General results for the Marshall and Olkin ’ s family of distributions

Abstract Marshall and Olkin (1997) introduced an interesting method of adding a parameter to a wellestablished distribution. However, they did not investigate general mathematical properties of their family of distributions. We provide for this family of distributions general expansions for the density function, explicit expressions for the moments and moments of the order statistics. Several especial models are investigated. We discuss estimation of the model parameters. An application to a real data set is presented for illustrative purposes.


INTRODUCTION
Adding parameters to a well-established distri bution is a time honored device for obtaining more flexible new families of distributions.(Marshall and Olkin 2007) introduced an interesting method of adding a new parameter to an existing distribution.The resulting distribution, known as Marshall-Olkin (M-O) extended distribution, includes the baseline distribution as a special case and gives more flexibility to model various types of data.The M-O family of distributions is also known as the proportional odds family (proportional odds model) or family with tilt parameter (Marshall and Olkin 2007).
Let F(x) =1-F(x) denote the survival function of a continuous random variable X which depends on a parameter vector β = (β 1 ,..., β q ) T of dimension q.Then, the corresponding M-O extended distribution has survival function defined by where α > 0 and α = 1-α.For α = 1, G(x) = F(x).Marshall and Olkin (1997) have noted that the method has a stability property, i.e., if the method is applied twice, nothing new is obtained in the second time around.
Additionally, the extended model is geometrically extremely stable.If X i = (i = 1, 2, ...) is a sequence of independent and identically distributed random variables with cdf F(x) and if N has a geometric distribution taken values {1, 2, ...}, then the random variables U = min {X 1 , ..., X N } and V = max{X 1 , ..., X N } are distributed as in (1).It implies that the new distribution is geometrically extremely stable.Marshall and Olkin (2007) have called the additional shape parameter "tilt parameter", since the hazard rate of the new family is shifted below (α ≥ 1) or above (0 < α ≤ 1) the hazard rate of the underlying distribution, that is, for all x ≥ 0, h(x) ≤ r(x) when α ≥ 1, and h(x) ≥ r(x) when 0 < α ≤ 1, where h(x) denotes the hazard rate of the transformed distribution and r(x) is that of the original distribution.Some special cases discussed in the literature include the M-O extensions of the Weibull distribution (Ghitany et al. 2005, Zhang andXie 2007), Pareto distribution (Ghitany 2005), gamma distribution (Ristic´ et al. 2007), Lomax distribution (Ghitany et al. 2007) and linear failure-rate distribution (Ghitany and Kotz 2007).More recently, Gómez-Déniz (2010) presented a new generalization of the geometric distribution using the M-O scheme.Economou and Caroni (2007) showed that the M-O extended distributions have a proportional odds property and Caroni (2010) presented some Monte Carlo simulations considering hypothesis testing on the parameter α for the extended Weibull distribution.Maximum likelihood estimation in M-O family is given in Lam and Leung (2001) and Gupta and Peng (2009).Gupta et al. (2010) compared this family and the original distribution with respect to some stochastic orderings and also investigate thoroughly the monotonicity of the failure rate of the resulting distribution when the baseline distribution is taken as Weibull.Nanda and Das (2012) investigated the tilt parameter of the M-O extended family.
The probability density function (pdf) of the M-O extended-F distribution, say g(x), is given by where f(x) = dF(x) / dx is the baseline density function corresponding to F(x).Here after, we refer to the family (2) as the M-O extended-F distribution.General mathematical properties of the M-O extended-F distribution were not derived by Marshall and Olkin (1997) such as moments and moments of order statistics.In this article, we derive some general structural properties of the M-O extended-F distribution including: (i) expansions for the pdf; (ii) general expressions for the moments; (iii) moments of order statistics; (iv) Rényi entropy.We propose several M-O extended-F distributions taken as baseline in the definitions the Weibull, Fréchet, Pareto, generalized exponential, Kumaraswamy and power function distributions.We discuss maximum likelihood estimation of the model parameters.
The article is organized as follows.Section 2 presents expansions for the density function and for the density function of the order statistics.Explicit expressions for the moments and moments of the order statistics of the M-O extended-F distribution are given in Section 3. Rényi entropy is derived in Section 4. Estimation of the model parameters by maximum likelihood is discussed in Section 5. Section 6 presents an alternative method to estimate the model parameters.In Section 7, we propose several M-O extended-F distributions and discuss some of their properties.Simulation results are performed in Section 8.An application of the current family to a real data set is explored in Section 9. Finally, some concluding remarks are presented in Section 10.
(2) MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS EXPANSIONS Consider the series representation which is valid for |z| < 1 and k > 0, where Г(•) is the gamma function.If α 2 (0,1) using ( 3) in (2), we obtain where The density function (2) can be expressed as Hence, for α > 1, using (3) in the last equation yields We now give the pdf of the ith order statistic X i:n , say g i:n (x), in a random sample of size n from the M-O extended-F distribution.The pdf of X i:n can be expressed as If α 2 (0,1) using expansion (??) in the last equation, we obtain where For α > 1, we write 1 -α F(x) = α{1-(α-1) F(x)/ α} and using (Ref: exp), g i:n (x) becomes where Equations ( 4)-( 7) reveal that the density functions of the M-O extended-F distribution and of its order statistics can be expressed as the baseline density f(x) multiplied by an infinite power series of F(x).They play an important role and will be used to obtain explicit expressions for the moments of the M-O extended-F distribution and of its order statistics in a general framework and for special models.

