# Mathematical properties, application and simulation for the exponentiated generalized standardized Gumbel distribution.

Introduction

The statistical literature is replete with illustrations in which the Gumbel model is used effectively to explain real phenomena. In the area of climate modeling, for example, some applications of the Gumbel model include global warming problems, offshore modeling, and rainfall and wind speed modeling (Nadarajah, 2006). Here, it is worth mentioning some recent works that consider the Gumbel distribution in different contexts: Nadarajah (2006), Cordeiro, Ortega, and Cunha (2013) and Andrade, Rodrigues, Bourguignon, and Cordeiro (2015). The cumulative distribution function (cdf) of the Gumbel (Gu) distribution is given by Equation 1:

[mathematical expression not reproducible], (1)

for t [member of] IR, [mu] [member of] IR and [sigma]>0. In a recent paper, Cordeiro et al. (2013) proposed a generalization for the Gumbel model using the so-called exponentiated generalized (EG) class of distributions defined by the cdf expressed as Equation 2:

F(x) = [{1 - [[1 - G(x)].sup.a]}.sup.b], (2)

where a > 0 and b > 0.

The probability density function (pdf) corresponding to Equation 2 is given by Equation 3:

f(x) = ab[[1 - G(x)].sup.a-1][{1 - [[1 - G(x)].sup.a]}.sup.b-1] g(x), (3)

where g(x) = dG(x)/dx is the baseline pdf. Thus, Cordeiro et al. (2013) studied the so-called exponentiated generalized Gumbel (EGGu, for short) distribution by inserting Equation 1 into Equation 2. Later, Andrade et al. (2015) investigated in detail several mathematical properties for the EGGu model. In this paper, we will follow the methodology used by Cordeiro et al. (2013) and Andrade et al. (2015), but we will consider a standardized version of the Gumbel distribution, the so-called standardized Gumbel (SGu) model. Let T be a random variable having cdf Equation 1. The SGu distribution is defined by a linear transformation X = (T - [mu])/[sigma], where [mu] [member of] IR and [sigma]>0. Without loss of generality, we can work with the SGu model, since T = [mu] + [sigma]X, where [mu] [member of] IR and [sigma]>0. The G(x) and pdf g(x) of the SGu distribution are given by Equation 4 and 5:

G(x) = exp[- exp(-x)] (4)

g(x) = exp[-x - exp(-x)] (5)

respectively, for x [member of] IR. The goal is to show that our simplified version of the EGGu model has great power of fit to real data and good simulation properties, with the advantage of having only two parameters. Therefore, we define the exponentiated generalized standardized Gumbel (EGSGu) distribution by inserting Equation 4 into Equation 2. The cdf F(x) and pdf f(x) of the EGSGu distribution are given by Equation 6 and 7:

F(x) = [(1 -[{1 - exp[-exp(-x)]}.sup.a]).sup.b] (6)

[mathematical expression not reproducible], (7)

where a > 0 and b > 0. The two extra parameters a and b in the density Equation 7 can control both tail weights, enabling the generation of flexible distributions, with heavier or lighter tails, as appropriate. There is also an attractive physical interpretation of the model Equation 7 when a and b are positive integers: Suppose initially that a certain device is composed of b components in a parallel system. Consider also that, for each component b, there exists a series of subcomponents a independent and distributed according to the SGu model. Suppose also that each component b fails if some a subcomponent fails. Let [X.sub.j1], ..., [X.sub.ja] denote the lifetimes of the subcomponents within the jth component, j = 1, ..., b with common cdf SGu. Let [X.sub.j] denote the lifetime of the j component and let X denote the lifetime of the device. Hence:

[mathematical expression not reproducible].

Thus, the lifetime of the device obeys the EGSGu family of distributions. Besides this introduction, the paper is organized as follows. In the Material and Methods section, we investigated the quantile function and its applications. Next, several mathematical properties of the new model are derived and numerical studies are detailed. In the Results and Discussion section, we used a real dataset to empirically show the power of fit of the EGSGu model and presented the Monte Carlo simulation study.

Material and methods

Quantile function

As an additional characterization of X, we define in this section the quantile function (qf) of the EGSGu model. This function comes directly from the inversion of the cdf Equation 6. Thus, the qf of X is given by Equation 8:

Q(u) = -log {-log [1 -[(1 - [u.sup.1/b]).sup.1/a]]}, (8)

where u [member of] (0,1). There are many important practical applications for Equation 8. For example, occurrences of the random variable X can be easily obtained from a uniform random variable U by X = Q(U). Next, we use Equation 8 to simulate 200 EGSGu (3, 2) occurrences. Figure 1 gives the EGSGu (3, 2) pdf, histogram, exact and empirical cdfs for these simulated data.

In addition, to illustrate the practical utility of Equation 8, it should be mentioned that it can be used to determine the median of X as Med = Q(1/2). Table 1 below presents a small simulation study, whose objective is to compare the empirical medians (Med) generated for different parameter values and random samples of size n = 50, 100, 150, with their corresponding theoretical medians (Med) obtained by Q(1/2). The simulation process is performed in the software R and, to ensure the reproducibility of the experiment, we use the seed for the random number generator: set.seed (103). As expected, the difference between EMed and Med decreases when n increases.

