^I\left(t;\mathit{\vartheta }\right)=exp\left[-{\left({\displaystyle \frac{t}{a}}\right)}^{b+ct}\right]{\left({\displaystyle \frac{t}{a}}\right)}^{b+ct-1}\left[{\displaystyle \frac{b+ct}{a}}+{\displaystyle \frac{tc}{a}}ln\left({\displaystyle \frac{t}{a}}\right)\right],$$

(13)

$$f^{II}\left(t;\mathit{\theta }\right)=exp\left[-u\left(t;\mathit{\theta }\right){\displaystyle \frac{t}{a}}\right]u\left(t;\mathit{\theta }\right)\left[{\displaystyle \frac{b+c\left(t-\tau \right)S_F\left(t-\tau \right)}{a}}+{\displaystyle \frac{{ctS}_F\left(t-\tau \right)}{a}}ln\left({\displaystyle \frac{t}{a}}\right)\right],$$

(14)

where $u\left(t;\mathit{\theta }\right)={\left({\displaystyle \frac{t}{a}}\right)}^{b+c\left(t-\tau \right)S_F\left(t-\tau \right)-1}$.

Proof of Proposition 2.

By computing the derivatives based on (9) and (10) regarding the lifetime t, we easily obtain Formulas (13) and (14). Recall that $t,a,b,c>0$ and the Formulas (13) and (14) are non-negative.

Let $g\left(z\right)=-z\left[1+ln\left(z\right)\right]$. $g\left(z\right)$ has a maximum value of 0.135335 for $z=0.1345$.

Let $z=t/a$; then we have from (13)

$${\displaystyle \frac{b}{a}}+cz+czln\left(z\right)\ge 0\Rightarrow {\displaystyle \frac{b}{ac}}\ge -z\left[1+ln\left(z\right)\right]=0.135335.$$

Let $t-\tau \ge 0$, then we have from (14)

$${\displaystyle \frac{b+c\left(t-\tau \right)}{a}}+{\displaystyle \frac{ct}{a}}ln\left({\displaystyle \frac{t}{a}}\right)\ge 0\Rightarrow {\displaystyle \frac{b-c\tau }{ac}}\ge -z\left[1+ln\left(z\right)\right]=0.135335.$$

 □

Figure 2 shows the RF of the WDSF^I and WDSF^II. Pseudo-bimodality or bimodality is visible here. The R codes (R Core Team 2021) for calculating the RF values of the WDSF^I and WDSF^II are provided in Appendix B.

Figure 3 shows the FDF of the WDSF^I and WDSF^II. We see the pseudo-bimodality (on the left) and bimodality (on the right). The R codes for calculating the FDF values of WDSF^I and WDSF^II are provided in Appendix B.

The WDSF^II with FDF given by (14) is identifiable in the parameter space $\mathit{\theta }=\left(a,b,c,\tau \right)$. This identifiability is apparent at the first glance. Particular parameters of the WDSF^II lifetime model have very strictly prescribed roles to impact failure density function. It causes that, for example, an impact of the scale parameter in no way can be compensated with shape and location parameters. In turn, it causes that it is impossible to obtain the same two values of the density function for two different sets of parameters. It is a meaningful fact that during over forty years of my activity in the reliability domain, I never encountered any validation of the Weibull lifetime model. Despite not being in any way validated, the model has for decades kept itself on the second place in the ranking of applicability.

Proposition 3.

The HRFs of the WDSF^I and WDSF^II have, respectively, forms

$$h^I\left(t;\mathit{\vartheta }\right)={\left({\displaystyle \frac{t}{a}}\right)}^{b+ct}\left[{\displaystyle \frac{b+ct}{t}}+cln\left({\displaystyle \frac{t}{a}}\right)\right],$$

(15)

$$h^{II}\left(t;\mathit{\theta }\right)=\left[{\displaystyle \frac{b+c\left(t-\tau \right)S_F\left(t-\tau \right)}{t}}+c{S}_F\left(t-\tau \right)ln\left({\displaystyle \frac{t}{a}}\right)\right]{\left({\displaystyle \frac{t}{a}}\right)}^{b+c{\left(t-\tau \right)S}_F\left(t-\tau \right)}.$$

(16)

Proof of Proposition 3.

The proof based on the HRF definition is trivial. □

Figure 4 shows the bathtub HRF of the WDSF^I and WDSF^II. The curves flatten as c decreases. The R codes for calculating the HRF values of WDSF^I and WDSF^II are provided in Appendix B.

Proposition 4.

The HRAFs of the WDSF^I and WDSF^II are, respectively, defined as [85]

$$h{a}^I\left(t;\mathit{\vartheta }\right)=-{\displaystyle \frac{1}{t}}ln\left[R^I\left(t;\mathit{\vartheta }\right)\right]={\displaystyle \frac{1}{t}}{\left({\displaystyle \frac{t}{a}}\right)}^{b+ct},$$

(17)

$$h{a}^{II}\left(t;\theta \right)=-{\displaystyle \frac{1}{t}}ln\left[R^{II}\left(t;\vartheta \right)\right]={\displaystyle \frac{1}{t}}{\left({\displaystyle \frac{t}{a}}\right)}^{b+c\left(t-\tau \right)S_F\left(t-\tau \right)}.$$

(18)

Proof of Proposition 4.