MOMENTS
Here after, suppose that X has the density function (2).We derive general expressions for the moments of X and its order statistics in terms of the probability weighted moments (PWMs) of the F distribution.The PWMs, first proposed by Greenwood et al. (1979), are expectations of certain functions of a random variable whose mean exists.A general theory for these moments covers the summarization and description of theoretical probability distributions and observed data samples, non parametric estimation of the underlying distribution of an observed sample, estimation of parameters, quantiles of probability distributions and hypothesis tests.The PWMs method can generally be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly.The PWMs for the baseline F distribution are formally defined by Thus, from equations (4) and (5), the sth moment of X for α 2(0,1) and α >1 can be written as Now, using equations ( 6) and ( 7), we can determine the sth moment of the ith order statistic X i:n in a random sample of size n from X for α 2 (0,1) and α >1 as respectively, where the quantities w j,k , v j , u j,l,k and c j,l are defined in Section 2. Thus, the moments of X and X i:n are obtained in terms of infinite weighted sums of PWMs of the baseline F distribution.

RéNYI ENTROPY
The entropy of a random variable is a measure of uncertainty variation and has been used in various situations in science and engineering.The Rényi entropy is defined by (9) MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS where δ > 0 and δ ≠ 1.For furthers details, the reader is referred to Song (2001).For α 2 (0,1), using expansion (3), we can write For α > 1, we obtain Thus, the Rényi entropy of X can be obtained for α 2 (0,1) and α > 1 as , respectively, where An interesting quantity based on the Rényi entropy is defined by S g = -2d I R (δ) / dδ| δ=1 .It is a location and scale-free positive functional and measures the intrinsic shape of a distribution (see, Song 2001).

MAXIMUM LIKELIHOOD
The model parameters of the M-O extended-F distribution can be estimated by maximum likelihood.Let x = (x 1 , ..., x n ) ┬ be a random sample of size n from X with unknown parameter vector θ = (α, β ┬ ) ┬ , where β = (β 1 ,..., β q ) ┬ corresponds to the parameter vector of the baseline distribution.The log-likelihood function By taking the partial derivatives of the log-likelihood function with respect to α and β, we obtain the components of the score vector Setting these equations to zero, U θ = 0, and solving them simultaneously yields the maximum likelihood estimate ┬ .These equations cannot be solved analytically and statistical software can be used to solve them numerically.For example, the BFGS method (see, Nocedal andWright 1999, Press et al. 2007) with analytical derivatives can be used for maximizing the log-likelihood function ℓ (θ).
The normal approximation for the θ can be used for constructing approximate confidence intervals and confidence regions for the parameters α and β.Under conditions that are fulfilled for the parameters in the interior of the parameter space, we have n √ (θ ^ -θ) α N q+1 (0, K(θ) -1 ), where α means approximately distributed and K(θ) is the unit expected information matrix given by whose elements are given by The asymptotic behavior remains valid if K(θ) = lim n →∞ n -1 J n (θ), where J n (θ) is is the observed information matrix, it is replaced by the average sample information matrix evaluated at θ ^, i.e. n -1 J n (θ ^).The observed information matrix is given by MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS whose elements are We can easily check if the fit of the M-O extended-F model is statistically "superior" to a fit using the F model by testing the null hypothesis H 0 : α≠1.For testing H 0 : α≠1 the likelihood ratio (LR) statistic is given by w = 2{ℓ(α ^, β ^) -ℓ(α ^, β ^)}, where α ^ and β ^ are the unrestricted MLEs obtained from the maximization of ℓ under H 1 and β .The limiting distribution of this statistic is x 1 2 under the null hypothesis.The null hypothesis is rejected if the test statistic exceeds the upper 100(1-γ)% quantile of the x 1 2 distribution.