Finally, we present a third application for qf Equation 8, which consists in obtaining in the classical measures of asymmetry and kurtosis of X the Bowley skewness (Kenney & Keeping, 1962) (B) and Moors kurtosis (Moors, 1988) (M). In Figure 2 and 3, we present 3D plots of the M and B measures for selected parameter values, respectively. These plots are obtained using the software 'Wolfram Mathematica'. Based on these plots, it is possible to conclude that changes in the additional parameters a and b have a considerable impact on the skewness and kurtosis of the EGSGu model, thus corroborating its greater flexibility. Hence, theses plots reinforce the importance of the additional parameters.

Properties of the EGSGu distribution

In this section, we study the structural properties of the EGSGu model.

Linear representations

For an arbitrary baseline cdf G(x), a random variable [Y.sub.c] has the exp-G distribution with power parameter c > 0 say [Y.sub.c] ~ exp-G if its cdf and pdf are given by [H.sub.c](x) = G[(x).sup.c] and [h.sub.c](x) = cg(x)G[(x).sup.c-1] respectively. For a comprehensive discussion about the exponentiated distributions, see a recent paper by Tahir and Nadarajah (2015). Based on some results in Cordeiro and Lemonte (2014), we can express the EG cdf Equation 2 as Equation 9.

F(x) = [[infinity].summation over (j=0)][w.sub.j+1][H.sub.j+1](x), (9)

where [mathematical expression not reproducible] and [H.sub.j+1](x) = G[(x).sup.j+1] is the exp-G cdf with power parameter j+1. By differentiating Equation 9, we obtain a similar linear representation for f(x) as Equation 10.

f(x) = [[infinity].summation over (j=0)][w.sub.j+1][h.sub.j+1](x), (10)

where [h.sub.j+1](x) = d[H.sub.j+1] (x)/dx. The expSGu pdf with power parameter j+1, [h.sub.j+1](x), (for j [greater than or equal to] 0) becomes Equation 11:

[h.sub.j+1](x) = (j + 1)exp[-x - (j + 1) exp(-x)]. (11)

Combining Equation 10 and 11, we have an important result: The EGSGu density function is a linear combination of expSGu densities. This result can be used to derive some mathematical properties of X.

Moments and probability weighted moments

We provide below two ways to compute the n-th moments of X with density Equation 7. Moreover, we go beyond and also present alternative ways to calculate the probability weighted moments, say PWM and denoted by [[tau].sub.s,r], for EGSGu model. The first formula for the n-th moments of X become by using Equation 7, follow Equation 12:

[mathematical expression not reproducible]. (12)

Alternatively, combining Equation 10 and 11, we can express E([X.sup.n]) in terms of expSGu moments. We write Equation 13:

[mathematical expression not reproducible]. (13)

It is very simple to calculate the n-th moment of X computationally, using the expressions Equation 12 and 13. To illustrate it, we provide next a small numerical study, comparing E([X.sup.n]) from both formulas. We consider several parameter values and n = 1, 2, 3, 4, 5, 6. The results are shown in Table 2. This table shows that the results agree at five decimal digits of precision for both methods. All computations are obtained using the 'Wolfram Mathematica' platform. The (s, r)th PWM of X is defined by [[delta].sub.s,r] = E[[X.sup.s] f[(x).sup.r]]. Clearly, the ordinary moments follow as [[delta].sub.s,0] = E([X.sup.s]). Next, we derive simple expressions for the PWMs of X defined by Equation 14:

[mathematical expression not reproducible]. (14)

Inserting Equation 6 and 7 into Equation 14, the PWMs of X can be expressed in a simple form Equation 15:

[mathematical expression not reproducible]. (15)

Under simple algebraic manipulation, we can write [[delta].sub.s,r] as Equation 16:

[mathematical expression not reproducible], (16)

where [h.sub.l+1](x) is the expSGu density with parameter (l+1). Equation 16 reveals that the PWMs of X can be expressed in terms of the ordinary moments of X ~ expSGu(l+1).

Table 3 gives from Equation 15 the values of [[delta].sub.s,r] for X ~ EGSGu(a, b) and some values of s and r. All computations are performed using the 'Wolfram Mathematica Software'. Based on the values in Table 3, we conclude that, for fixed r, the PWMs increase when s increases. The opposite happens when we fix parameter s and r increases.

Stochastic comparisons of sample maximum

In this section, we compare the maximum of two independent and heterogeneous samples each following EG class with the same baseline distributions. Let [X.sub.1:n] [less than or equal to] [X.sub.2:n] [less than or equal to] ... [less than or equal to] [X.sub.n:n] be the order statistics to the random variables [X.sub.1], [X.sub.2], ..., [X.sub.n] with [X.sub.1:n] and [X.sub.n:n] the sample minimum and sample maximum, respectively. The study of Parallel systems plays a prominent role in reliability theory and is equivalent to the study of the largest order statistics. The following two definitions and notation will be needed to prove our main result.

