The Shifted-Exponential Variation Property for the Weibull and Log-Logistic Models

Abstract

In this paper, the recent shifted-exponential variation property which is defined as the ratio of variance to the squared of shifted expectation is investigated for both three-parameter Weibull and log-logistic models. These nonnegative semicontinuous models are widely considered in engineering, economics, hydrology, demography and many other fields. It is shown that the log-logistic distribution corresponds to over-, equi-, and under-varied if and only if its only positive shape parameter is greater, equal and less than the determined value , respectively. Similar result holds for the Weibull distribution with and extends the one of two-parameter model. The Newton–Raphson method is used to determine the approximative value of the log-logistic model; it can thus lead to the reference shifted-exponential model, as for of the Weibull one. The relative variation between Weibull and log-logistic is also mentioned. Finally, two illustrative applications are provided.

INTRODUCTION

The three-parameter Weibull as well as log-logistic distributions can be viewed as flexible and very useful parametric models. They have various important properties compared to many other parametric distributions used in many different fields. These fields include the survival and reliability analysis, economics for the distribution of wealth or income inequality, demography for modeling population growth, hydrology for modeling stream flow rates and precipitation, and many other ones. Among many other works, one can refer to Muse et al. [20], Kokonendji and Sawadogo [15], Silva et al. [23], ter Berg [26], Ashkar and Mahdi [5], Rowinski et al. [22], Singh et al. [24], Ahmad et al. [4], Tanaka and Ichikawa [25], Barlow and Proschan [7], Weibull [27] for some applications, properties and connexions. It is to be noted that the two-parameter log-logistic distribution is also known as the fisk distribution in economics (e.g., Kleiber and Kotz [13], Section 6.4). Furthermore, the log-logistic stands in the same relationship to the logistic model as the Weibull does to the extreme value.

Regarding the model selection (assistance) tool for the preliminary variability properties to discriminate between and within models which are well-known for counting (e.g., Kokonendji and Puig [14]; Abid and Kokonendji [2]), we here consider the more recents for particular continuous models or data (e.g., Abid and Kokonendji [1]; also Ghosh [12], and Ristić et al. [17], for other approaches). For that, let be a nonnegative semicontinuous random variable on . As the analogue of the Fisher or Poisson dispersion index for count model, the so-called Jørgensen or exponential variation index (e.g., Abid et al. [3]; Kokonendji et al. [16]) of is defined by the ratio of two variances:

1

where its denominator corresponds to the variance of the expected exponential random variable. For instance, let be the two-parameter Weibull distribution on with . Then:

2

See Kokonendji and Sawadogo ([15], Theorem 1.1) for the detailed proof of (2) and, also, for its interpretation in reliability with respect to the classical failure rate (or bathtub) curve.

Bourguignon and Kokonendji [9] have recently introduced, among many other indexes, the coherent shifted exponential variation index for nonnegative semicontinuous random variable on with . More precisely, this positive quantity extends the previous (1) by

3

i.e., indicates over- (equi- and under-)varied compared to the -shifted exponential random variable with the same expectation if [ and ], respectively. Notice that in (3) leads to (1). In fact and similar to the count models from zero value, certain authors sometimes and (un)consciously allow themselves to use the unappropriate index (1) for models on the support , , as instead of (3). They could falsely argue that the variability indexes cannot change under the shifted operation if one does not take into account the suitable denominator in (3). Note that the square root of can be viewed as a coherent coefficient of variation of on .

The main goal of this paper is to study the -exponential variation of both three-parameter Weibull and log-logistic distribution in the sense of over-, equi-, and under-variation according to (3). Thus, we produce novel properties. Also, we provide the equivalent of (2) for the three-parameter Weibull distribution. The rest of the paper is laid out as follows. Section 2 first gives the result of the three-parameter Weibull model and then it states the shifted exponential variation property for the three-parameter log-logistic model, including some concluding comments. Section 3 illustrates the shifted-exponential variation property with two applications for both the Weibull and log-logistic models. Section 4 is devoted to the detailed proof of the log-logistic result.

RESULTS AND COMMENTS

The probability density function (pdf) of the shifted Weibull variable with threshold parameter , scale parameter and shape parameter is expressed as

where denotes the indicator function of any given event . In particular, if then denotes the shifted-exponential distribution which is here the reference model of . Recall that the expectation and variance of are provided by

4

respectively, where is the classical gamma function. Thus, its shifted-exponential variation property is presented in the following proposition, which extends (2).