The proof based on Formulas (11) and (12) and definition of the HRAF is trivial. □

Figure 5 shows the bathtub HRAF of the WDSF^I and WDSF^II. HRAF curves are flatter than HRF curves. The R codes for calculating the HARF values of WDSF^I and WDSF^II are provided in Appendix B.

Figure 6 shows the logarithm of theoretical and empirical HRAFs for the WDSF^I and sample size $n=30$.

Proposition 5.

Let $0<p<1$. The Qs $q_p^I$ and $q_p^{II}$ of the WDSF^I and WDSF^II are, respectively, solutions of the equations

$${\left({\displaystyle \frac{q_p^I}{a}}\right)}^{b+c{q}_p^I}+ln\left(1-p\right)=0,$$

(19)

$${\left({\displaystyle \frac{q_p^{II}}{a}}\right)}^{b+c\left(q_p^{II}-\tau \right)S_F\left(q_p^{II}-\tau \right)}+ln\left(1-p\right)=0.$$

(20)

Proof of Proposition 5.

The proof based on (9) and (10) is easy. □

The R codes for calculating the $q_p^I$ and $q_p^{II}$ values of WDSF^I and WDSF^II are provided in Appendix B.

Proposition 6.

Let $R\sim Unif\left(0,1\right)$, $T^I$ and $T^{II}$ follow the WDSF^I and WDSF^II, respectively. We can obtain $T^I$ and $T^{II}$ in two ways.

• The first way. The $T^I$ and $T^{II}$ are respectively solutions of the equations

  $${\left({\displaystyle \frac{T^I}{a}}\right)}^{b+c{T}^I}+ln\left(1-R\right)=0,$$

  (21)

  $${\left({\displaystyle \frac{T^{II}}{a}}\right)}^{b+c\left(T^{II}-\tau \right)S_F\left(T^{II}-\tau \right)}+ln\left(1-R\right)=0.$$

  (22)
• The second way. The algorithm for obtaining the $T^I$ and $T^{II}$ is as follows:

  1. 

  Let $\epsilon ={10}^{-10}$, $k=0$, $T_0^I=0,T_0^{II}=0$.

  2. 

  Let $k=k+1$

  3. 

  $T_{k+1}^I={a\left[-ln\left(R\right)\right]}^{{\left[b+c{T}_k^I\right]}^{-1}}$, $T_{k+1}^{II}={a\left[-ln\left(R\right)\right]}^{{\left[b+c{S}_F\left(T_{k+1}^{II}-\tau \right)\left(T_k^{II}-\tau \right)\right]}^{-1}}$.

  4. 

  If $\left|T_{k+1}^I-T_k^I\right|\ge \epsilon$, $\left|T_{k+1}^{II}-T_k^{II}\right|\ge \epsilon$ then go to step 2.

  5. 

  Return $T^I=T_{k+1}^I$, $T^{II}=T_{k+1}^{II}$.

Proof of Proposition 6.

Based on (9), we get

$$exp\left[-{\left({\displaystyle \frac{T^I}{a}}\right)}^{b+c{T}^I}\right]=1-R$$

(23)

and after taking logarithm on both sides (23), we have (21). Similarly, based on (10), we obtain (22). Recursive formulas in step 3 of the algorithm are obtained based on (9) and (10) using the CFF inversion method. □

The R codes for the PRNG of WDSF^I and WDSF^II are provided in Appendix B.

In this passage, we focus on Figure 7 and Figure 8 to answer such questions: what is a small sample used for and what is a large one used for? Conventionally, small samples are those of size $n\le 30$. Analysts of lifetime data as a rule deal with small samples. Thus, Figure 7 presents the empirical and theoretical reliability functions for sample size n = 30. This figure impresses on readers how far empirical points may depart from the actual reliability function. Figure 8 presents the empirical and theoretical reliability functions for sample size n = 1000. Empirical points practically do not depart from the actual reliability function. This figure impresses on readers that the Weibull-sf random number generator works satisfactorily.

4. Estimation Methods

An additional goal for this article is outlined in this section.

Looking through the literature searching for distributions that are modifications of the Weibull distribution, we find that the most dominant method of parameter estimation is the maximum-likelihood (ML) method. However, the question remains whether this choice is right. The younger the paper, the more often the parameters are estimated using other methods, e.g., the ordinary least-squares (LS) and weighted least-squares (WLS) ones; see, e.g., [48,86,87,88,89].

What relates to the WDSF^II is that the main fraction of the probability mass is concentrated on the “contractual” end of the distribution. Thus, for small and medium values of t, the Weibull and WDSF^II are practically indistinguishable in terms of the reliability or density functions. Censoring appears to be inapplicable to the Weibull-sf except in some special cases. This is the reason why in the reviewed paper a problem of censoring has been omitted. Please notice that this is the same inconvenient situation as with the compound Weibull lifetime model. Finally, one should not marvel at this because the WDSF^II has been designed as a “lighter” version of the compound Weibull lifetime model.