Definition 1: (Shaked & Shanthikumar, 2007): Let Y and Z be two random variables with, respectively, absolutely continuous distribution functions M(*) and H(*) densities m(*) and h(*), and reversed hazard rate functions [??](*) and [??](*). Hence, Y is said to be smaller than Z in (i) 'usual stochastic order', denoted by Y [[less than or equal to].sub.st] Z, if M(t) [greater than or equal to] H(t), [for all]t; (ii) 'reversed hazard rate order', denoted by [mathematical expression not reproducible].

We consider in our main result stochastic comparison in terms of the reversed hazard rate order. Note that reversed hazard rate order implies usual stochastic order. Let [I.sup.n] be an n-dimensional Euclidean space with I [subset or equal to] IR.

Definition 2: (Marshall, Olkin, & Arnold, 2011): Let x = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and y = ([y.sub.1], [y.sub.2], ..., [y.sub.n]) be two vectors in [I.sup.n]. The vector x is said:

(i) to majorize the vector [mathematical expression not reproducible] if [mathematical expression not reproducible], and [mathematical expression not reproducible];

(ii) to weakly supermajorize the vector [mathematical expression not reproducible] if [mathematical expression not reproducible], for j = 1, 2, ..., n;

(iii) to weakly submajorize the vector y (say, x [[+ or -].sub.w] y) if [mathematical expression not reproducible], for j = 1, 2, ..., n.

Notation: Let us include the following notations.

(i) D = {([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] I[R.sup.n]: [x.sub.1] [greater than or equal to] [x.sub.2] [greater than or equal to] ... [greater than or equal to] [x.sub.n]};

(ii) E = {([x.sub.1], [x.sub.2], ..., [x.sub.n]) [member of] I[R.sup.n]: [x.sub.1] [less than or equal to] [x.sub.2] [less than or equal to] ... [less than or equal to] [x.sub.n]};

(iii) [D.sub.+] = {([x.sub.1], ..., [x.sub.n]) [member of] I[R.sup.n]: [x.sub.1] [greater than or equal to] ... [greater than or equal to] [x.sub.n] > 0};

(iv) [E.sub.+] = {([x.sub.1], [x.sub.2] ..., [x.sub.n]) [member of] I[R.sup.n]: 0 < [x.sub.1] [less than or equal to] ... [less than or equal to] [x.sub.n]}.

Consider X a random variable with continuous distribution function G(x) and pdf g(x). For I = 1, 2, ..., n, consider also [X.sub.i] ~ EG([a.sub.i], [b.sub.i], G) and [Y.sub.i] ~ EG([c.sub.i], [d.sub.i], G) two sets of n independent random variables where the baseline G(x) is homogenous and common to both sets of random variables. Suppose that [F.sub.n:n](*) and [H.sub.n:n](*) are the cdfs of [X.sub.n:n] and [Y.sub.n:n], respectively. Hence, [mathematical expression not reproducible].

We can also notice that if [[??].sub.n:n](*) and [[??].sub.n:n](*) are, respectively, reversed hazard rate functions of [X.sub.n:n] and [Y.sub.n:n], then [mathematical expression not reproducible] and [mathematical expression not reproducible].

The following lemmas will be needed to prove our main result.

Lemma 1 (Marshall et al., 2011, p. 86): Let [phi]:I [right arrow] IR. Then, ([a.sub.1], [a.sub.2], ..., [a.sub.n]) [[+ or -].sub.w] ([b.sub.1], [b.sub.2], ..., [b.sub.n]) implies [phi]([a.sub.1], [a.sub.2], ..., [a.sub.n]) [greater than or equal to] [phi]([b.sub.1], [b.sub.2], ..., [b.sub.n]) if, and only if, [phi] is decreasing and Schur-convex on [I.sup.n].

Lemma 2 (Kundu, Chowdhury, Nanda, & Hazra, 2016): Let [phi](x) = [n.summation over (i=1)][u.sub.i]g([x.sub.i]) with x [member of] D and let I [subset or equal to] IR be an interval. Consider a function g:I [right arrow] IR. If u=([u.sub.1], [u.sub.2], ..., [u.sub.n]) [member of] [E.sub.+] and g(*) is decreasing and convex then [phi](x) is Schur-convex on D.

Consider the following vectors belonging to [I.sup.n]: a = ([a.sub.1], [a.sub.2], ..., [a.sub.n]), b = ([b.sub.1], [b.sub.2], ..., [b.sub.n]), and c = ([c.sub.1], [c.sub.2], ..., [c.sub.n]).

Theorem (Main result of the section): Suppose [X.sub.i] ~ EG([a.sub.i], [b.sub.i], G) and [Y.sub.i] ~ EG([c.sub.i], [d.sub.i], G) with [X.sub.i] and [Y.sub.i] two sets of mutually independent random variables and I = 1, 2, ..., n. Also suppose that a, c [member of] [D.sub.+] and b [member of] [E.sub.+] Then, [mathematical expression not reproducible] implies [X.sub.n:n] [[greater than or equal to].sub.rhr] [Y.sub.n:n].

Proof. Let [bar.G](x) = 1 - G(x). Differentiating [mathematical expression not reproducible] partially with respect to [a.sub.i], we have

[mathematical expression not reproducible].

