The Transmuted Marshall-Olkin Extended Lomax Distribution

SILVA, RENILMA P. DA; CYSNEIROS, AUDREY HELEN M.A.; CORDEIRO, GAUSS M.; TABLADA, CLAUDIO JAVIER

doi:10.1590/0001-3765202020180777

Abstract

The transmuted family of distributions has been receiving increased attention over the last few years. In this paper, we generalize the Marshall-Olkin extended Lomax distribution using the quadratic rank transmutation map to obtain the transmuted Marshall-Olkin extended Lomax distribution. Several properties of the new distribution are discussed including the hazard rate function, ordinary and incomplete moments, characteristic function and order statistics. We provide an estimation procedure by the maximum likelihood method and a simulation study to assess the performance of the new distribution. We prove empirically the flexibility of the new model by means of an application to a real data set. It is superior to other three and four parameter lifetime distributions.

Key words
generalized distribution; lifetime analysis; Lomax distribution; Marshall-Olkin extended; transmuted family

INTRODUCTION

Non-negative random variables are used to model a wide variety of applications in survival analysis, demography, reliability, actuarial study and other areas. For this reason, there is a growing interest in constructing new distributions with positive real support to model lifetime data in several fields. One of the most useful methods to generate new distributions is the integral transform of existing distributions, usually referred to as generalized \(G\) classes (^{Tahir & Nadarajah 2015}B33 TAHIR MH & NADARAJAH S. 2015. Parameter induction in continuous univariate distributions: Well-established G families. An Acad Bras Cienc 87: 539-568.). The principal reason for this is the ability of these generalized distributions to be more flexible than the baseline \(G\) distribution and therefore provide better fits to skewed data (^{Pescim et al. 2010}B23 PESCIM RR, DEMÉTRIO CGB, CORDEIRO GM, ORTEGA EMM & URBANO MR. 2010. The beta generalized half-normal distribution. Comput Stat Data Anal 54: 945-957.). The second reason is the powerful computational facilities available in several analytical platforms, which facilitate handling and computing complex mathematical expressions.

Some of the best known generalized \(G\) classes of distributions are: the Marshall-Olkin extended (MOE) family (^{Marshall & Olkin 1997}B21 MARSHALL AW & OLKIN I. 1997. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84: 641-652.), exponentiated-generated (exp-G) families (^{Cordeiro et al. 2013}B9 CORDEIRO GM & CASTRO M. 2011. A new family of generalized distributions. J Stat Comput Simul 81: 883-898., ^{Gupta et al. 1998}B15 GUPTA RC, GUPTA PL & GUPTA RD. 1998. Modeling failure time data by Lehmann alternatives. Commun Stat Theory Methods 27: 887-904.), beta-generated (beta-G) family (^{Eugene et al. 2002}B12 EUGENE N, LEE C & FAMOYE F. 2002. Beta-normal distribution and its applications. Commun Stat Theory Methods 31: 497-512.), Kumaraswamy-generated (Kw-G) family (^{Cordeiro & Castro 2011}B8 CORDEIRO G, ORTEGA EM & DA CUNHA D. 2013. The exponentiated generalized class of distributions. J Data Sci 11: 1-27.), gamma-generated (gamma-G) families (^{Nadarajah et al. 2015}B22 NADARAJAH S, CORDEIRO GM & ORTEGA EMM. 2015. The Zografos-Balakrishnan-G Family of Distributions: Mathematical Properties and Applications. Commun Stat Theory Methods 44: 186-215., ^{Ristić & Balakrishnan 2012}B27 RISTIć MM & BALAKRISHNAN N. 2012. The gamma-exponentiated exponential distribution. J Stat Comput Simul 82: 1191-1206., ^{Zografos & Balakrishnan 2009}B34 ZOGRAFOS K & BALAKRISHNAN N. 2009. On families of beta-and generalized gamma-generated distributions and associated inference. Stat Methodol 6: 344-362.), McDonald-generated (Mc-G) family (^{Alexander et al. 2012}B3 ALEXANDER C, CORDEIRO GM, ORTEGA EMM & SARABIA JM. 2012. Generalized beta-generated distributions. Comput Stat Data Anal 56: 1880-1897.) and \(T\!\!-\!\!X\) family (^{Alzaatreh et al. 2013}B4 ALZAATREH A, LEE C & FAMOYE F. 2013. A new method for generating families of continuous distributions. METRON 71: 63-79.). A detailed compilation of these families can be found in ^{Tahir & Nadarajah (2015)}B33 TAHIR MH & NADARAJAH S. 2015. Parameter induction in continuous univariate distributions: Well-established G families. An Acad Bras Cienc 87: 539-568..

Shaw & Buckley (2009) pioneered an interesting method by adding a new parameter to an existing distribution that would offer more distributional flexibility. They used the quadratic rank transmutation map (QRTM) to generate a flexible family. The generated class called the transmuted extended family includes as a special case the baseline distribution and gives more flexibility to model various types of data. General results for this family and new models are discussed in Bourguignon et al. (2016).

In this paper, we adopt the transmuted generated (T-G) family to define a new distribution called the transmuted Marshall-Olkin extended Lomax (TMOELx) distribution by taking the Marshall-Olkin extended Lomax distribution (^{Ghitany et al. 2007}B13 GHITANY ME, AL-AWADHI FA & ALKHALFAN LA. 2007. Marshall-Olkin Extended Lomax Distribution and Its Application to Censored Data. Commun Stat Theory Methods 36: 1855-1866.)) as the baseline \(G\) model. We obtain the TMOELx density function as a linear combination of exponentiated-Lomax (exp-Lx) densities. Given that the new distribution has positive real support, our objective is to define a flexible distribution for lifetime applications. Also, we present explicit expressions for the quantile function (qf), moments, characteristic function and order statistics. In addition, we consider a study of the maximum likelihood estimates of the model for complete samples and a simulation study to verify the performance of these estimates. Finally, we consider an application of the TMOELx distribution and compare it with others distributions based on some goodness-of-fit statistics.

THE NEW DISTRIBUTION

The cumulative distribution function (cdf) of the Lomax distribution, say \(\mbox{Lx}(\beta,\gamma)\), also known as the Pareto distribution of the second kind, is

\begin{aligned} \label{eq:lomax} R(x) = 1- (1 + \beta x)^{-\gamma}, \ x>0, \ \beta, \gamma>0,\end{aligned}

where \(\beta\) and \(\gamma\) are, respectively, the scale and shape parameters.

The Marshall-Olkin extended Lomax (MOELx) distribution (^{Ghitany et al. 2007}B13 GHITANY ME, AL-AWADHI FA & ALKHALFAN LA. 2007. Marshall-Olkin Extended Lomax Distribution and Its Application to Censored Data. Commun Stat Theory Methods 36: 1855-1866.) is obtained by taking the Lomax distribution (1) as the baseline model in the MOE family. Its cdf has the form

\begin{aligned} \label{eq:MOELx} {G}(x)= 1 - \frac{\alpha}{(1+\beta x)^\gamma - \bar{\alpha}}, \quad x > 0,\ \alpha, \beta, \gamma > 0,\ \bar{\alpha} = 1 - \alpha.\end{aligned}

The cdf and probability density function (pdf) of the T-G family are, respectively,

\begin{aligned} \label{eq:cdftransmuted} T(x)= T(x;\bm\xi, \lambda)= (1+\lambda)G(x;\bm\xi) - \lambda G(x;\bm\xi)^2,\quad \lambda \in [-1,1]\end{aligned}

and

\begin{aligned} \label{eq:pdftramuted} t(x)= t(x;\bm\xi, \lambda)= [1 + \lambda - 2\lambda G(x;\bm\xi)]g(x;\bm\xi),\ x \in \mathcal{D} \subseteq \mathbb{R},\end{aligned}

where \(G(x;\bm\xi)\) and \(g(x;\bm\xi)\) are, respectively, the baseline cdf and pdf, and \(\bm\xi\) is the parameter vector of the baseline distribution. For \(\lambda = 0\), it reduces to the baseline model. In ^{Bourguignon et al. (2016)}B6 BOURGUIGNON M, GHOSH I & CORDEIRO GM. 2016. General results for the transmuted family of distributions and new models. J Probabil Stat 2016: 1-12., the authors proved that the pdf (4) can be expressed as a linear combination of exp-G densities.