Let $\mathit{\vartheta }=\left(a,b,c\right),\mathit{\theta }=\left(a,b,c,\tau \right)$ be parameter vectors and $t_1^{*},t_2^{*},\dots ,t_n^{*}$ be a random sample of size n from the WDSF^I and WDSF^II. To estimate unknown values of parameters, we use estimation methods such as the ML, LS, WLS and least absolute values (LAWs). The LAW, which is the first additional goal of the work, measures the absolute values of the differences between the empirical and theoretical RFs.

The likelihood functions (LFs) of the WDSF^I and WDSF^II, based on (13) and (14), are given by, respectively,

$$L^I\left(t_i^{*};\mathit{\vartheta }\right)=\prod _{i=1}^{n}exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}}\right]{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}-1}\left[{\displaystyle \frac{b+c{t}_i^{*}}{a}}+{\displaystyle \frac{c{t}_i^{*}}{a}}ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)\right],$$

(24)

$$\begin{array}{ccc}\hfill L^{II}\left(t_i^{*};\mathit{\theta }\right)& =& \prod _{i=1}^{n}exp\left[-u\left(t_i^{*};\mathit{\theta }\right){\displaystyle \frac{t_i^{*}}{a}}\right]·u\left(t_i^{*};\mathit{\theta }\right)·\hfill \\ & & ·\left[{\displaystyle \frac{b+c\left(t_i^{*}-\tau \right)S_F\left(t_i^{*}-\tau \right)}{a}}+{\displaystyle \frac{c{t}_i^{*}{S}_F\left(t_i^{*}-\tau \right)}{a}}ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)\right],\hfill \end{array}$$

(25)

where $u\left(t_i^{*};\mathit{\theta }\right)={\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c\left(t_i^{*}-\tau \right)S_F\left(t_i^{*}-\tau \right)-1}$.

The log-likelihood functions (LLFs) of the WDSF^I and WDSF^II, based on (25) and (26), are defined as, respectively,

$$l^I\left(t_i^{*};\mathit{\vartheta }\right)=-\sum _{i=1}^n{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}}+\sum _{i=1}^n\left(b+c{t}_i^{*}-1\right)ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)+\sum _{i=1}^{n}ln\left[{\displaystyle \frac{b+c{t}_i^{*}}{a}}+{\displaystyle \frac{c{t}_i^{*}}{a}}ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)\right],$$

(26)

$$\begin{array}{cc}\hfill l^{II}=& -\sum _{i=1}^n{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c\left(t_i^{*}-\tau \right)S_F\left(t_i^{*}-\tau \right)}+\hfill \\ \hfill & +\sum _{i=1}^n\left(b+c\left(t_i^{*}-\tau \right)S_F\left(t_i^{*}-\tau \right)-1\right)ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)+\hfill \\ \hfill & +\sum _{i=1}^{n}ln\left[{\displaystyle \frac{b+c\left(t_i^{*}-\tau \right)S_F\left(t_i^{*}-\tau \right)}{a}}+{\displaystyle \frac{c{t}_i^{*}{S}_F\left(t_i^{*}-\tau \right)}{a}}ln\left({\displaystyle \frac{t_i^{*}}{a}}\right)\right].\hfill \end{array}$$

(27)

Formulas ${\displaystyle \frac{d{l}^I}{d\mathit{\vartheta }}},{\displaystyle \frac{d{l}^{II}}{d\mathit{\theta }}}$ have complex forms, so in practice we maximize the LF (27) and (28) or the LLFs (29) and (30) to obtain the ML estimates.

To obtain the OLS estimates of the WDSF^I and WDSF^II parameters, we minimize the following objective functions, respectively.

$${OLS}^I=\sum _{i=1}^n{\left\{{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}}\right]\right\}}^2,$$

(28)

$${OLS}^{II}=\sum _{i=1}^n{\left\{{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{\left(t_i^{*}-\tau \right)S}_F\left(t_i^{*}-\tau \right)}\right]\right\}}^2.$$

(29)

To obtain the WLS estimates of the WDSF^I and WDSF^II parameters, we minimize the following objective functions, respectively.

$${WLS}^I=\sum _{i=1}^n{\displaystyle \frac{{\left(n+1\right)}^2\left(n+2\right)}{i\left(n-i+1\right)}}{\left\{{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}}\right]\right\}}^2,$$

(30)

$${WLS}^{II}=\sum _{i=1}^n{\displaystyle \frac{{\left(n+1\right)}^2\left(n+2\right)}{i\left(n-i+1\right)}}{\left\{{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{\left(t_i^{*}-\tau \right)S}_F\left(t_i^{*}-\tau \right)}\right]\right\}}^2.$$

(31)

Let $R_e\left(i\right)=1-{\displaystyle \frac{i}{n+1}}$ be the empirical RF. To obtain the LAW estimates of the WDSF^I and WDSF^II parameters, we minimize the following objective functions, respectively.