We note that log [mathematical expression not reproducible] since log x [less than or equal to] x - 1, for all x > 0. Hence, v'([a.sub.i]) [less than or equal to] 0 and v([a.sub.i]) is decreasing in [a.sub.i]. By differentiating again v'([a.sub.i]) with respect to [a.sub.i], we obtain [mathematical expression not reproducible], with [mathematical expression not reproducible]. Taking the derivative of [v.sub.1]([a.sub.i]) with respect to [a.sub.i], we obtain [v'.sub.1]([a.sub.i]) = [v.sub.2]([a.sub.i])log [bar.G](x), where [mathematical expression not reproducible]. Finally, by differentiating [v.sub.2]([a.sub.i]), also with respect to [a.sub.i], we have [mathematical expression not reproducible].

It is easy to see, for [a.sub.i] > 0, the following chain of implications, [mathematical expression not reproducible] decreasing in [a.sub.i] [mathematical expression not reproducible] convex in [a.sub.i].

Thus, by Lemma 1 and Lemma 2, the proof is obvious.

Considering that the theorem holds for any continuous baseline G we naturally have the following corollary.

Corollary (Result applied to EGSGu distribution): Suppose [X.sub.i] ~ EGSGu([a.sub.i], [b.sub.i]) and [Y.sub.i] ~ EGSGu([c.sub.i], [b.sub.i]) with [X.sub.i] and [Y.sub.i] two sets of mutually independent random variables and i = 1, 2, ..., n. Also suppose that a, c [member of] [D.sub.+] and b [member of] [E.sub.+] and Then, [mathematical expression not reproducible] implies [X.sub.n:n] [[greater than or equal to].sub.rhr] [Y.sub.n:n].

Dual generalized order statistics

The dual generalized order statistics (dgos) were introduced in Burkschat, Cramer, and Kamps (2003) as a model for descending ordered random variables and admits as special cases reversed ordered order statistics, lower k-records and lower Pfeifer records (Arnold, Balakrishnan, & Nagaraja, 1998). In this section, we present general expressions for dgos from the EG class. Next, we present results for the EGSGu distribution. We derive an explicit expression for the density of the mth (1 [less than or equal to] m [less than or equal to] n) dual generalized order statistic [X.sup.*](m, n, v, k), say [mathematical expression not reproducible], in a random sample of size n from the EG class. By definition we have Equation 17:

[mathematical expression not reproducible] (17)

where

[mathematical expression not reproducible] and [mathematical expression not reproducible]. According to Equation 17, we can rewrite it in two cases, follow Equation 18:

[mathematical expression not reproducible]. (18)

Case I: For v [not equal to] -1.

Using the binomial expansion in the first sentence of Equation 18 and inserting cdf Equation 2 and pdf Equation 3, we readily obtain Equation 19:

[mathematical expression not reproducible]. (19)

For any real non-integer [beta], ([absolute value of z]<1), Equation 20:

[mathematical expression not reproducible]. (20)

By applying the last equation twice in Equation 19 and after simple algebraic manipulation, we write [mathematical expression not reproducible] as Equation 21:

[mathematical expression not reproducible], (21)

where

[mathematical expression not reproducible].

Case II: For v = -1.

By expanding the logarithm function in power series and then using an equation for a power series raised to a positive integer given in Gradshteyn and Ryzhik (2007) (Section 0.314), we have Equation 22,

[mathematical expression not reproducible], (22)

where [bar.F](x) = 1 - F(x) and the coefficients [c.sub.m-1,p] (for p = 1, 2, ...) are determined from the recurrence equation [mathematical expression not reproducible], with [c.sub.m-1,0] = [a.sup.m-1.sub.0]. From Equation 22, the second sentence of Equation 18 reduces to [mathematical expression not reproducible].

Analogously, inserting Equation 2 and 3 in the previous equation and applying Equation 20 twice, the pdf [mathematical expression not reproducible] can be expressed as Equation 23,

[mathematical expression not reproducible], (23)

where

[mathematical expression not reproducible].

Hence, cases I (Equation 21) and II (Equation 23) can be summarized as (for baseline G distribution) Equation 24:

[mathematical expression not reproducible]. (24)

Based on the previous equation, we can easily obtain the dgos of EGSGu distribution since it depends on exp-G densities. Furthermore, several mathematical quantities of the EG dgos can be obtained from those quantities of exp-G distributions. We provide an example for the dgos moments from EGSGu distribution. Before that, an additional comment on the moments of the expSGu is in order. Inspired by Andrade et al. (2015), we can find a formula for the tth moment of [Z.sub.c] ~expSGu with power parameter c > 0. By setting w = exp{-z}, and using Andrade et al. (2015)'s Equation 14, the tth moment of [Z.sub.c] ~expSGu can be written as Equation 25:

[mathematical expression not reproducible]. (25)

Thus, the tth moment dgos of the EGSGu distribution can be expressed from Equation 24 and 25 as:

[mathematical expression not reproducible].

It is well known that if v = 0, k = 1 then [X.sup.*](m, n, v, k) reduces to the (n-m+1)th order statistics [X.sub.n-m+1 : n] from the sample [X.sub.1], ..., [X.sub.n], and when v = -1, then [X.sup.*](m, n, v, k) reduces to the mth lower k-record value. The main result of this section is given by Equation 24.