Based on the T-G family and the MOELx distribution, we propose a new four-parameter distribution so-called the TMOELx distribution. By inserting (2) as the baseline distribution in equation (3), the cdf of the TMOELx distribution (for \(x > 0\)) can be expressed as

\begin{aligned} \label{eq:cdftmoel} F(x)=\frac{[(1 + \beta x)^\gamma - 1][(1+\beta x)^\gamma + \alpha\lambda - \bar{\alpha}]}{[(1 + \beta x)^\gamma - \bar{\alpha}]^2},\end{aligned}

whereas its pdf is

\begin{aligned} \label{eq:pdftmoel} f(x) = \frac{\alpha\,\beta\,\gamma\, (1 + \beta x)^{\gamma-1}\{[1-(1+\beta x)^\gamma](\lambda-1) + \alpha(\lambda+1)\}}{[(1+\beta x)^\gamma - \bar{\alpha}]^3},\end{aligned}

where \(\alpha>0,\ \beta>0,\ \gamma>0\) and \(\lambda \in [-1,1]\). Hereafter, a random variable \(X\) having pdf (6) will be denoted by \(X\sim \mathrm{TMOELx}(\alpha, \beta, \gamma, \lambda)\).

In lifetime analysis, a useful function is the hazard rate function (hrf) \(h(x)\). So, the hrf of \(X\) is given by

\begin{aligned} \label{eq:funcaorisco} h(x) = \frac{\beta\gamma(1+\beta x)^{\gamma-1} \{ (\lambda -1)[(1+\beta x)^{\gamma} -1]- \alpha(1 + \lambda)\} }{[(1+\beta x)^\gamma -\bar{\alpha}]\{ (\lambda-1)[(1+\beta x)^{\gamma} -1]-\alpha\}}.\end{aligned}

For selected values of the parameters \(\alpha\), \(\beta\), \(\gamma\) and \(\lambda\), some sub-models of the TMOELx distribution published in the literature are listed in Table I.

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Table I
Some TMOELx sub-models. MOELx: Marshall-Olkin extended Lomax, TLx: Transmuted Lomax, Lx: Lomax.

SHAPES OF THE DENSITY AND HAZARD RATE FUNCTIONS

The shapes of the pdf (6) can be described analytically by examining the roots of the equation \(f'(x)=0\) and analyzing its limits when \(x\rightarrow0\) or \(x\rightarrow\infty\). Since \(f(x)\) is the pdf of a continuous random variable, then \(\lim_{x\rightarrow\infty}f(x)=0\). Further, we have

\begin{aligned} \lim_{x\rightarrow 0}\,f(x)=\frac{\beta\,\gamma\,(\lambda+1)}{\alpha}\end{aligned}

and, therefore, \(\lim_{x\rightarrow 0}f(x)=0\) if and only if \(\lambda=-1\). Some plots of the TMOELx pdf, for different parameter values, are displayed in . These plots reveal that the pdf of \(X\) can be strictly decreasing or unimodal with mode \(x=x_0\) at

\begin{aligned} x_0=\frac{1}{\beta}\left[-1 + \left(\frac{-1+\alpha+\lambda+2\alpha\gamma\lambda-\{\alpha^2\lambda^2+4\alpha\gamma\lambda\,(-1+\alpha+\lambda)+\gamma^2[2\alpha\,(\lambda-1)+(\lambda-1)^2+\alpha^2(3\lambda^2+1)]\}^{1/2}}{(1+\gamma)(\lambda-1)}\right)^{1/\gamma}\right].\end{aligned}

Figure 1
Plots of the pdf (6) for selected parameters.

Further, we obtain the conditions of the behavior of the TMOELx density in terms of the parameters. In fact, from (4), the density \(t(x)\) is decreasing when \(t'(x)=g'(x)[1+\lambda-2\,\lambda\,G(x)]-2\,\lambda\,g^2(x)<0\) for all \(x>0\), where \(G(x)\) and \(g(x)\) are, respectively, the cdf and pdf of the MOELx distribution. So, \(t(x)\) is decreasing when

\begin{aligned} \frac{g'(x)}{g(x)}[1+\lambda-2\,\lambda\,G(x)]<2\,\lambda\,g(x),\quad \forall x>0.\end{aligned}

After some algebraic manipulation and considering \(x\rightarrow0^+\), the above inequality is equivalent to

\begin{aligned} \alpha+2\,\gamma-\alpha\gamma+\alpha\lambda+4\,\gamma\lambda-\alpha\gamma\lambda>0.\end{aligned}

Therefore, \(t(x)\) is unimodal if and only if

\begin{aligned} \alpha+2\,\gamma-\alpha\gamma+\alpha\lambda+4\,\gamma\lambda-\alpha\gamma\lambda<0.\end{aligned}

Thus, the parameters \(\alpha\), \(\gamma\) and \(\lambda\) control the shapes of the pdf, while \(\beta\) is the scale parameter. The shape parameters allow extensive control on the right tail, providing more heavy (light) tails when \(\alpha\) increases (decreases) and \(\gamma\) and \(\lambda\) decrease (increase).

The corresponding hrf can have shapes such as decreasing and unimodal as shown in Figure 2. Thus, the new distribution can be appropriate for different applications in lifetime analysis. For the conditions of the behavior of the hrf \(h(x)\), note that \(h'(x)=\frac{t'(x)[1-T(x)]+t^2(x)}{[1-T(x)]^2}\) and, therefore, \(h(x)\) is decreasing if and only if

\begin{aligned} t'(x)[1-T(x)]+t^2(x)<0,\quad\forall x.\end{aligned}

Figure 2
Plots of the hrf (7) for selected parameters.

From (4) and considering \(x\rightarrow0^+\), the above inequality is equivalent to

\begin{aligned} \alpha(\gamma-1)(\lambda+1)+\gamma(\lambda^2-2\,\lambda-1)<0.\end{aligned}

Then, \(h(x)\) is unimodal if and only if

\begin{aligned} \alpha(\gamma-1)(\lambda+1)+\gamma(\lambda^2-2\,\lambda-1)>0.\end{aligned}

Thus, the parameters \(\alpha\), \(\gamma\) and \(\lambda\) control the shapes of the hrf of \(X\).

[fig:subfig1.1][fig:subfig1.2][fig:subfig1.3][fig:subfig1.4]

USEFUL EXPANSIONS

We can obtain a power series for the cdf of the TMOEL distribution from eqs. (1), (2) and (3) using the generalized binomial expansion (see appendix A)

\begin{aligned} \label{eq:cdf_expansion01} F(x)=\sum_{k=0}^\infty p_k\, H_{k+1}(x),\end{aligned}

where \(H_{k+1}(x)=R^{k+1}(x)\) is the exp-Lx cdf with power parameter \(k+1\). The coefficients are

\begin{aligned} p_k=\left\{\begin{array}{ll} (1+\lambda)\,b_k - \lambda\,c_k, & \mbox{ if } 0<\alpha \leq 1/2,\\ a_k, & \mbox{ if } \alpha > 1/2, \end{array}\right.\end{aligned}

and

\begin{aligned} a_k&=&\frac{[(1+\lambda)(\alpha-1)-k\lambda]}{\alpha^2}\!\left(\frac{\alpha-1}{\alpha}\right)^{k-1},\ k=0,1,2,\ldots\\[0.3cm] b_k&=&\sum_{j=k}^\infty{j \choose k}(1-\alpha)^j,\ k=0,1,2,\ldots\\[0.3cm] c_0&=&b_0^2\ \mbox{ and }\ c_m=\frac{1}{m\,b_0}\sum_{k=1}^m(3k-m)\,b_k\,c_{m-k},\ m=1,2,\ldots\end{aligned}

Differentiating (8) gives

\begin{aligned} \label{eq:pdf_expansion} f(x) = \sum_{k=0}^\infty p_k\,h_{k+1}(x),\end{aligned}

where \(h_{k+1}(x)=\frac{d}{dx}H_{k+1}(x)\) is the exp-Lx density with power parameter \(k+1\).

Equation (9) reveals that the pdf of \(X\) can be expressed as a linear combination of exp-Lx densities. Thus, some structural properties of the TMOELx distribution can be determined from those of the exp-Lx distribution (^{Salem 2014}B28 SALEM HM. 2014. The Exponentiated Lomax Distribution: Different Estimation Methods. Am J Math Stat 2: 364-368.).