Proposition 1.Let be the three-parameter Weibull distribution on with,, and. Then:

Proof 1. Apply (3), to and from (4), one easily gets

which does not depend on and . Using the relation , , one deduces that

and, therefore,

5

For any fixed , this last expression (5) above is exactly the same obtained in the proof of (2) (see Kokonendji and Sawadogo ([15], Eq. 2.1)) from which the desired result follows, with quite fine analysis of the gamma, digamma, and trigamma functions, and their derivatives.

Concerning the pdf of the three-parameter log-logistic distribution denoted by on where the parameters , , and , respectively, are the location, scale and shape for our investigation, it can be expressed as

see, e.g., Ahmad et al. [4]. At this level, the expectation and variance of are, respectively, given for and by

6

The function stands for the Euler beta function. Its shifted-exponential variation property is stated as follows.

Theorem 2.Letbe the three-parameter log-logistic distribution on with,and. Then, there existssuch that:

with. Furthermore, one approximatively has.

Obviously, our Theorem 2 holds too for the standard two-parameter log-logistic model on as well as Proposition 1 for the corresponding Weibull case in (2). In practice, if an estimation of is equal or close to , then one can consider as .

More generally, we end this section by the relative variation index between Weibull and log-logistic themselves. Under the same supports and same means in (4) and in (6), one can state the following remark without detailed proof.

Remark 3. Let and be, respectively, three-parameter Weibull and log-logistic distributions on with , , , and such that their equality of means corresponds to

7

Then:

8

In fact, for given , there always exist and such that and . From (7), if then and coincide in the sense of equi-variation (8). For , the similarity between Weibull and log-logistic is pure; i.e., it is not in terms of the shifted exponential distribution (such as above). The right inequalities in (8), only depending on and , are directly deduced from (5) and (10) below. Thus, Remark 3 provides the tools for relative variation index.

TWO ILLUSTRATIVE APPLICATIONS

In the literature, there are many data sets that can be used for illustrative purposes. Herein, two real data sets are retained. The first is the Bearing data set from McCool [18] which is concerned with on the fatigue lives (in h) of bearings of a certain kind. They are 152.7, 172.0, 172.5, 173.3, 193.0, 204.7, 216.5, 234.9, 262.6, 422.6. The second one is the Kelvin data set which had been used by Ahmad et al. [4]. This data set comprises annual maximum series between 1948 and 1982 for Killermont Scottish catchment. They are given as follows 98.3, 94.1, 90.1, 105.6, 76.4, 98.3, 128.2, 77.5, 79.9, 69.6, 60.4, 68.5, 78.7, 107.1, 114.9, 80.6, 68.3, 87.2, 91.8, 80.6, 68.3, 91.0, 64.5, 65.8, 53.9, 57.3, 91.0, 76.4, 86.5, 72.3, 73.6, 80.6, 65.1, 77.0, 77.0. Although the two data sets had been used in the context of log-logistic models, we here have considered to fit the Weibull models to them.

Let be a random sample from a semi-continuous random variable with support on with . An empirical estimator of directly derived from (3) is determined by

where , and are the empirical versions of , the theoretical mean and variance of , respectively.

In finite samples, we shall suggest the use of a bootstrapped approach as for Table 8 of Kokonendji et al. [16] for reducing the estimated variances and their corresponding lengths of confidence intervals. We have made use of the boot package (Canty and Ripley [10]) for computing the confidence intervals of . The number of bootstrap replicates is set to

Table 1 contains the bootstrap estimated value of the -variation index for the two data sets. From the results, one notices that the Bearing data set exhibits an overvariation since . Nevertheless, the Kelvin data set presents an undervariation as . Hence, any distribution namely log-logistic or Weibull which fits well to any of the data sets should have its shape parameter being subjected to the conditions given by Proposition 1 or Theorem 2.

Table 1. Bootstrap point estimation of with associated confidence intervals with significance level of using the percentile method and the empirical estimator of ; se stands for standard error

Data set			bias	se
Bearing
Kelvin

In what follows, the log-logistic and Weibull models are considered for the study of these data sets. The models parameters have been estimated by means of the maximum likelihood method. To this end, we used some own codes in conjunction with the nlminb() function and the FAdist package (Aucoin [6]) of R. In all the optimizations, successful convergence is indicated by code 0. The maximum number of iterations is set to and the relative tolerance to . In order to evaluate the goodness-of-fit test that the two data sets come from specified log-logistic and Weibull distributions, we had recourse to the Cramér-Von Mises tests. For this purpose, the goftest package (Faraway et al. [11]) of R had been used. The results of both the log-logistic and Weibull parameters estimated values and the Cramer-Von Mises tests are presented in Table 2.