Results and discussion

Real data illustration

We adjusted the EGSGu model given in Equation 7, which contains just two parameters, and compared the results with other important models in the literature. We considered the EGSGu distribution and two Gumbel Lehmann's alternatives sub-models, denoted by EISGu and EIISGu, respectively. In addition, we have adjusted the models proposed by Mahmoudi (2011) (BGP distribution), Nadarajah and Eljabri (2013) (KwGP distribution), and Silva, Ortega, and Cordeiro (2010) (BMW distribution). Their respective densities are given by

[mathematical expression not reproducible]

and [mathematical expression not reproducible],

where u = [[1 + [xi](x - t)/[sigma]].sup.-1/[xi]] and [mathematical expression not reproducible] is the beta function.

The data set was obtained from Murthy, Xie, and Jiang (2004), and consists of the times between failures for repairable items. In Table 4 we provide the MLEs (and their standard errors in parentheses) for all fitted models.

Table 4 also lists the values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and corrected Akaike information criterion (Caic) statistics. In general, it is considered that lower values of these criteria indicate better fit to the data. With the exception of the Caic of the EISGu model, the figures in Table 4 reveal that the EGSGu model has the lowest AIC, BIC and Caic values among all fitted models. Thus, the proposed EGSGu distribution is the best model to explain these data. Plots of the estimated pdf and cdf of the EGSGu distribution and the histogram of the data are displayed in Figure 4. These plots clearly reveal that the EGSGu model fits the data adequately and then it can be chosen for modeling these data.

Simulated data illustration

We investigated, by means of a simulation study, the behavior of the MLEs for the parameters of the EGSGu model by generating from qf Equation 8 with samples sizes n = 50, 100, 150, 200 and selected values for a and b. The simulation process is based on 10,000 Monte Carlo replications, performed in the software R using the simulated-annealing (SANN) maximization method in the maxLik script. The results of these new simulations are presented in Table 5 and 6, which contain the estimates and their estimated asymptotic variances in parentheses. These results reveal that, for all estimates, in general, the biases and variances decrease as the sample size increases.

Conclusion

In this article, we propose and study a new two-parameter distribution with real support called the exponentiated generalized standardized Gumbel distribution (EGSGu). Our proposal includes both Lehmann's type I and II transformations as special cases. We study some mathematical properties of the new model. The model's parameters are estimated by the maximum likelihood method. A simulation study reveals that the estimators have desirable properties. We empirically prove that the new distribution provides a better fit to a real dataset than other competitive models.

Doi: 10.4025/actascitechnol.v41i1.39867

Received on October 1, 2017.

Accepted on December 11, 2017.

Acknowledgements

We thank two anonymous referees and the associate editor for their valuable suggestions, which certainly contributed to the improvement of this paper. Additionally, Thiago A. N. de Andrade is grateful for the financial support from Capes (Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior).

References

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Murthy, D., Xie, M., & Jiang, R. (2004). Weibull models. Wiley series in probability and statistics. New Jersey, NJ: John Wiley and Sons.

Nadarajah, S. (2006). The exponentiated Gumbel distribution with climate application. Environmetrics, 17(1), 13-23. doi: 10.1002/env.739

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Silva, O. G., Ortega, E. M. M., & Cordeiro, G. (2010). The beta modified Weibull distribution. Lifetime Data Anal, 16(3), 409-430. doi: 10.1007/s10985-010-9161-1

Tahir, M. H., & Nadarajah, S. (2015). Parameter induction in continuous univariate distributions: Well-established G families. Anais da Academia Brasileira de Ciencias, 87(2), 539-568. doi: 10.1590/0001-3765201520140299

Thiago Alexandro Nascimento de Andrade (1), Frank Gomes-Silva (2) * and Luz Milena Zea (3)

(1) Departamento de Estatistica, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil. (2) Departamento de Estatistica e Informatica, Universidade Federal Rural de Pernambuco, Avenida Professor Moraes Rego, 1235, 50670-901, Recife, Pernambuco, Brazil. (3) Departamento de Estatistica, Universidade Federal do Rio Grande do Norte, Natal, Rio Grande do Norte, Brazil. * Author for correspondence. E-mail: franksinatrags@gmail.com

Caption: Figure 1. Plots of the EGSGu (3, 2) pdf, histogram, exact and empirical cdfs for simulated data with n = 200.

Caption: Figure 2. Plots of the Moors kurtosis for the EGSGu distribution.

Caption: Figure 3. Plots of the Bowley skewness for the EGSGu distribution.