QUANTILE FUNCTION

Since the cdf \(F(x)\) given in (5) is continuous and strictly increasing, the qf of \(X\) is \(Q(u)=F^{-1}(u)\), for \(0<u<1\). From ^{Bourguignon et al. (2016)}B6 BOURGUIGNON M, GHOSH I & CORDEIRO GM. 2016. General results for the transmuted family of distributions and new models. J Probabil Stat 2016: 1-12., we obtain the qf of the TMOELx distribution as

\begin{aligned} \label{eq:qf} Q(u;\bm\xi,\lambda)=\left\{\begin{array}{lc} Q_G\left(\frac{1+\lambda-\sqrt{(1+\lambda)^2-4\lambda u}}{2\lambda};\bm\xi\right), & \mbox{if}\ \lambda\neq 0, \\ Q_G(u;\bm\xi), & \mbox{if}\ \lambda=0, \end{array} \right.\end{aligned}

where \(\bm\xi=(\alpha,\beta,\gamma)^\top\) is the parameter vector and \(Q_G(u;\bm\xi)\) is the qf of the MOELx distribution

\begin{aligned} Q_G(u;\bm\xi)=\beta^{-1}\left[\left(1 - \frac{\alpha u}{u-1}\right)^{1/\gamma} - 1\right].\end{aligned}

Using (10), we can generate random numbers from the TMOELx distribution as follows. If \(U\sim\mathcal{U}(0,1)\), then

\begin{aligned} Q(U;\alpha,\beta,\gamma,\lambda)\sim\mathrm{TMOELx}\,(\alpha,\beta,\gamma,\lambda).\end{aligned}

Another alternative to generate random numbers from the TMOELx distribution can be based on random extrema in transmuted distributions, which are given in ^{Kozubowski & Podgórski (2016)}B18 KOZUBOWSKI TJ & PODGÓRSKI K. 2016. Transmuted distributions and random extrema. Stat Probabil Lett 116: 6-8. (Proposition 2.1). Let \(X_1\) and \(X_2\) be i.i.d. random variables from the MOELx distribution and let \(N_p\) be an integer-valued random variable such as \(N_p-1\sim \mathrm{Bernoulli(p)}\), \(0\leq p\leq1\). Further, suppose that \(N_p\) and \(X_i\), \(i=1,2\), are independent random variables. Then, an observation \(y\) from the TMOELx distribution can be generated in the following way:

Generate \(u_1\) and \(u_2\) independently from the uniform distribution \(\mathcal{U}(0,1)\).
Calculate \(x_i=Q_G(u_i)\), \(i=1,2\).
If \(\lambda\in[-1,0]\), define \(p=-\lambda\in[0,1]\) and generate \(n_p-1\) from the \(\mathrm{Bernoulli}(p)\) distribution.
Then, obtain \(y=\displaystyle\vee_{i=1}^{n_p}x_i\).
If \(\lambda\in[0,1]\), define \(p=\lambda\) and generate \(n_p-1\) from the \(\mathrm{Bernoulli}(p)\) distribution.
Finally, obtain \(y=\displaystyle\wedge_{i=1}^{n_p}x_i\).

For \(\lambda=0\) in equation (3), we have \(n_p=1\) almost surely and the steps 4 and 6 are satisfied simultaneously with their right-hand-sides reducing to \(x_1\). Further, in the extreme cases \(\lambda=\pm1\), we have \(n_p=2\) almost surely and the steps 4 and 6 are reduced to \(y=\max(x_1,x_2)\) and \(y=\min(x_1,x_2)\), respectively.

In Figure 3, we compare the exact TMOELx densities and histograms from two simulated data sets for selected parameters which show the consistent of the simulated values from the above algorithm with the TMOELx distribution. We simulate the data using the R software (version 3.2.3).

Figure 3
Plots of the exact TMOELx densities and histograms of the simulated data for some parameter values.

Skewness and kurtosis

Useful skewness and kurtosis measures are given by \(\alpha_3=\mu_3/\sigma^3\) and \(\alpha_4=\mu_4/\sigma^4\), respectively, where \(\mu_j\) is the \(j\)-th central moment and \(\sigma\) is the standard deviation.

For some distributions in the T-G family, it could be difficult to find the third and fourth moments. Alternative measures for the skewness and kurtosis based on quantiles are sometimes more appropriate. The measure of skewness \(S\) of Bowley and the measure of kurtosis \(K\) of Moors are given by

\begin{aligned} \label{eq:skewness} S=\frac{Q(6/8) + Q(2/8) - 2\,Q(4/8)}{Q(6/8) - Q(2/8)} \end{aligned}

and

\begin{aligned} \label{eq:kurtosis} K=\frac{Q(7/8) - Q(5/8) + Q(3/8) - Q(1/8)}{Q(6/8) - Q(2/8)},\end{aligned}

respectively, where \(Q(\cdot)\) is given by (10). These measures exist even for distributions without moments.

The plots in Figure 4 display the skewness (11) and kurtosis (12) as functions of \(\lambda\) for some parameter values. They reveal that the skewness and kurtosis of \(X\) decrease rapidly when \(\lambda\) converges to one.

Figure 4
Plots of the skewness and kurtosis for selected parameters.

MOMENTS AND CHARACTERISTIC FUNCTION

Moments are important in any statistical analysis. For example, some characteristics of a distribution can be described using measures such as the mean, variance, skewness and kurtosis, which are determined from the first four ordinary moments.

For \(r\in\mathbb{N}\), let \(\mu'_r=E(X^r)\) be the \(r\)-th ordinary moment of \(X\). From equation (9), we can express \(\mu'_r\) as a linear combination of the \(r\)-th ordinary moments of exp-Lx random variables. In fact, for \(r<\gamma\),

\begin{aligned} \label{eq:expectation1} \mu'_r = \sum_{k=0}^\infty p_k \,E(Y_{k}^r),\end{aligned}

where \(Y_{k}\sim\mathrm{exp\!-\!Lx}(k+1,\beta,\gamma)\).

The \(r\)-th ordinary moment of \(Y_{k}\) is given by ^{Salem (2014)}B28 SALEM HM. 2014. The Exponentiated Lomax Distribution: Different Estimation Methods. Am J Math Stat 2: 364-368.(for \(r<\gamma\)) as

\begin{aligned} \label{eq:exp-Lomax_expectation} E(Y_{k}^r) = \frac{(k+1)}{\beta^r}\sum_{j=0}^r (-1)^j{r \choose j}\,B\!\left(1-\frac{1}{\gamma}(r-j),k+1\right),\end{aligned}

where \(B(a,b)=\Gamma(a)\,\Gamma(b)/ \Gamma(a+b)\) is the beta function and \(\Gamma(\cdot)\) is the gamma function. Inserting (14) in equation (13) gives (for \(r<\gamma\))

\begin{aligned} \label{eq:expectation_expansion} \mu'_r = \sum_{k=0}^\infty\sum_{j=0}^r b_r(k,j)\,B\!\left(1-\frac{1}{\gamma}(r-j),k+1\right),\end{aligned}

where \(b_r(k,j)=\displaystyle(-1)^j {r \choose j} \frac{(k+1)}{\beta^r}\,p_k\). From equation (15), we note that \(\mu'_r<\infty\) for \(r<\gamma\), a condition that also holds for the Lomax distribution.

Numerical results

In this section, we use the expansions for the \(r\)-th ordinary moment of \(X\) to compare the numerical results (for some parameter values) for the mean, variance, skewness and kurtosis of the TMOELx distribution obtained by the methods of truncation, numerical integration and Monte Carlo simulation (\(50,000\) replications). For reasons of simplicity, we consider \(\alpha>1/2\) since, in this case, the expansions for the moments are easier to obtain.

For the truncation method, the \(r\)-th truncated ordinary moment of \(X\) follows from (15) as

\begin{aligned} %\label{eq:truncated_expectation_expansion} \mu'_{r,N} = \sum_{k=0}^N\sum_{j=0}^r b_r(k,j)\,B\!\left(1-\frac{1}{\gamma}(r-j),k+1\right),\quad r<\gamma,\ N\in\mathbb{N}.\end{aligned}

For the truncation and Monte Carlo methods, we use the Ox plataform (version 7.10, see ^{Doornik (2007)}B11 DOORNIK JA. 2007. Object-Oriented Matrix Programming Using Ox. www.doornik.com. 3rd ed., London.
www.doornik.com ... ). For the numerical integration method, we adopt algorithms in the Mathematica software for recursively subdivide the integration region.