$${LAW}^I=\sum _{i=1}^n\left|R_e\left(i\right)-R^I\left(t;\mathit{\vartheta }\right)\right|=\sum _{i=1}^n\left|{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{t}_i^{*}}\right]\right|,$$

(32)

$${LAW}^{II}=\sum _{i=1}^n\left|R_e\left(i\right)-R^{II}\left(t;\mathit{\theta }\right)\right|=\sum _{i=1}^n\left|{\displaystyle \frac{i-n-1}{n+1}}+exp\left[-{\left({\displaystyle \frac{t_i^{*}}{a}}\right)}^{b+c{\left(t_i^{*}-\tau \right)S}_F\left(t_i^{*}-\tau \right)}\right]\right|.$$

(33)

The following three books with many real-world data examples are particularly interesting. The first one is the Lai and Xie’s book [90]. The analysed examples are characterized by sample sizes with $min=18,max=406,mean=84,125$. The second one is the Lee and Wang’s book [91]. The analysed examples are characterized by sample sizes with $min=21,max=3765,mean=973,28$. The third one is the Nelson’s book [92]. The analysed examples are characterized by sample sizes with $min=3,max=70,ave=17,400$.

Based on the above sample size data, a simulation study was performed with ${10}^4$ samples with a size of $10,25,50,75,100,250,500,1000$. The samples were drawn from the WDSF^II with $\mathit{\theta }=\left(1,1,c,0\right)$, where $c=\left(1,2,3\right)$. To obtain highly accurate parameter estimates, the optimization procedure was run ${10}^2$ times with random initial values $Unif(1-0.25,1+0.25)$ for $a,b$ and $Unif(c-0.25,c+0.25)$ for c and $Unif(0,0.25)$ for $\tau$. Our estimates for a given sample are the values that minimize (29), (31) and (33) and maximize (25) or (27).

The biases (Bs) and the root mean squared errors (RMSEs) of the WDSF^II estimates are shown in

Table 2. Biases and RMSEs of the estimates obtained using various methods (M). Samples of size n were drawn from WDSF^II with $\mathit{\theta }=(1,1,c=1,0)$.

n	M	$\widehat{\mathit{a}}$		$\widehat{\mathit{b}}$		$\widehat{\mathit{c}}$		$\widehat{\mathit{\tau }}$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
10	ML	−0.0205	0.1322	0.0616	0.1743	0.0717	0.1659	0.1099	0.1333
	LS	0.0065	0.1810	0.1663	0.7298	1.1725	4.2181	0.4838	0.7441
	WLS	0.0119	0.1817	0.1684	0.7136	1.9641	6.8872	0.5260	0.7813
	LAW	0.0138	0.1820	0.1528	0.6781	0.6359	1.8316	0.4337	0.6849
25	ML	−0.0112	0.0993	0.0601	0.1602	0.0520	0.1611	0.1093	0.1328
	LS	0.0040	0.1160	0.1310	0.4143	0.5184	1.7261	0.3160	0.5317
	WLS	0.0049	0.1156	0.1267	0.4165	0.7341	2.5792	0.3404	0.5672
	LAW	0.0041	0.1178	0.1423	0.4278	0.4388	1.2526	0.3388	0.5320
50	ML	−0.0067	0.0756	0.0608	0.1487	0.0426	0.1543	0.1057	0.1293
	LS	0.0037	0.0825	0.1168	0.2943	0.2645	0.8340	0.2149	0.3674
	WLS	0.0072	0.0830	0.1002	0.2770	0.3537	1.0173	0.2300	0.3980
	LAW	0.0061	0.0833	0.1443	0.3315	0.4012	1.0325	0.2890	0.4586
75	ML	−0.0051	0.0645	0.0608	0.1394	0.0374	0.1504	0.1027	0.1257
	LS	0.0027	0.0666	0.1075	0.2406	0.1660	0.5696	0.1715	0.2936
	WLS	0.0046	0.0679	0.0872	0.2267	0.2197	0.6502	0.1820	0.3209
	LAW	0.0034	0.0676	0.1216	0.2619	0.2157	0.6200	0.2081	0.3367
100	ML	−0.0070	0.0761	0.0625	0.1485	0.0419	0.1535	0.1062	0.1296
	LS	0.0019	0.0586	0.0989	0.2078	0.1276	0.4553	0.1502	0.2536
	WLS	0.0040	0.0583	0.0814	0.1963	0.1587	0.4730	0.1508	0.2679
	LAW	0.0029	0.0579	0.1209	0.2415	0.2238	0.6223	0.2060	0.3372
250	ML	−0.0036	0.0576	0.0654	0.1362	0.0322	0.1449	0.0996	0.1226
	LS	0.0015	0.0364	0.0809	0.1435	0.0516	0.2351	0.1021	0.1676
	WLS	0.0013	0.0366	0.0573	0.1235	0.0541	0.2045	0.0862	0.1510
	LAW	0.0023	0.0374	0.0901	0.1603	0.0815	0.2801	0.1252	0.2133
500	ML	−0.0022	0.0411	0.0626	0.1181	0.0222	0.1300	0.0903	0.1120
	LS	0.0005	0.0257	0.0647	0.1068	0.0196	0.1468	0.0730	0.1182
	WLS	0.0006	0.0256	0.0436	0.0895	0.0266	0.1228	0.0587	0.0987
	LAW	0.0012	0.0261	0.0695	0.1157	0.0306	0.1581	0.0819	0.1363
1000	ML	−0.0005	0.0287	0.0587	0.0963	0.0132	0.1085	0.0755	0.0935
	LS	0.0001	0.0183	0.0522	0.0809	0.0103	0.0973	0.0566	0.0878
	WLS	0.0007	0.0181	0.0358	0.0676	0.0134	0.0836	0.0429	0.0703
	LAW	0.0004	0.0183	0.0529	0.0848	0.0126	0.1024	0.0586	0.0948