Caption: Figure 4. Estimated pdf and cdf of the EGSGu model for the times between failures for repairable items.
```Table 1. Theoretical and empirical medians (for n = 50, 100, 150)
of X for some parameter values.

a       b        Med         EMed [right arrow]     n = 100
[down arrow]           n = 50

1.5    3.0      0.8452              0.7252           0.8830
3.0    3.0      0.1123              0.0342           0.1367
3.0    1.5      -0.2355            -0.3218          -0.2087

a      n = 150

1.5     0.8620
3.0     0.1232
3.0    -0.2236

Table 2. First sixth moments of X for several a and b values.

a    b     E(X)      E([X.sup.2])    E([X.sup.3])    E([X.sup.4])

1    1    0.57722       1.97811         5.44487         23.5615
2    1.27036       3.25876         10.7232         45.3600
3    1.67583       4.45333         15.3804         66.3318
4    1.96351       5.50031         19.6637         86.4083

2    1   -0.11593       0.69747         0.16651         1.76299
2    0.34165       0.72200         1.03513         2.52093
3    0.59545       0.92435         1.59047         3.63454
4    0.77030       1.14191         2.07825         4.75215

3    1   -0.40361       0.61140        -0.45472         0.93635
2    0.02993       0.37204         0.10596         0.51195
3    0.17170       0.36710         0.31147         0.60373
4    0.30846       0.41307         0.45178         0.75980

4    1   -0.57351       0.67294        -0.70212         1.00506
2   -0.24490       0.33249        -0.14312         0.29771
3   -0.07069       0.24739         0.01646         0.20673
4    0.04630       0.22709         0.09714         0.20835

a    b   E([X.sup.5])    E([X.sup.6])

1    1      117.839         715.067
2      233.071         1418.73
3      345.015         2113.31
4      454.276         2798.98

2    1      2.60769         11.4045
2      6.50153         20.6720
3      9.74663         30.7523
4      12.8901         40.7867

3    1     -0.68052         2.31902
2      0.57386         1.68848
3      1.02392         2.32992
4      1.38992         3.06458

4    1     -1.28616         2.13690
2     -0.09027         0.46581
3      0.12820         0.38704
4      0.23144         0.45403

Table 3. The PWM of X for fixed r = 1 and several values of a, b
and s.

a    b    [[delta].sub.1,1]     [[delta].sub.2,1]

1    1         0.63518               1.62938
2         0.98176               2.75015
3         1.18449               3.62849
4         1.32833               4.35138

2    1         0.17082               0.36100
2         0.38515               0.57095
3         0.50519               0.77208
4         0.58839               0.94653

3    1        -0.01497               0.18602
2         0.15423               0.20654
3         0.24681               0.26923
4         0.31014               0.33295

4    1        -0.12245               0.16625
2         0.02315               0.11354
3         0.10163               0.12300
4         0.15486               0.14437

a    b    [[delta].sub.3,1]     [[delta].sub.4,1]

1    1         5.36162               22.6800
2         9.83185               43.2041
3         13.6947               62.1381
4         17.1322               79.8149

2    1         0.51756               1.26046
2         1.03912               2.37608
3         1.48086               3.45325
4         1.88113               4.47512

3    1         0.05298               0.25598
2         0.22589               0.37990
3         0.33986               0.54915
4         0.44066               0.71657

4    1        -0.07156               0.14885
2         0.04857               0.10418
3         0.09790               0.13451
4         0.13469               0.17284

a    b    [[delta].sub.5,1]     [[delta].sub.6,1]

1    1         116.536               709.365
2         227.138               1399.49
3         332.862               2072.90
4         434.481               2731.50

2    1         3.25077               10.3360
2         6.44506               20.3933
3         9.50317               30.2954
4         12.4662               40.0248

3    1         0.28693               0.84429
2         0.69496               1.53229
3         1.03731               2.27785
4         1.36863               3.01745

4    1        -0.04514               0.23290
2         0.11572               0.22702
3         0.18804               0.32561
4         0.25175               0.43164

Table 4. MLEs (and the corresponding standard errors in
parentheses), AIC, BIC and CAIC statistics for number of successive
failures for the air conditioning system.

Distributions          [??]                [??]