The results for the three methods are given in Table II. For the truncation method, we consider the values \(N=5,\ 10\) and \(20\). We note that the values for the truncation method are more accurate when \(N\) increases in agreement with \(\mu'_r=\lim_{N\rightarrow\infty}\mu'_{r,N}\). Further, these values can be considered sufficiently precise when \(N=20\) compared with those values from numerical integration method. For the Monte Carlo method, the results are less accurate when we consider large-order moments.

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Table II
Results for the mean, variance, skewness and kurtosis for some parameter values.

The script (in Ox language) for calculating the values corresponding to truncation and Monte Carlo methods in Table II is given in Appendix B.

Incomplete moments

The \(r\)-th incomplete moment of \(X\) is determined from (9) as

\begin{aligned} \label{eq:incomplete_moment1} m_X^{(r)}(z) = \int_0^z\!x^rf(x)\,dx = \sum_{k=0}^\infty p_k\,m_{Y_k}^{(r)}(z),\end{aligned}

where

\begin{aligned} m_{Y_k}^{(r)}(z) = \int_0^z\!x^r\,h_{k+1}(x)\,dx = \beta\,\gamma\,(k+1)\int_0^z\!x^r\left[1 - (1+\beta x)^{-\gamma}\right]^k(1+\beta x)^{-\gamma-1}\,dx\end{aligned}

is the \(r\)-th incomplete moment of \(Y_k\).

Using the binomial expansion gives

\begin{aligned} \label{eq:expansion_k} \left[1 - (1+\beta x)^{-\gamma}\right]^k = \sum_{j=0}^k(-1)^j{k \choose j} (1+\beta x)^{-j\gamma}.\end{aligned}

By replacing this expansion in \(m_{Y_k}^{(r)}(z)\) and interchanging \(\sum_{k=0}^\infty\sum_{j=0}^k\) by \(\sum_{j=0}^\infty\sum_{k=j}^\infty\), we have

\begin{aligned} \label{eq:incomplete_moment_TMOELx} m_X^{(r)}(z) = \sum_{j=0}^\infty e_j\,m_{V_{j}}^{(r)}(z),\end{aligned}

where \(V_{j}\sim\mathrm{Lx}(\beta, (j+1)\gamma)\), \(e_j=\displaystyle\sum_{k=j}^\infty(-1)^j{k \choose j}\frac{(k+1)}{(j+1)}\,p_k\),

\begin{aligned} m_{V_{j}}^{(r)}(z) = \frac{(j+1)\beta\,\gamma\,z^{r+1}\,\Hypergeometric{2}{1}{r+1,1+(j+1)\gamma}{r+2}{-\beta z}}{r+1},\end{aligned}

and \(\Hypergeometric{p}{q}{a}{b}{x}\) is the hypergeometric function (^{Gradshteyn & Ryzhik 2007}B14 GRADSHTEYN IS & RYZHIK IM. 2007. Table of integrals, series, and products. Amsterdam: Elsevier/Academic Press, 7th ed.).

Characteristic function

The generating and characteristic functions are useful tools, since they can be used for computing the moments and cumulants of a distribution. For the Lomax distribution, the generating function is defined only for \(t\leq 0\). Consequently, the TMOELx generating function is also defined only for \(t\leq 0\). However, the characteristic function (chf) of a distribution exists for all \(t\in\mathbb{R}\). The chf of \(X\) is

\begin{aligned} \phi_X(t) = E({\mathrm e}^{itX}) = \int_0^\infty\!{\mathrm e}^{itx}f(x)\,dx,\end{aligned}

where \(i=\sqrt{-1}\) and \(t\in\mathbb{R}\).

Using equation (9), we obtain

\begin{aligned} \phi_X(t) = \sum_{k=0}^\infty p_k\,\phi_{Y_k}(t).\end{aligned}

Further, using equation (16), we can write

\begin{aligned} \label{eq:cf_expansion1} \phi_X(t) = \sum_{j=0}^\infty e_j\,\phi_{V_j}(t),\end{aligned}

where

\begin{aligned} \label{eq:cf_expansion2} \phi_{V_j}(t) = (j+1)\,\gamma\,{\mathrm e}^{-it/\beta}(-it)^{(j+1)\gamma}\beta^{-(j+1)\gamma}\,\Gamma\!\left(-(j+1)\gamma,-\frac{it}{\beta}\right)\end{aligned}

and \(\Gamma(s,z)=\int_{z}^{\infty}t^{s-1}\,{\mathrm e}^{-t}\,dt\) is the upper incomplete gamma function, which, by analytic continuation, is defined for almost all combinations of complex \(s\) and \(z\). Replacing (17) in equation (16) and setting

\begin{aligned} d_j(t)=(j+1)\,\gamma\,e_j\,{\mathrm e}^{-i\,t/\beta}(-i \,t)^{(j+1)\gamma}\beta^{-(j+1)\gamma},\end{aligned}

we obtain

\begin{aligned} \label{eq:cf_expansion} \phi_X(t) = \sum_{j=0}^\infty d_j(t)\,\Gamma\!\left(-(j+1)\,\gamma,-\frac{i\,t}{\beta}\right).\end{aligned}

Equation (18) is the main result of this section.

ORDER STATISTICS

Let \(X_1,\ldots,X_n\) be a random sample of size \(n\) from a distribution \(F(x)\). Then, for \(1\leq m\leq n\), the pdf of the \(m\)-th order statistic, \(X_{(m)}\), can be expressed as (^{Severini 2005}B30 SEVERINI T. 2005. Elements of Distribution Theory. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.)

\begin{aligned} f_{(m)}(x)= M\,F(x)^{m-1}(1-F(x))^{n-m}f(x)=M\,f(x)\sum_{j=0}^{n-m}(-1)^j{\displaystyle n-m \choose j}F(x)^{m+j-1},\end{aligned}

where \(M=n!/[(m-1)!\,(n-m)!]\).

Based on (8) and using the expansion for a power series raised to positive integer powers (^{Gradshteyn & Ryzhik 2007}B14 GRADSHTEYN IS & RYZHIK IM. 2007. Table of integrals, series, and products. Amsterdam: Elsevier/Academic Press, 7th ed.), we have

\begin{aligned} F(x)^{m+j-1}=\left(\sum_{k=0}^{\infty} p_k\,H_{k+1}(x)\right)^{m+j-1}=\sum_{k=0}^{\infty}c_{m+j-1,k}\,H_{k+1}(x),\end{aligned}

where \(c_{m+j-1,0}=p_0^{m+j-1}\) and, for \(i\geq 1\),

\begin{aligned} c_{m+j-1,i}=\frac{1}{i\,p_0}\sum_{s=1}^{i}[(m+j)s-i]\,p_s\,c_{m+j-1,i-s}.\end{aligned}

Therefore, we obtain

\begin{aligned} f_{(m)}(x)= Mf(x)\sum_{j=0}^{n-m}\sum_{k=0}^{\infty}(-1)^j{\displaystyle n-m \choose j}c_{m+j-1,k}\,H_{k+1}(x),\end{aligned}

Replacing \(f(x)\) by the expansion (9) in the last equation and, after some algebraic manipulation, we can writte

\begin{aligned} f_{(m)}(x)= M \sum_{j=0}^{n-m}\sum_{k,\ell = 0}^{\infty} (-1)^j{\displaystyle n-m \choose j}\frac{(\ell+1)}{(\ell+k+2)}\,c_{m+j-1,k}\,p_\ell\,h_{\ell+k+2}(x).\end{aligned}

We note that \(f_{(m)}(x)\) in the above equation is a linear combination of exp-Lx densities. So, some properties of the TMOELx order statistics can be easily obtained from those of the exp-Lx.

MAXIMUM LIKELIHOOD ESTIMATION

In this section, we consider the estimation of the parameters of the TMOELx distribution by the maximum likelihood method. Let \(x=(x_1,\ldots,x_n)^\top\) be a sample of size \(n\) from \(X\sim \mathrm{TMOELx}(\alpha,\beta,\gamma,\lambda)\) and \(\boldsymbol\theta=(\alpha,\beta,\gamma,\lambda)^\top\) be the parameter vector. The log-likelihood for \(\boldsymbol\theta\), denoted by \(\ell(\boldsymbol\theta)\), is given by

\begin{aligned} \label{eq:veross} \ell(\boldsymbol\theta) &=& n[\log(\alpha)+ \log(\beta) + \log(\gamma)] + (\gamma - 1)\sum_{i=1}^{n}\log(1 + \beta x_i) + \sum_{i=1}^{n} \log\{[1 - (1 + \beta x_i)^\gamma](\lambda - 1)\nonumber\\ &+& \alpha(\lambda + 1)\} - 3\sum_{i=1}^{n}\log[(1 + \beta x_i)^\gamma - \bar{\alpha}].\end{aligned}

The MLE \(\hat{\boldsymbol\theta}\) of \(\boldsymbol\theta\) can be obtained by maximizing (19) directly by using the SAS (PROC NLMIXED), R (optim and MaxLik functions) and Ox program (MaxBFGS sub-routine).