Table 2. Estimated values of parameters and -values for the Cramér-Von Mises tests for both the log-logistic and Weibull (in parentheses) distributions

Data set				-value
Bearing
Kelvin

Since all the -values are much greater than for the log-logistic distributions, we fail to reject the log-logistic in both cases. However, all the -values of the goodness-of-fit are less than with regards to the Weibull distributions in parenthese of Table 2; thus, the Weibull distributions are rejected for fitting these data sets.

According to Theorem 2, since the Bearing (resp. Kelvin) data set exhibits overvariation (resp. undervariation), the shape parameter of the corresponding log-logistic distributions should be subjected to (resp. ) with . One can notice that although the Bearing and Kelvin data sets do not fit by the Weibull distributions, their corresponding shape parameters are such that and , respectively; and, from Proposition 1, this indicates overvariation for the Bearing data set and undervariation for the Kelvin data set. What is true.

In this paper, we have considered the coherent shifted-exponential variation property introduced by Bourguignon and Kokonendji [9] for the investigation of the 3-parameter log-logistic and 3-parameter Weibull distributions. We stated some results which have been proved analytically and numerically. For illustrative purposes of the stated results, we have considered two real data sets. The results obtained from the data sets are in accordance with the theoretical stated results. For both log-logistic and Weibull models, only the shape parameter controls the -shifted exponential variation property; and, it evolves in the opposite directions compared to the reference value according to these two models. See again Proposition 1, Theorem 2, Tables 1 and 2.

PROOF OF THEOREM 2

Applying (3) to and considering (6), one successively has

for which the last equality above does not depend on and . Now, we use the well-known relationship between the beta and gamma functions to obtain the below expression of in terms of parameter and gamma function:

9

Figure 1 represents the graph of in (9) as a function of which visually guarantees the announced behaviour in the theorem.

[See PDF for image]

Fig. 1

Graph of in (9).

Let us recall, without proof, the following key result we need.

Lemma 4 (e.g., Moll [12], Corollary 16.5.7).For, the reflection formula

holds.

Using Lemma 4 in (9) it successively follows that, for all with ,

Thus, for all , and , on can write

10

For fixed and , the right inequalities of (10) are equivalent to

11

Setting , Inequalities (11) becomes

Now, let us introduce the function defined by

12

Figure 2 depicts the look of in (12) and of its first derivative on which visually allow to follow the two below lemmas.

[See PDF for image]

Fig. 2

Graphs of both functions (in full line) and (in dotted line) from (12).

The following lemma shows some asymptotic behaviours of .

Lemma 5.The real-valued functiondefined onviasatisfies:

1. is continuous on,

2. ,

3. , which means that the line of the equationindicates a vertical asymptote of.

Proof 2.Trivial. In what follows, we show through the lemma below that there exits a unique value such that the function defined on in (12) is negative on , null at and positive on , respectively.

Lemma 6.The continuous functioninis also differentiable on. Its first derivative given byverifies:

1. with,

2. and.

Proof 3.Easy. From both previous Lemmas 5 and 6 and in order to determine the unique value such that , one considers the Newton–Raphson method, also known as a tangent method, to find its good approximation. To this end, the pracma package (Borchers, 2023) of the R software (R Core Team, 2023) had been used. The maximum number of iterations is set to and the tolerance to . The result yields in four iterations and, also, one retains . Thus, from (11) and (12), a good approximation of satisfying is given by . This concludes the proof of the theorem.

ACKNOWLEDGMENTS

We sincerely thank the two anonymous referees and the associate editor for their valuable comments that significantly improved the article.

FUNDING

CNPq Grants (CNPq grant no. 304140/2021-0): Professor Marcelo Bourguignon; this work has been done in the frame of EIPHI Graduate School (contract ANR-17-EURE-0002) and supported by Bourgogne-Franche-Comté Region and, also, Brazilian-French Network in mathematics: Professor Célestin C. Kokonendji.

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.

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