,

Table 3. Biases and RMSEs of the estimates obtained using various methods (M). Samples of size n were drawn from WDSF^II with $\mathit{\theta }=(1,1,c=2,0)$.

n	M	$\widehat{\mathit{a}}$		$\widehat{\mathit{b}}$		$\widehat{\mathit{c}}$		$\widehat{\mathit{\tau }}$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
10	ML	−0.0204	0.1029	0.0606	0.1710	0.0545	0.1549	0.0963	0.1216
	LS	0.0004	0.1217	0.4674	1.2624	1.2525	4.8709	0.4122	0.6212
	WLS	0.0044	0.1238	0.4013	1.1117	2.4625	8.3951	0.4199	0.6330
	LAW	0.0041	0.1199	0.3712	1.0770	0.7364	2.3315	0.3509	0.5513
25	ML	−0.0121	0.0700	0.0615	0.1602	0.0397	0.1517	0.0918	0.1167
	LS	−0.0016	0.0754	0.3269	0.7495	0.4629	2.4316	0.2488	0.4249
	WLS	0.0011	0.0766	0.2829	0.7138	0.8034	3.2729	0.2481	0.4248
	LAW	0.0018	0.0761	0.3387	0.7396	0.4181	1.5841	0.2655	0.4161
50	ML	−0.0083	0.0520	0.0668	0.1514	0.0295	0.1482	0.0831	0.1070
	LS	0.0008	0.0535	0.2473	0.5208	0.1890	1.1678	0.1526	0.2686
	WLS	0.0018	0.0528	0.1959	0.4556	0.3386	1.3953	0.1483	0.2697
	LAW	0.0018	0.0544	0.2963	0.5798	0.4046	1.4745	0.2128	0.3550
75	ML	−0.0062	0.0437	0.0732	0.1477	0.0278	0.1465	0.0779	0.1008
	LS	0.0005	0.0437	0.2008	0.3945	0.1208	0.7561	0.1177	0.2040
	WLS	0.0010	0.0433	0.1525	0.3437	0.1963	0.8617	0.1106	0.2036
	LAW	0.0016	0.0444	0.2371	0.4510	0.1620	0.8155	0.1448	0.2430
100	ML	−0.0049	0.0389	0.0729	0.1431	0.0233	0.1445	0.0729	0.0947
	LS	0.0001	0.0376	0.1773	0.3330	0.0788	0.5702	0.1005	0.1726
	WLS	0.0007	0.0378	0.1323	0.2844	0.1205	0.5751	0.0880	0.1588
	LAW	0.0008	0.0380	0.2268	0.4140	0.1874	0.7833	0.1419	0.2456
250	ML	−0.0029	0.0284	0.0747	0.1316	0.0152	0.1379	0.0591	0.0761
	LS	0.0002	0.0240	0.1267	0.2091	0.0301	0.3086	0.0668	0.1087
	WLS	0.0004	0.0237	0.0840	0.1662	0.0460	0.2623	0.0517	0.0882
	LAW	0.0002	0.0244	0.1386	0.2392	0.0543	0.3625	0.0772	0.1354
500	ML	−0.0014	0.0236	0.0739	0.1232	0.0082	0.1338	0.0511	0.0644
	LS	0.0004	0.0170	0.0969	0.1507	0.0146	0.2058	0.0488	0.0778
	WLS	0.0000	0.0167	0.0603	0.1147	0.0227	0.1699	0.0354	0.0589
	LAW	0.0004	0.0172	0.0966	0.1573	0.0211	0.2203	0.0512	0.0861
1000	ML	−0.0010	0.0205	0.0718	0.1161	0.0047	0.1303	0.0466	0.0582
	LS	−0.0001	0.0121	0.0736	0.1092	0.0079	0.1420	0.0377	0.0579
	WLS	−0.0001	0.0118	0.0432	0.0813	0.0113	0.1177	0.0252	0.0412
	LAW	0.0002	0.0120	0.0699	0.1098	0.0073	0.1446	0.0355	0.0590

and

Table 4. Biases and RMSEs of the estimates obtained using various methods (M). Samples of size n were drawn from WDSF^II with $\mathit{\theta }=(1,1,c=3,0)$.