Alternatively, the components of the score vector \(U_{\boldsymbol\theta}=(U_{\alpha}, U_{\beta}, U_{\gamma}, U_{\lambda})^{\top}\) are

\begin{aligned} U_{\alpha} &=& \displaystyle\frac{\partial\,\ell(\boldsymbol\theta)}{\partial\alpha}= n\alpha^{-1} - 3\sum_{i=1}^{n}\displaystyle(\alpha-1+u_i^\gamma)^{-1} + (\lambda + 1)\sum_{i=1}^{n}\displaystyle[\alpha (\lambda + 1) + (\lambda - 1) (1-u_i^\gamma) ]^{-1}, \\[0.5cm] % % U_{\beta} &=& \displaystyle\frac{\partial\,\ell(\boldsymbol\theta)}{\partial\beta}= {n}{\beta}^{-1}+ \gamma(\gamma-1)\sum_{i=1}^{n}x_iu_i^{-1} - 3\gamma\sum_{i=1}^{n} x_i u_i^{\gamma-1}(\alpha-1+u_i^\gamma)^{-1} \\ &-& \gamma(\lambda-1) \sum_{i=1}^{n} x_iu_i^{\gamma-1}[\alpha (\lambda + 1) + (\lambda - 1) (1-u_i^\gamma) ]^{-1}, \\[0.5cm] U_{\gamma} &=& \displaystyle\frac{\partial\,\ell(\boldsymbol\theta)}{\partial\gamma} = {n}{\gamma}^{-1} + (\gamma-1)\sum_{i=1}^{n} (\log u_i) + \sum_{i=1}^{n} (\log u_i^\gamma) - 3 \sum_{i=1}^{n}u_i^\gamma (\log u_i)(\alpha-1+u_i^\gamma)^{-1} \\ &-& (\lambda - 1)\sum_{i=1}^{n} u_i^\gamma (\log u_i) [\alpha (\lambda + 1) + (\lambda - 1) (1-u_i^\gamma) ]^{-1}, \\[0.5cm] U_{\lambda} &=&\displaystyle\frac{\partial\,\ell(\boldsymbol\theta)}{\partial\lambda}=\displaystyle\sum_{i=1}^{n} (1+\alpha - u_i^\gamma) [\alpha (\lambda + 1) + (\lambda - 1) (1-u_i^\gamma) ]^{-1},\end{aligned}

where \(u_i = 1 + \beta x_i\).

The MLE \(\hat{\boldsymbol\theta}\) can also be determined by solving the nonlinear equations \(U_\alpha=U_\beta=U_\gamma=U_\lambda=0\) simultaneously. In this case, these equations should be evaluated numerically using Newton-Raphson algorithms.

Under general regularity conditions, we have \((\hat{\boldsymbol\theta}-\boldsymbol\theta)\stackrel{a}{\sim} N_4(0,K(\boldsymbol\theta)^{-1})\), where \(K(\boldsymbol\theta)\) is the \(4\times4\) expected information matrix and \(\stackrel{a}{\sim}\) denotes asymptotic distribution. For large \(n\), \(K(\boldsymbol\theta)\) can be approximated by the observed information matrix. This normal approximation for the MLE \(\hat{\boldsymbol\theta}\) can be used for determining approximate confidence intervals and for testing hypotheses on the parameters \(\alpha,\beta,\gamma\) and \(\lambda\).

Suppose that the parameter vector is partitioned as \(\boldsymbol\theta=(\bm{\psi}_1^\top,\bm{\psi}_2^\top)^\top\), where \(\dim(\bm{\psi}_1)+\dim(\bm{\psi}_2)=\dim(\boldsymbol\theta)\). The likelihood ratio (LR) statistic for testing the null hypothesis \(\mathcal{H}_0:\ \bm{\psi}_1=\bm{\psi}_1^{(0)}\) against the alternative hypothesis \(\mathcal{H}_1:\ \bm{\psi}_1\neq\bm{\psi}_1^{(0)}\) is given by \(LR=2\,\{\ell(\hat{\boldsymbol\theta})-\ell(\tilde{\boldsymbol\theta})\}\), where \(\hat{\boldsymbol\theta}=(\hat{\bm{\psi}}_1^\top,\hat{\bm{\psi}}_2^\top)^\top\), \(\tilde{\boldsymbol\theta}=(\bm{\psi}_1^{(0)^\top},\tilde{\bm{\psi}}_2^\top)^\top\), \(\hat{\bm{\psi}}_i\) and \(\tilde{\bm{\psi}}_i\) are the MLEs under the alternative and null hypotheses, respectively, and \(\bm{\psi}_1^{(0)}\) is a specified parameter vector. Based on the first-order asymptotic theory, we know that \(LR\stackrel{a}{\sim}\chi_k^2\) (chi-square distribution with \(k\) degrees of freedom), where \(k=\dim(\bm{\psi}_1)\). Therefore, to the significance level \(\nu\), we reject \(\mathcal{H}_0\) if \(LR>\chi_{(1-\nu,k)}^2\), where \(\chi_{(1-\nu,k)}^2\) is the quantile \(1-\nu\) of the \(\chi_k^2\). Thus, we can test sub-models of the TMOELx distribution and analyze how significant are the parameters tested for modeling a given data set.

SIMULATION STUDY

In this section, we perform a Monte Carlo simulation experiment in order to evaluate the behavior of the MLE \(\hat{\boldsymbol\theta}=(\hat{\alpha},\hat{\beta},\hat{\gamma},\hat{\lambda})\) and estimate the relative biases and mean squared errors (MSEs) for sample sizes \(n=100,200\) and \(250\).

We consider \(10,000\) Monte Carlo replications and use the BFGS method in the Ox plataform (version 7.10, MaxBFGS function) to maximize the log-likelihood function (19). We set the parameter values \(\beta=0.25\), \(\gamma=0.3\) and vary \(\alpha\) and \(\lambda\). Some computational aspects related to the simulation study are detailed in Appendix C.

The results, given in Table III, reveal generally that the relative biases and MSE values decrease when \(n\) increases. The minimum absolute values for the relative biases and MSEs are equal to \(0.001\) and \(0.003\), respectively, whereas the maximum absolute values for the relative bias and MSE are \(1.632\) and \(4.467\), respectively. Moreover, we note in Table III that the parameter \(\lambda\) was underestimated in most cases (negative relative biases).

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Table III
Relative biases and MSE values for the MLEs

(\hat{α}, \hat{β}, \hat{γ}, \hat{λ})

(

β

=0.25 and

γ

=0.3).

APPLICATION

In this section, we present two applications of the TMOELx distribution.

First application: We compare the TMOELx distribution with its sub-models: the TLx, MOELx and Lx distributions (see Table I). We use an uncensored data set corresponding to \(128\) intervals between the times where vehicles pass a point on a road (traffic data). The data are given in ^{Jorgensen (2012)}B17 JORGENSEN B. 2012. Statistical properties of the generalized inverse Gaussian distribution. Volume 9. Springer Science & Business Media..

Since the parameter \(\lambda\) in the TMOELx distribution is such that \(|\lambda|\leq1\), we employ the SQP method (MaxSQP function of the Ox language) to maximizing the log-likelihood function (19). For maximizing the log-likelihood for the sub-models, we employ the R software (^{R Core Team 2018}B25 R CORE TEAM. 2018. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria. https://www.R-project.org/.
https://www.R-project.org/ ... ), AdequacyModel package (^{Diniz Marinho et al. 2016}B10 DINIZ MARINHO PR, BOURGUIGNON M & BARROS DIAS CR. 2016. AdequacyModel: Adequacy of Probabilistic Models and General Purpose Optimization. https://CRAN.R-project.org/package=AdequacyModel. R package version 2.0.0.
https://CRAN.R-project.org/package=Adeq... ). For checking the uniqueness of the solution to the score equations, we have perturbed the initial values, besides consider different methods: quasi-Newton methods (BFGS and Nelder-Mead) and heuristic methods (simulated annealing and particle swarm optimization). The MLEs of the model parameters of the TMOELx distribution and its sub-models are listed in Table IV.