EGSGu             1.340/ (0.247)      4.800/ (1.670)
EISGu                1/ (----)        3.118/ (0.569)
EIISGu            0.589/ (0.108)        1/ (----)
[??]                [??]
KwGP              1.970/ (0.837)     14.432/ (12.687)
BGP               1.979/ (0.460)     13.850/ (1.983)
[??]                [??]
BMW               2.901/ (1.903)      0.344/ (0.168)

Distributions

EGSGu
EISGu
EIISGu
[??]               [??]
KwGP              1.119/ (2.371)     4.972/ (4.870)
BGP               0.156/ (1.960)     11.061/ (3.852)
[??]               [??]
BMW               2.374/ (0.164)     0.854/ (0.210)

Distributions                        AIC      BIC     CAIC

EGSGu                               86.181   88.983   86.826
EISGu                               86.334   87.735   86.477
EIISGu                              103.73   105.13   103.87

KwGP                                87.283   92.888   88.883
BGP                                 87.257   92.862   88.857
[??]
BMW               0.093/ (0.109)    89.239   96.245   91.739

Table 5. MLEs for several a and b parameter values (variances
in parentheses).

n = 50                 n = 100

a     b     [??]        [??]        [??]        [??]

1     1    1.0521      1.0601      1.0262      1.0287
(0.040)    (0.0447)    (0.0185)    (0.0189)
1     2    1.0416      2.1546      1.0210      2.0733
(0.0280)    (0.2526)    (0.0131)    (0.1029)
1     3    1.0380      3.2710      1.0191      3.1277
(0.0240)    (0.7163)    (0.0113)    (0.2837)
1     4    1.0361      4.4045      1.0182      4.1898
(0.0219)    (1.5130)    (0.0103)    (0.5873)
1     5    1.0350      5.5516      1.0176      5.2585
(0.0206)    (2.6931)    (0.0097)    (1.0349)
2     1    2.1041      1.0601      2.0524      1.0287
(0.1599)    (0.0446)    (0.0739)    (0.0189)
2     2    2.0832      2.1545      2.0419      2.0732
(0.1122)    (0.2526)    (0.0526)    (0.1028)
2     3    2.0760      3.2711      2.0382      3.1277
(0.0960)    (0.7158)    (0.0451)    (0.2837)
2     4    2.0723      4.4044      2.0363      4.1900
(0.0875)    (1.5131)    (0.0412)    (0.5872)
2     5    2.0699      5.5510      2.0352      5.2581
(0.0821)    (2.6869)    (0.0387)    (1.0351)
3     1    3.1562      1.0601      3.0786      1.0287
(0.3598)    (0.0447)    (0.1663)    (0.0189)
3     2    3.1248      2.1546      3.0628      2.0732
(0.2525)    (0.2527)    (0.1182)    (0.1028)
3     3    3.1141      3.2710      3.0573      3.1277
(0.2159)    (0.7162)    (0.1014)    (0.2837)
3     4    3.1085      4.4044      3.0545      4.1898
(0.1968)    (1.5079)    (0.0927)    (0.5872)
3     5    3.1047      5.5492      3.0527      5.2580
(0.1846)    (2.6673)    (0.0871)    (1.0340)
4     1    4.2082      1.0601      4.1048      1.0287
(0.6396)    (0.0446)    (0.2956)    (0.0189)
4     2    4.1663      2.1544      4.0838      2.0732
(0.0487)    (0.2527)    (0.2101)    (0.1028)
4     3    4.1521      3.2711      4.0765      3.1277
(0.3839)    (0.7160)    (0.1804)    (0.2838)
4     4    4.1447      4.4046      4.0727      4.1899
(0.3502)    (1.5122)    (0.1647)    (0.5872)
4     5    4.1398      5.5499      4.0703      5.2581
(0.3283)    (2.6737)    (0.1549)    (1.0347)

n = 150                n = 200

a       [??]        [??]        [??]        [??]

1      1.0170      1.0192      1.0131      1.0145
(0.0121)    (0.0122)    (0.0089)    (0.0091)
1      1.0135      2.0487      1.0105      2.0368
(0.0087)    (0.0661)    (0.0064)    (0.0484)
1      1.0123      3.0845      1.0096      3.0640
(0.0075)    (0.1811)    (0.0055)    (0.1314)
1      1.0117      4.1254      1.0091      4.0947
(0.0068)    (0.3728)    (0.0050)    (0.2687)
1      1.0114      5.1705      1.0088      5.1284
(0.0064)    (0.6539)    (0.0047)    (0.4695)
2      2.0340      1.0193      2.0262      1.0146
(0.0485)    (0.0122)    (0.0355)    (0.0091)
2      2.0271      2.0488      2.0210      2.0368
(0.0348)    (0.0661)    (0.0255)    (0.0484)
2      2.0247      3.0847      2.0192      3.0640
(0.0299)    (0.1812)    (0.0219)    (0.1314)
2      2.0234      4.1254      2.0182      4.0947
(0.0273)    (0.3726)    (0.0200)    (0.2685)
2      2.0227      5.1702      2.0176      5.1282
(0.0257)    (0.6539)    (0.0189)    (0.4691)
3      3.0509      1.0192      3.0392      1.0145
(0.1093)    (0.0122)    (0.0799)    (0.0091)
3      3.0405      2.0487      3.0315      2.0369
(0.0782)    (0.0661)    (0.0573)    (0.0484)
3      3.0371      3.0847      3.0287      3.0639
(0.0673)    (0.1811)    (0.0493)    (0.1314)
3      3.0352      4.1255      3.0273      4.0947
(0.0615)    (0.3726)    (0.0450)    (0.2686)
3      3.0341      5.1702      3.0265      5.1285