n	M	$\widehat{\mathit{a}}$		$\widehat{\mathit{b}}$		$\widehat{\mathit{c}}$		$\widehat{\mathit{\tau }}$
		BIAS	RMSE	BIAS	RMSE	BIAS	RMSE	BIAS	RMSE
10	ML	−0.0173	0.0814	0.0577	0.1654	0.0446	0.1511	0.0798	0.1065
	LS	−0.0007	0.0921	0.7692	1.7676	1.3052	6.0734	0.3940	0.5789
	WLS	0.0009	0.0893	0.7173	1.6221	2.8120	11.6308	0.4092	0.5942
	LAW	0.0016	0.0908	0.6334	1.5279	0.7654	3.2369	0.3508	0.5536
25	ML	−0.0111	0.0538	0.0599	0.1572	0.0355	0.1495	0.0789	0.1032
	LS	−0.0005	0.0570	0.5873	1.1862	0.3761	2.9283	0.2520	0.4217
	WLS	0.0008	0.0557	0.5149	1.1200	0.8422	4.1144	0.2479	0.4207
	LAW	0.0014	0.0565	0.5792	1.1466	0.3686	1.9532	0.2628	0.4163
50	ML	−0.0079	0.0403	0.0648	0.1504	0.0213	0.1462	0.0685	0.0911
	LS	−0.0006	0.0400	0.4168	0.8292	0.0923	1.3141	0.1483	0.2641
	WLS	−0.0006	0.0394	0.3359	0.7441	0.2887	1.6475	0.1377	0.2601
	LAW	−0.0005	0.0411	0.4771	0.8903	0.2062	1.4312	0.1878	0.3127
75	ML	−0.0055	0.0339	0.0718	0.1465	0.0160	0.1449	0.0607	0.0817
	LS	−0.0006	0.0327	0.3528	0.6791	0.0450	1.0698	0.1128	0.2011
	WLS	−0.0004	0.0322	0.2437	0.5365	0.1575	0.9933	0.0945	0.1847
	LAW	−0.0004	0.0329	0.4038	0.7336	0.1251	1.1392	0.1428	0.2469
100	ML	−0.0047	0.0304	0.0739	0.1457	0.0108	0.1430	0.0555	0.0743
	LS	−0.0002	0.0286	0.2971	0.5415	0.0140	0.7771	0.0925	0.1560
	WLS	−0.0003	0.0278	0.1994	0.4361	0.0994	0.7371	0.0744	0.1465
	LAW	0.0000	0.0285	0.3486	0.6386	0.0836	0.9194	0.1176	0.2082
250	ML	−0.0026	0.0218	0.0750	0.1356	0.0067	0.1414	0.0429	0.0553
	LS	0.0000	0.0178	0.1854	0.3046	0.0048	0.4205	0.0594	0.0962
	WLS	0.0000	0.0174	0.1031	0.2051	0.0343	0.3255	0.0391	0.0676
	LAW	−0.0001	0.0178	0.2034	0.3517	0.0251	0.4881	0.0646	0.1160
500	ML	−0.0015	0.0185	0.0773	0.1294	0.0040	0.1366	0.0382	0.0481
	LS	−0.0001	0.0127	0.1387	0.2094	0.0018	0.2794	0.0444	0.0691
	WLS	0.0000	0.0124	0.0713	0.1356	0.0219	0.2170	0.0270	0.0459
	LAW	−0.0001	0.0127	0.1333	0.2182	0.0104	0.3005	0.0435	0.0760
1000	ML	−0.0010	0.0165	0.0777	0.1267	−0.0008	0.1359	0.0355	0.0443
	LS	0.0000	0.0090	0.1058	0.1517	0.0058	0.1870	0.0349	0.0523
	WLS	0.0001	0.0087	0.0534	0.0952	0.0117	0.1486	0.0195	0.0323
	LAW	0.0001	0.0089	0.0931	0.1469	0.0031	0.2002	0.0306	0.0532

. We observe that as the sample size increases, the estimates approach the true values, which means that the estimates are consistent. Bs and RMSE values are the lowest for $\widehat{a}$. The Bs are the largest for $\widehat{\tau }$ ($c=1)$ and $\widehat{b}$ ($c=2,3)$. The RMSEs are the highest for $\widehat{c}$. The Bs increase with the value of c for $\widehat{b}$. The RMSEs increase with the value of c for $\widehat{b},\widehat{c}$ and decrease for $\widehat{a}$. The Bs and RMSEs are the lowest (highest) for the ML (LAW) method associated with $\widehat{\tau },\widehat{b}$ and $\widehat{c}$. The ML method is not suitable for estimating scale parameters. In this case, the lowest Bs and RMSE values are for the WLS method.