We compare the fitted models by means of some goodness-of-fit statistics: Akaike Information Criterion (AIC) (^{Akaike 1974}B2 AKAIKE H. 1974. A new look at the statistical model identification. IEEE Trans Automat Contr 19: 716-723.), Bayesian Information Criterion (BIC) (^{Schwarz 1978}B29 SCHWARZ G. 1978. Estimating the Dimension of a Model. Ann Stat 6: 461-464.), Hannan-Quinn Information Criterion (HQIC) (^{Hannan & Quinn 1979}B16 HANNAN EJ & QUINN BG. 1979. The Determination of the Order of an Autoregression. J R Stat Soc B 41: 190-195.), Cramér-von Mises Criterion (W*) and Anderson-Darling Criterion (A*) (^{Chen & Balakrishnan 1995}B7 CHEN G & BALAKRISHNAN N. 1995. A general purpose approximate goodness-of-fit test. J Qual Technol 27: 154-161.). In general, small values of these statistics indicate better fits. We employ the R software (AdequacyModel package) to calculate these statistics. The goodness-of-fit values of the fitted distributions are listed in Table V.

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Table IV
MLEs (standard errors).

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Table V
Goodness-of-fit statistics (first application).

The values in Table V indicate that the TMOELx distribution presents the smallest values of the AIC, HQIC, W* and A* statistics among the fitted models. Therefore, according to these statistics, we can conclude that the TMOELx distribution gives the best fit to the current data.

To analyze how significant is the parameter \(\lambda\) of the TMOELx distribution in modeling these data, we use the LR statistic for testing the MOELx model against the TMOELx model, that is, we test \(\mathcal{H}_0: \lambda=0\) against \(\mathcal{H}_1: \lambda\neq0\). We obtain an approximate p-value of \(0,0019\). Therefore, at the 5% significance level, the test rejects the null hypothesis, that is, we reject the MOELx model. Thus, we have evidence of the potential need for including the parameter \(\lambda\) to model these data.

Second application: In this case, we compare the TMOELx distribution with other non-nested models proposed in the literature: exponentiated Lomax (ELx) (^{Lemonte & Cordeiro 2013}B19 LEMONTE AJ & CORDEIRO GM. 2013. An extended Lomax distribution. Statistics 47: 800-816.), exponentiated standard Lomax (EsLx) (^{Lemonte & Cordeiro 2013}B19 LEMONTE AJ & CORDEIRO GM. 2013. An extended Lomax distribution. Statistics 47: 800-816.), transmuted Marshall-Olkin Fréchet (TMOFr) (^{AFIFY et al. 2015}B1 AFIFY AZ, HAMEDANI G, GHOSH I & MEAD M. 2015. The Transmuted Marshall-Olkin Fr\'echet Distribution: Properties and Applications. Int J Stat Probabil 4: 132-148.), beta Lomax (BLx) (^{Rajab et al. 2013}B26 RAJAB M, ALEEM M, NAWAZ T & DANIYAL M. 2013. On Five Parameter Beta Lomax Distribution. J Stat 20: 102-118.) and Kumaraswamy Lomax (KwLx) (^{Shams 2013}B31 SHAMS TM. 2013. The Kumaraswamy-Generalized Lomax Distribution. Middle East J Sci Res 17: 641-646.) distributions, whose densities are given in Appendix D. The uncensored data set refers to \(213\) times of successive failures of air conditioning system of airplanes available in ^{Proschan (1963)}B24 PROSCHAN F. 1963. Theoretical explanation of observed decreasing failure rate. Technometrics 5(3): 375-383..

Since all considered models are non-nested, we compare them by using the statistics W* and A*, since the AIC, BIC and HQIC criterions are useful only for nested models. The goodness-of-fit values of the fitted distributions are listed in Table VI. We can note that the TMOELx distribution has the smallest values of the W* and A* among the fitted models. Therefore, the TMOELx model gives the best fit to the current data.

The plots of the estimated TMOELx, TMOFr and KwLx densities are displayed in Figure 5.

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Table VI
Goodness-of-fit statistics (second application).

Figure 5
Comparison of the TMOELx, TMOFr and KwLx estimated densities.

CONCLUSIONS

In this paper, we study a new four-parameter lifetime model, named the transmuted Marshall–Olkin extended Lomax (TMOELx) distribution, obtained from the transmuted-G (T-G) family (^{Shaw & Buckley 2009}B32 SHAW WT & BUCKLEY IRC. 2009. The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. ArXiv e-prints .) when the baseline model is the Marshall-Olkin extended Lomax (MOELx) distribution (^{Ghitany et al. 2007}B13 GHITANY ME, AL-AWADHI FA & ALKHALFAN LA. 2007. Marshall-Olkin Extended Lomax Distribution and Its Application to Censored Data. Commun Stat Theory Methods 36: 1855-1866.). We present some sub-models of the new distribution. We obtain simple expressions for the cumulative and density functions. We demonstrate that the TMOELx density can be expressed as a linear combination of exponentiated-Lomax densities and then some of its structural properties can be determined from those of these models. We obtain explicit expressions for the quantile function, ordinary and incomplete moments, characteristic function and order statistics. We determine the maximum likelihood estimates for complete samples and perform a Monte Carlo study to evaluate the behavior of these estimates in finite samples. We compare the performance of the new model with other distributions using classical goodness-of-fit statistics. The overall results confirm that the TMOELx model is very appropriate for lifetime applications.

ACKNOWLEDGMENTS

This research was supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001. Also, it was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and FACEPE, Brazil.

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APPENDIX A

Here, we provide the mathematical development to derive eq. (8). The Lomax cdf (1) is given by

\begin{aligned} R(x)=1- (1 + \beta x)^{-\gamma} = \frac{(1 + \beta x)^{\gamma}-1}{(1 + \beta x)^{\gamma}},\end{aligned}

and then \((1+\beta x)^{-\gamma} = 1-R(x)\).

The MOEL cdf can be expressed in terms of \(R(x)\) from eq. (2) as

\begin{aligned} \label{eq:MOEL_exp} G(x) &=& 1 - \frac{\alpha}{(1+\beta x)^\gamma - (1-\alpha)} = 1 - \frac{\alpha(1+\beta x)^{-\gamma}}{[(1+\beta x)^\gamma - 1](1+\beta x)^{-\gamma} + \alpha(1+\beta x)^{-\gamma}}\nonumber\\[0.2cm] &=& 1 - \frac{\alpha(1-R(x))}{R(x)+\alpha(1-R(x))}=1 - \frac{\alpha(1-R(x))}{(1-\alpha)R(x) + \alpha} =\frac{R(x)}{\alpha\left[\frac{1-\alpha}{\alpha}\,R(x) + 1\right]}.\end{aligned}

By considering the cdf (3) of the transmuted family, the TMOEL cdf (for \(\lambda\in[-1,1]\)) is

\begin{aligned} \label{eq:cdf_sum} F(x)=(1+\lambda)\,G(x) - \lambda\,G^2(x)=\frac{(1+\lambda)\,R(x)}{\alpha\left[1 + \left(\frac{1-\alpha}{\alpha}\right)R(x)\right]} - \frac{\lambda\,R^2(x)}{\alpha^2\left[1 + \left(\frac{1-\alpha}{\alpha}\right)R(x)\right]^2}.\end{aligned}

For \(\alpha > 1/2\), we have \(\left|\frac{1-\alpha}{\alpha}\right|<1\). Thus, since \(0<R(x)<1\) (for \(x>0\)), using the generalized binomial expansion ( , p. 25, subsection 1.112), we have (for \(\alpha>1/2\))

\begin{aligned} \label{eq:cdf_expansion1} \left[1 + \left(\frac{1-\alpha}{\alpha}\right)R(x)\right]^{-1}=\sum_{k=1}^\infty(-1)^{k-1}\left(\frac{1-\alpha}{\alpha}\right)^{k-1} R^{k-1}(x)=\sum_{k=0}^\infty\left(\frac{\alpha-1}{\alpha}\right)^k R^k(x)\end{aligned}

and

\begin{aligned} \label{eq:cdf_expansion2} \left[1 + \left(\frac{1-\alpha}{\alpha}\right)R(x)\right]^{-2}=\sum_{k=1}^\infty(-1)^{k-1}k\left (\frac{1-\alpha}{\alpha}\right)^{k-1} R^{k-1}(x)=\sum_{k=1}^\infty k\left (\frac{\alpha-1}{\alpha}\right)^{k-1} R^{k-1}(x).\end{aligned}