(0.0579)    (0.6537)    (0.0424)    (0.4689)
4      4.0679      1.0192      4.0523      1.0146
(0.1945)    (0.0122)    (0.1421)    (0.0091)
4      4.0541      2.0487      4.0419      2.0368
(0.1390)    (0.0661)    (0.1019)    (0.0484)
4      4.0495      3.0848      4.0383      3.0640
(0.1195)    (0.1811)    (0.0876)    (0.1314)
4      4.0470      4.1255      4.0365      4.0946
(0.1092)    (0.3724)    (0.0801)    (0.2686)
4      4.0455      5.1704      4.0353      5.1284
(0.1028)    (0.6539)    (0.0754)    (0.4691)

Table 6. MLEs for several a and b parameter values (variances
in parentheses).

a    b           n = 10                 n = 20

[??]         [??]        [??]        [??]

1    1    1.2922       1.4124      1.1364      1.1665
(0.3835)     (0.8753)    (0.1291)    (0.1816)
1    2    1.2207       3.0650      1.1079      2.4438
(0.2182)     (4.5985)    (0.0861)    (1.1979)
1    3    1.1844       4.6307      1.0975      3.7774
(0.1543)     (8.5029)    (0.0714)    (3.2781)
1    4    1.1540       6.0016      1.0901      5.1139
(0.1154)    (11.2756)    (0.0617)    (6.0177)
1    5    1.1261       7.1680      1.0825      6.3945
(0.0895)    (12.9804)    (0.0535)    (8.6183)
2    1    2.5842       1.4121      2.2728      1.1665
(1.5318)     (0.8739)    (0.5161)    (0.1815)
2    2    2.4408       3.0571      2.2158      2.4432
(0.8700)     (4.4186)    (0.3442)    (1.1851)
2    3    2.3679       4.6131      2.1946      3.7732
(0.6192)     (8.1938)    (0.2844)    (3.2076)
2    4    2.3042       5.9499      2.1784      5.0913
(0.4616)    (10.6918)    (0.2433)    (5.6459)
2    5    2.2427       7.0540      2.1610      6.3515
(0.3550)    (11.9683)    (0.2085)    (8.1286)
3    1    2.1358       1.0780      3.4091      1.1665
(0.2130)     (0.0608)    (1.1614)    (0.1816)
3    2    3.6590       3.0530      3.3236      2.4429
(1.9302)     (4.3600)    (0.7739)    (1.1783)
3    3    3.5521       4.6151      3.2920      3.7749
(1.3818)     (8.1101)    (0.6405)    (3.2347)
3    4    3.4579       5.9566      3.2681      5.0951
(1.0376)    (10.5722)    (0.5486)    (5.6934)
3    5    3.3682       7.0695      3.2416      6.3449
(0.7999)    (11.7716)    (0.4708)    (7.9364)
4    1    5.1642       1.4101      4.5455      1.1665
(6.0063)     (0.8514)    (2.0640)    (0.1815)
4    2    4.8702       3.0296      4.4313      2.4427
4    2   (3.3608)     (4.0290)    (1.3750)    (1.1782)
4    3    4.7246       4.5771      4.3891      3.7720
(2.3826)     (7.7049)    (1.1352)    (3.1640)
4    4    4.6035       5.9348      4.3588      5.1031
(1.7977)    (10.3640)    (0.9799)    (5.7925)
4    5    4.4874       7.0556      4.3254      6.3667
(1.3902)    (11.5550)    (0.8448)    (8.2116)

a            n = 30                n = 40

[??]        [??]        [??]        [??]

1     1.0893      1.1032      1.0679      1.0780
(0.0763)    (0.0876)    (0.0533)    (0.0608)
1     1.0709      2.2699      1.0542      2.2022
(0.0521)    (0.5299)    (0.0369)    (0.3524)
1     1.0646      3.4785      1.0495      3.3564
(0.0443)    (1.5574)    (0.0315)    (1.0175)
1     1.0611      4.7132      1.0471      4.5328
(0.0399)    (3.2699)    (0.0286)    (2.1581)
1     1.0581      5.9510      1.0453      5.7225
(0.0365)    (5.4658)    (0.0267)    (3.7875)
2     2.1786      1.1032      2.1358      1.0780
(0.3052)    (0.0876)    (0.2130)    (0.0608)
2     2.1418      2.2701      2.1083      2.2021
(0.2087)    (0.5337)    (0.1476)    (0.3523)
2     2.1290      3.4772      2.0989      3.3562
(0.1768)    (1.5407)    (0.1259)    (1.0163)
2     2.1220      4.7113      2.0940      4.5316
(0.1592)    (3.2351)    (0.1143)    (2.1459)
2     2.1161      5.9501      2.0905      5.7206
(0.1459)    (5.5010)    (0.1066)    (3.7591)
3     3.2680      1.1032      3.2037      1.0780
(0.6863)    (0.0875)    (0.4793)    (0.0608)
3     3.2126      2.2701      3.1626      2.2023
(0.4694)    (0.5335)    (0.3322)    (0.3523)
3     3.1937      3.4778      3.1484      3.3562
(0.3982)    (1.5472)    (0.2832)    (1.0168)
3     3.1830      4.7108      3.1411      4.5327
(0.3587)    (3.2458)    (0.2575)    (2.1610)
3     4.2310      5.9381      3.1355      5.7184
(0.5797)    (5.2623)    (0.2393)    (3.7228)
4     4.3571      1.1032      4.2715      1.0780
(1.2202)    (0.0876)    (0.8520)    (0.0607)
4     4.2836      2.2699      4.2168      2.2024
4    (0.8342)    (0.5277)    (0.5907)    (0.3527)
4     4.2584      3.4784      4.1979      3.3560
(0.7086)    (1.5609)    (0.5032)    (1.0154)
4     4.2443      4.7129      4.1880      4.5314
(0.6392)    (3.2742)    (0.4573)    (2.1404)
4     4.3254      6.3667      4.1805      5.7169
(0.8448)    (8.2116)    (0.4245)    (3.7004)
```