To examine the accuracy of the coverage probability of the asymptotic confidence intervals (CIs), another simulation study was performed with ${10}^4$ samples using sample sizes of $10,25,50,75,100,250,500,1000$. The study focused on the parameters $a,b,c,\tau$ and samples drawn from the WDSF^II with $\mathit{\theta }=\left(1,0.8,0.2,0\right)$ and $\mathit{\theta }=\left(1,1.5,1,0\right)$. The coverage probabilities of the obtained $95\%$ CIs for $\mathit{\theta }$ reported in

Table 5. Coverage probability for the standard asymptotic $95\%$ CIs. Samples of size n were drawn from WDSF^II with ${\mathit{\theta }}_{\mathbf{1}}=(1,0.8,0.2,0)$ and ${\mathit{\theta }}_{\mathbf{2}}=(1,1.5,1,0)$.

n	${\mathit{\theta }}_{\mathbf{1}}=(1,0.8,0.2,0)$				${\mathit{\theta }}_2=(1,1,5,1,0)$
	a	b	c	$\mathit{\tau }$	a	b	c	$\mathit{\tau }$
10	0.9492	0.9517	0.9523	0.9512	0.9543	0.9501	0.9506	0.9457
25	0.9518	0.9490	0.9560	0.9524	0.9551	0.9487	0.9521	0.9551
50	0.9501	0.9515	0.9453	0.9498	0.9546	0.9427	0.9458	0.9509
75	0.9496	0.9496	0.9471	0.9518	0.9527	0.9525	0.9517	0.9513
100	0.954	0.9456	0.9545	0.9531	0.9547	0.9528	0.951	0.9496
250	0.95	0.955	0.9476	0.9504	0.9515	0.9436	0.9504	0.9507
500	0.957	0.9476	0.9488	0.9501	0.9525	0.9493	0.9473	0.9462
1000	0.9505	0.9508	0.9426	0.9451	0.9454	0.9489	0.9499	0.9498

are very close to the nominal level. The results suggested that the obtained standard errors and hence the asymptotic CIs are reliable.

5. Applications

In this section, we illustrate the importance of the WDSFI and WDSFII distributions using three real-life datasets. The new models are compared with bimodal LTMs, characterized by unimodal, increasing, decreasing and bathub HRF, such as exponentiated additive Weibull, McDonald Weibull, McDonald modified Weibull, McDonald extended Weibull, McDonald generalized Power Weibull, Compound Weibull, Kumaraswamy transmuted exponentiated modified Weibull, and Kumaraswamy transmuted exponentiated additive Weibull. CFFs of the mentioned models are presented in “Appendix A”.

All calculations for comparison were performed in R software, version 4.3.1. To avoid local maxima, the optimization procedure was run ${10}^3$ times with random starting model parameter values that are widely scattered in the parameter space. The final parameter estimates are those parameter values that best maximize the log-likelihood function. AIC, BIC, HQIC criteria and the Kolmogorov–Smirnov (KS) statistic are calculated. Let us remind the reader that

$$AIC=-2l+2p,BIC=-2l+pln\left(n\right),HQIC=-2l+2pln\left(ln\left(n\right)\right)$$

where l is the log-likelihood function, n is the sample size and p is the number of model parameters. The KS statistic is given by

$$max\left\{\underset{1\le i\le n}{max}\left[{\displaystyle \frac{i}{n}}-F\left(t_{\left(i\right)}\right)\right],\underset{1\le i\le n}{max}\left[F\left(t_{\left(i\right)}\right)-{\displaystyle \frac{i-1}{n}}\right]\right\}.$$

5.1. Failure Times of Devices

As the first real dataset, 50 failure times of the devices [1,90] presented in Appendix C are used. ML estimates (MLEs), AIC, BIC, HQIC and KS values are given in

Table 6. MLE, IC and KS values of models fitted to 50 failure times of devices.

LTM	MLE								AIC	BIC	HQIC	KS
	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{d}}$	$\widehat{\mathit{e}}$	$\widehat{\mathit{f}}$	$\widehat{\mathit{g}}$	$\widehat{\mathit{h}}$
WDSFI	63.296	0.591	0.026						448.826	454.562	451.011	0.133
WDSFII	67.308	0.689	1.097	81.999					421.516	429.164	424.428	0.098
EAW	3.660	0.046	0.002	1.571	107.658				469.737	479.297	473.377	0.154
McW	0.023	3.948	0.581	0.113	0.123				453.758	463.318	457.399	0.129
McMW	0.011	0.070	1.400	99.689	0.122	110.808			465.340	476.812	469.709	0.148
McEW	0.005	0.365	0.342	5.639	22.194	67.675			513.666	525.138	518.035	0.246
McGPW	0.004	0.975	3.756	30.574	0.026	43.588			473.988	485.460	478.357	0.269
CW	72.727	4.505	19.101	0.762	0.005	0.545			454.997	466.469	459.366	0.140
KTEMW	15.268	33.904	15.729	0.009	0.002	2.499	2.099		470.777	484.161	475.874	0.146
KTEAW	0.092	1.608	0.0002	0.120	2.290	3.200	0.744	2.197	488.172	503.468	493.997	0.280

. Better values are marked in bold. Figure 9 shows the estimated FDFs for compared models. The WDSF^II model is more appropriate based on information criteria and KS values. The adaptability of the WDSF^II model can also be observed in Figure 9.