Inserting (21) and (22) in eq. (21) gives

\begin{aligned} \label{eq:TMOEL_exp1} F(x)&=&\frac{(1+\lambda)}{\alpha}\,R(x) \sum_{k=0}^\infty\left(\frac{\alpha-1}{\alpha}\right)^k R^k(x) - \frac{\lambda}{\alpha^2}\,R^2(x) \sum_{k=1}^\infty k\left (\frac{\alpha-1}{\alpha}\right)^{k-1} R^{k-1}(x)\nonumber\\[0.2cm] &=&\frac{(1+\lambda)}{\alpha}\,R(x)\!\left[1 + \sum_{k=1}^\infty\left(\frac{\alpha-1}{\alpha}\right)^k R^k(x)\right] - \frac{\lambda}{\alpha^2}\sum_{k=1}^\infty k\left (\frac{\alpha-1}{\alpha}\right)^{k-1} R^{k+1}(x)\nonumber\\[0.2cm] %&=&\frac{(1+\lambda)}{\alpha}\,R(x) + \sum_{k=1}^\infty\left[\frac{(1+\lambda)}{\alpha}\left(\frac{\alpha-1}{\alpha}\right)^k R^{k+1}(x) - \frac{k\lambda}{\alpha^2}\left (\frac{\alpha-1}{\alpha}\right)^{k-1} R^{k+1}(x)\right]\nonumber\\[0.2cm] &=&\frac{(1+\lambda)}{\alpha}\,R(x) + \sum_{k=1}^\infty\left[\frac{((1+\lambda)(\alpha-1)-k\lambda)}{\alpha^2}\left(\frac{\alpha-1}{\alpha}\right)^{k-1}\right]R^{k+1}(x)\nonumber\\[0.2cm] &=& \sum_{k=0}^\infty a_k\,R^{k+1}(x),\end{aligned}

where \(a_k=\frac{[(1+\lambda)(\alpha-1)-k\lambda]}{\alpha^2}\!\left(\frac{\alpha-1}{\alpha}\right)^{k-1}\) for \(k=0,1,2,\ldots\) (with \(a_0=0\) when \(\alpha=1\)).

To obtain an expansion for \(F(x)\) when \(0 <\alpha \leq 1/2\), we can rewrite equation (20) as

\begin{aligned} \label{eq:MOEL_exp2} G(x)=\frac{R(x)}{1-(1-\alpha)(1-R(x))}.\end{aligned}

Since \(|(1-\alpha)(1-R(x))|<1\), we consider the generalized binomial expansion (for \(0<\alpha\leq1/2\))

\begin{aligned} ^{-1}=\sum_{j=1}^\infty(1-\alpha)^{j-1} \,(1-R(x))^{j-1} = \sum_{j=0}^\infty(1-\alpha)^j \,(1-R(x))^j.\end{aligned}

For \(j\in\mathbb{N}\) fixed, expanding \((1-R(x))^j=\sum_{k=0}^j{j \choose k}R^k(x)\) gives

\begin{aligned} ^{-1}= \sum_{j=0}^\infty\sum_{k=0}^j{j \choose k}(1-\alpha)^jR^k(x) =\sum_{k=0}^\infty\sum_{j=k}^\infty{j \choose k}(1-\alpha)^jR^k(x),\end{aligned}

where the last equality follows by rearranging terms in the summations.

By replacing the above equation in eq. (25), we obtain (for \(0<\alpha\leq1/2\))

\begin{aligned} G(x)= \sum_{k=0}^\infty\sum_{j=k}^\infty{j \choose k}(1-\alpha)^jR^{k+1}(x)=\sum_{k=0}^\infty b_k\,R^{k+1}(x),\end{aligned}

where \(b_k=\sum_{j=k}^\infty{j \choose k}(1-\alpha)^j\) (for \(k=0,1,2,\ldots\))

Inserting the last equation in (3) gives

\begin{aligned} \label{eq:cdf_expansion4} F(x)&=&(1+\lambda)\,G(x) - \lambda\,G^2(x)=(1+\lambda)\sum_{k=0}^\infty b_k\,R^{k+1}(x) - \lambda\left[\sum_{k=0}^\infty b_k\,R^{k+1}(x)\right]^2\end{aligned}

By using the expansion for a power series raised to positive integer powers ( , p. 17, section 0.314) we have

\begin{aligned} \left [\sum_{k=0}^\infty b_k\,R^{k+1}(x)\right]^2 = \sum_{k=0}^\infty c_k\,R^{k+1}(x),\end{aligned}

where \(c_0=b_0^2\) and \(c_m=\frac{1}{m\,b_0}\sum_{k=1}^m(3k-m)\,b_k\,c_{m-k}\), for \(m\geq1\). Thus, for \(0<\alpha\leq1/2\), we can write

\begin{aligned} \label{eq:TMOEL_exp2} F(x)=(1+\lambda)\sum_{k=0}^\infty b_k\,R^{k+1}(x) - \lambda\sum_{k=0}^\infty c_k\,R^{k+1}(x) =\sum_{k=0}^\infty[(1+\lambda)\,b_k - \lambda\,c_k] \,R^{k+1}(x).\end{aligned}

Finally, from equations (24) and (26), the cdf \(F(x)\) (for all \(\alpha>0\)) can be expressed as

\begin{aligned} %\label{eq:cdf_expansion} F(x)=\sum_{k=0}^\infty p_k\, H_{k+1}(x),\end{aligned}

where \(H_{k+1}(x)=R^{k+1}(x)\) is the exp-Lx cdf with power parameter \(k+1\) and

\begin{aligned} p_k=\left\{\begin{array}{ll} (1+\lambda)\,b_k - \lambda\,c_k, & \mbox{ if } 0<\alpha \leq 1/2,\\ a_k, & \mbox{ if } \alpha > 1/2. \end{array}\right.\end{aligned}

APPENDIX B

The routine (in Ox language) for calculating the values of the mean, variance, skewness and kurtosis of the TMOEL distribution in Table II is given below:

APPENDIX C

In this appendix, we detail some computational aspects related to the section of simulation study . All Monte Carlo simulations are performed using scripts implemented in the Ox programming language. A free version of Ox is available in https://www.doornik.com.

For maximizing the log-likelihood function (19), we use the routine MaxBFGS implemented in Ox:

func

In: the function to be maximized with \(p\) parameters (the log-likelihood function in this case).
avP

In: matrix of order \(p \times 1\) with the initial values.

Out: matrix of order \(p \times 1\) with the values maximizing func.
adFunc

In: address to object.

Out: maximum value of func.
amHessian

In: 0 or the Hessian matrix addressed (we used 0).
fNumDer

In: 0, to use the first analytic derivatives or 1, to use the first numeric derivatives (we used 1).

The log-likelihood function (19) is implemented by using the Ox language as follows:

Finally, the function (19) is maximized by using the routine:

General comments:

the initial values (\(\theta_0\)) used for maximizing the log-likelihood function were obtained from the true value of the parameter \(\theta\) by adding a small arbitrary constant \(\delta<1\), that is, \(\theta_0=\theta+\delta\);
For \(10,000\) Monte Carlo replications, the convergence rate, in almost all scenarios considered, was greater than \(85\%\). For \(\beta>0.5\) or \(n<100\), the BFGS method exhibits poor convergence. Therefore, it is not recommended to work with samples smaller than 100.

APPENDIX D

The model densities used for comparison with the TMOELx distribution are given below:

The TMOFr pdf is

\begin{aligned} f(x) = \frac{\alpha \beta \sigma^{\beta} x^{-(\beta + 1)} \mathrm{e}^{{-\left(\frac{\sigma}{x}\right)}^\beta}}{\left[\alpha + (1 - \alpha) \mathrm{e}^{{-\left(\frac{\sigma}{x}\right)}^\beta}\right]^{2}} \left[1 + \lambda - \frac{2\lambda \mathrm{e}^{{-\left(\frac{\sigma}{x}\right)}^\beta}}{\alpha + (1 - \alpha) \mathrm{e}^{{-\left(\frac{\sigma}{x}\right)}^\beta}}\right], \ x>0,\end{aligned}

where \(\beta, \alpha, \sigma\) are positive parameters and \(\mid\lambda\mid\leq 1\).