5.2. 500 MW Generators

As the second real dataset, 36 times to the first failure of 500 MW generators collected over a 6-year period [90,93], presented in Appendix C, are used. ML estimates (MLE), AIC, BIC, HQIC and KS values are given in

Table 7. MLE, IC and KS values of models fitted to 36 times to the first failure of 500 MW generators.

LTM	MLE								AIC	BIC	HQIC	KS
	$\widehat{\mathit{a}}$	$\widehat{\mathit{b}}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{d}}$	$\widehat{\mathit{e}}$	$\widehat{\mathit{f}}$	$\widehat{\mathit{g}}$	$\widehat{\mathit{h}}$
WDSFI	2615.140	0.725	0.00003						639.431	644.181	641.089	0.100
WDSFII	2567.828	0.731	0.00003	644.846					641.410	647.744	643.621	0.100
EAW	1.858	0.092	0.002	0.766	46.659				644.581	652.498	647.344	0.116
McW	0.016	0.212	64.343	7.5670	32.539				642.722	650.639	645.485	0.114
McMW	0.0007	0.068	3.047	4.772	8.793	0.386			652.555	662.056	655.871	0.175
McEW	0.073	0.371	21.007	0.104	0.126	23.577			645.083	654.584	648.399	0.109
McGPW	0.025	0.676	1.008	3.438	0.535	0.247			646.477	655.978	649.793	0.165
CW	4343.488	8.000	549.686	34.468	55.643	0.935			645.011	654.512	648.327	0.111
KTEMW	0.068	0.248	8.454	0.048	0.001	0.025	0.890		651.727	662.812	655.596	0.114
KTEAW	0.923	1.417	3 $\times {10}^{-6}$	0.096	4.952	0.282	2.850	3.101	650.268	662.936	654.689	0.082

. The WDSF^I model is more appropriate based on information criteria. The adaptability of the WDSF^I model can also be observed in Figure 10.

6. Discussion

The results obtained in this study demonstrate that the proposed WDSF^I and WDSF^II models provide a flexible and accurate framework for modelling full-life-cycle data with a wide range of hazard ratio shapes, including monotonic, bathtub, and pseudobimodal patterns. Parameter estimation using OLS, WLS, and LAW methods yielded consistent results, with the LAW approach often demonstrating improved robustness in the presence of irregularities in the empirical reliability function.

From the perspective of previous research, our findings are consistent with, and even extend, previous work on generalizations of the Weibull distribution and its modified forms. Similar to these studies, we observed that introducing additional parameters can significantly improve model fit, allowing for more complex hazard ratio shapes. However, the WDSF^I and WDSF^II models differ in structure, offering better adaptation to datasets where existing models still exhibit systematic biases.

Regarding our working hypotheses, empirical analyses support the assumption that additional flexibility in scale structure and distribution shape leads to better data representation without sacrificing interpretability. The models’ performance on real-world datasets suggests they may be particularly well-suited for applications in engineering reliability, biomedical survival analysis, and materials degradation studies, where multiphase failure mechanisms are present.

In a broader context, these findings contribute to the growing literature on extended-life models and highlight the continuing need for statistical tools capable of capturing various hazardous behaviours. Such tools are crucial for risk assessment, preventive maintenance planning, and decision-making in safety-critical systems.

Future research directions include te following:

• Extending the proposed models to more precisely process censored, truncated, and interval data.
• Investigation of Bayesian inference methods to incorporate a priori information and quantify uncertainty.
• Development of multivariate or concatenated life-cycle models to capture inter-component dependencies.
• Incorporation of models into regression frameworks, such as proportional hazards or accelerated failure time models, to assess the impact of interdependent variables.

Overall, this study confirms the practical value of the WDSF^I and WDSF^II models and opens promising avenues for both theoretical development and applied research in reliability and survival analysis.

7. Conclusions

This article presents a three- and four-parameter flexible modified Weibull LTM called the Weibull distribution with a linear shape function. An innovative idea is to replace the Weibull shape parameter with a shape function. An estimation method based on theoretical and empirical reliability functions is proposed. An extensive literature review was performed, considering the modalities and shapes of the risk rate function. The simulation study is carried out using ML, LS, WLS and LAW methods. Furthermore, to check the suitability and flexibility, the new LTMs are validated with two real datasets and compared with other bimodal LTMs with a bathtub HRF.

The paper shows that even a three-parameter distribution can compete in data modelling with LTM distributions that have two or even almost three times more parameters.

Regarding further research, there is a lot to be accomplished. It appears that there is only one way to succeed—laborious numerical experiments. Research is to be carried out in two directions. The first is to strive to elongate and flatten the central segment of the hazard rate function. To do so, replace a linear shape function with a nonlinear one. Theoretically, any function that is strictly increasing and convex downward may be useful, but probably to different degrees. The second is to carry out Monte Carlo experiments to assess the accuracy of the parameter estimation.