The ELx pdf is

\begin{aligned} f(x) = \frac{\alpha\beta^{\alpha}(\beta + x)^{-(\alpha + 1)}}{B(a,1)} \left[1 -\left(\frac{\beta}{\beta + x}\right)^{\alpha} \right]^{a-1},\ x>0,\end{aligned}

where \(B(a,b)\) is the beta function and \(\alpha\), \(\beta\) and \(a\) are positive parameters.

The EsLx pdf is

\begin{aligned} f(x) = \frac{\alpha(1+ x)^{-(\alpha + 1)}}{B(a,1)} \left[1 -\left(\frac{1}{1+ x}\right)^{\alpha} \right]^{a-1},\ x>0.\end{aligned}

where \(\alpha\) and \(a\) are positive parameters.

The BLx pdf is

\begin{aligned} f(x) = \frac{\alpha}{\lambda B(a,b)}\left\{1 - \left[1 + \left(\frac{x}{\lambda}\right)\right]^{-\alpha}\right\}^{a-1}\left[1 + \left(\frac{x}{\lambda}\right)\right]^{-(\alpha b+1)},\ x>0,\end{aligned}

where \(\alpha\), \(\lambda\), \(a\) and \(b\) are positive parameters.

Finally, the KwLx pdf is

\begin{aligned} f(x) = ab\frac{\alpha}{\lambda}\left(1 + \frac{x}{\lambda}\right)^{-(\alpha+1)}\left[1 - \left(1 + \frac{x}{\lambda}\right)^{-\alpha}\right]^{a-1}\left\{1 - \left[1 - \left(1 + \frac{x}{\lambda}\right)^{-\alpha}\right]^a\right\}^{b-1},\ x>0,\end{aligned}

where \(\alpha\), \(\lambda\), \(a\) and \(b\) are positive parameters.

Publication Dates

Publication in this collection
23 Oct 2020
Date of issue
2020

History

Received
26 July 2018
Accepted
12 Jan 2019

This is an open-access article distributed under the terms of the Creative Commons Attribution License

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$α$	$β$	$γ$	$λ$	Model	Reference
$α$	$β$	$γ$	0	MOELx ( $α, β, γ$ )	^{Ghitany et al. (2007)}B13 GHITANY ME, AL-AWADHI FA & ALKHALFAN LA. 2007. Marshall-Olkin Extended Lomax Distribution and Its Application to Censored Data. Commun Stat Theory Methods 36: 1855-1866.
1	$β$	$γ$	$λ$	TLx ( $β, γ, λ$ )	^{Ashour & Eltehiwy (2013)}B5 ASHOUR SK & ELTEHIWY MA. 2013. Transmuted Lomax distribution. Am J Math Stat 1: 121-127.
1	$β$	$γ$	0	Lx ( $β, γ$ )	^{Lomax (1954)}B20 LOMAX KS. 1954. Business Failures: Another Example of the Analysis of Failure Data. J Am Stat Assoc 49: 847-852.

Parameter	Method	$N$	mean	variance	skewness	kurtosis
$α$ =0.75	Truncation	5	0.7356	0.7407	5.1369	134.01
$β$ =0.5			10	0.6960	0.6916	5.2047
$γ$ =4.5			20	0.6963	0.6923	5.2040
$λ$ =-0.75	Num. int.		0.6963	0.6923	5.2040	138.07
	Monte Carlo		0.6949	0.7069	5.7817	99.360
$α$ =1.5	Truncation	5	0.3861	0.1688	3.8940	55.140
$β$ =1.0			10	0.3933	0.1715	3.8878
$γ$ =5.0			20	0.3934	0.1715	3.8877
$λ$ =-0.5	Num. int.		0.3934	0.1715	3.8877	54.930
	Monte Carlo		0.3927	0.1735	4.3059	56.020
$α$ =3.0	Truncation	5	0.1621	0.0344	3.1759	32.507
$β$ =2.0			10	0.2060	0.0402	3.0934
$γ$ =5.5			20	0.2174	0.0420	3.0889
$λ$ =-0.25	Num. int.		0.2178	0.0421	3.0887	31.238
	Monte Carlo		0.2175	0.0424	3.3848	35.354
$α$ =5.0	Truncation	5	0.0956	0.0117	2.8857	24.900
$β$ =3.0			10	0.1239	0.0137	2.7908
$γ$ =6.0			20	0.1314	0.0143	2.7898
$λ$ =0.25	Num. int.		0.1304	0.0142	2.7891	23.886
	Monte Carlo		0.1302	0.0142	3.0287	27.900
$α$ =7.0	Truncation	5	0.0587	0.0043	2.4756	17.285
$β$ =4.0			10	0.0820	0.0051	2.3795
$γ$ =7.0			20	0.0888	0.0053	2.4155
$λ$ =0.5	Num. int.		0.0828	0.0048	2.3729	17.031
	Monte Carlo		0.0827	0.0048	2.5454	20.037

			relative bias				MSE
$λ$	$α$	$n$	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$
-0.5	0.25	100	0.954	0.975	0.057	-0.086	0.687	0.932	0.012	0.142
		200	0.549	0.696	0.045	-0.049	0.144	0.337	0.007	0.124
		250	0.471	0.632	0.040	-0.049	0.111	0.301	0.006	0.117
	0.5	100	0.742	0.884	0.021	-0.158	2.155	1.025	0.007	0.153
		200	0.403	0.695	0.020	-0.073	0.521	0.290	0.004	0.139
		250	0.348	0.671	0.020	-0.059	0.225	0.279	0.003	0.133
0.25	0.25	100	1.245	1.632	$0.098$	-0.349	2.260	3.752	0.021	0.151
		200	0.495	0.751	$0.063$	-0.312	0.215	0.432	0.013	0.126
		250	0.382	0.570	$0.049$	-0.272	0.117	0.238	0.010	0.109
	0.5	100	0.802	1.122	$0.043$	-0.110	3.145	1.638	0.013	0.122
		200	0.376	0.528	$0.001$	$0.062$	1.097	0.452	0.008	0.119
		250	0.277	0.429	-0.011	$0.076$	0.301	0.144	0.007	0.122
0.5	0.25	100	0.931	1.131	0.144	-0.292	1.436	1.867	0.028	0.136
		200	0.379	0.462	0.087	-0.202	0.227	0.297	0.018	0.105
		250	0.308	0.385	0.062	-0.154	0.153	0.221	0.015	0.096
	0.5	100	0.772	1.057	0.101	-0.324	4.467	2.560	0.019	0.121
		200	0.237	0.348	0.040	-0.222	0.609	0.288	0.011	0.101
		250	0.144	0.249	0.022	-0.196	0.379	0.122	0.009	0.098

	MLE
Distribution	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$
TMOELx	2,087	0,775	1,062	-0,99
		(0.497)	(0.247)	(0.141)
MOELx	118,067	7,962	1,227	-
		(179.861)	(10.360)	(0.095)
TLx	-	0,067	1,679	0,189
		-	(0.040)	(0.467)
Lx	-	0,083	1,610	-
		-	(0.031)	(0.392)

Distribution	AIC	BIC	HQIC	W*	A*
TMOELx	928,168	939,576	932,803	0,256	1,528
MOELx	933,116	941,672	936,593	0,334	2,029
TLx	935,166	943,722	938,642	0,361	2,213
Lx	933,298	939,002	935,616	0,360	2,207

Brasil

Brasil

The Transmuted Marshall-Olkin Extended Lomax Distribution

Abstract

INTRODUCTION

THE NEW DISTRIBUTION

SHAPES OF THE DENSITY AND HAZARD RATE FUNCTIONS

USEFUL EXPANSIONS

QUANTILE FUNCTION

Skewness and kurtosis

MOMENTS AND CHARACTERISTIC FUNCTION

Numerical results

Incomplete moments

Characteristic function

ORDER STATISTICS

MAXIMUM LIKELIHOOD ESTIMATION

SIMULATION STUDY

APPLICATION

CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

Publication Dates

History

Distribution	W*	A*
TMOELx	0,034	0,249
TMOFr	0,116	0,813
EsLx	0,698	4,487
ELx	0,051	0,370
BLx	0,401	2,662
KwLx	0,085	0,632