ESTIMATION-TYPE METHOD OF MOMENTS
We now present an alternative method to estimate the model parameters.Since the moments cannot be obtained in closed form, the estimation by the method of moments is complicated.However, after some algebra, we obtain Thus, we can use (12) to construct a new method of estimation, i.e., if x 1 , ..., x 2 is a random sample with survival function (1), we can estimate the model parameters from the equation In Section, we apply the two methods (maximum likelihood and estimation-type method of moments) to estimate the model parameters of the M-O extended family.

SPECIAL M-O EXTENDED MODELS
We motivate the study of Marshall and Olkin's distributions by considering some special models to illustrate the applicability of the previous results.Here, we obtain the moments and Rényi entropy for some special M-O extended-F distributions when the baseline F distribution follows the Weibull, Fréchet, Pareto, generalized exponential, Kumaraswamy and power distributions.Some others M-O extended-F distributions could be proposed and our general results applied to them.Clearly, the quantities ¿ p,r are determined from the baseline F cdf.
The M-O-EW distribution was studied by Ghitany et al. (2005); see also Barreto-Souza et al. (2011).We obtain where the last equation holds for (δ -1)(° -1) ˃ -1 From these quantities, we immediately obtain explicit expressions for the moments, moments of the order statistics and Rényi entropy.If ° = 1, the results correspond to the M-O extended exponential distribution.

M-O EXTENDED FRéCHET DISTRIBUTION
Here, we consider the Fréchet distribution (for x, σ, λ ˃ 0) with cdf and pdf given by F(x)= e -(σ/x) ° and f(x) = λ σ λ x -(λ +1) e -(σ +1) λ , respectively.The pdf and survival function of the M-O extended Fréchet (M-O-EF) distribution (for x ˃ 0) reduce to respectively.After some algebra, we obtain which is valid for p < λ.Applying this result in ( 8) and ( 9), it follows simple expressions for the moments and moments of the order statistics of the M-O-EF distribution.An expression for the Rényi entropy of this distribution is obtained by inserting in ( 10) and ( 11).

M-O EXTENDED GENERALIZED EXPONENTIAL DISTRIBUTION
The pdf and cdf of the generalized exponential distribution, introduced by Gupta and Kundu (1999), for x ˃ 0 and λ, ° ˃ 0, are given by f δ (x) = ° λe -(λx) (1 -e -λx ) ° -] 1 and F(x) = (1 -e -λx ) °, respectively.By replacing these quantities in (1) and ( 2), the pdf and survival function of the M-O extended generalized exponential (M-O-EGE) distribution reduce respectively to (for x ˃ 0) From (4) and ( 5), we obtain the moment generating function (mgf) of the M-O-EGE distribution as for α 2 (0,1) and , BARRETO-SOUZA, ARTUR J. LEMONTE and GAUSS M. CORDEIRO for α ˃ 1.Both formulas hold for t < λ min {1,1+°}.Hence, their moments can be obtained from the derivatives of the mgf at t = 0. We also have which leads to a simple expression for the Rényi entropy.

M-O EXTENDED POWER DISTRIBUTION
Our final special case concentrates on the M-O extended power (M-O-EPo) distribution defined (for x 2(0, 1/θ) and θ > 0) by taking F(x) = (θx) k in (1).For x 2(0, 1/θ), the pdf and cdf of the M-O-EPo distribution are given respectively by Their moments and moments of the order statistics can be obtained by setting in ( 8) and ( 9).We also have , MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS valid for δ (1-k) -1 .Replacing the last expression in ( 10) and ( 11), it follows simple expressions for the Rényi entropy.

SIMULATION RESULTS
In what follows, we shall present Monte Carlo simulation results.All the Monte Carlo simulation experiments are performed using the Ox matrix programming language (Doornik 2006).Ox is freely distributed for academic purposes and available at http://www.doornik.com.The number of Monte Carlo replications was R = 20,000.We apply the estimation methods before discussed in order to estimate the model parameters of the M-O extended-F distribution.We adopt the M-O-EE distribution with pdf and hazard rate given by respectively, where α ˃ 0 and λ ˃ 0. According to Marshall and Olkin (1997), this distribution may sometimes be a competitor to the Weibull and gamma distributions.The authors derived several properties of the M-O-EE distribution.For example, they showed that h(x) is decreasing in x for 0 < α ≤ 1 and that h(x) is increasing in 1 and e -λx ≤ G(x) ≤ e -λx/α for α ≥ 1. Rao et al. ( 2009) developed a reliability test plan for acceptance/rejection of a lot of products submitted for inspection with lifetimes governed by the M-O-EE distribution.
For a random sample of size n from this distribution, the total log-likelihood function for the parameter vector θ = (α, λ) ┬ is given by The components of the score vector x i e -λxi 1 -αe -λxi .
Setting U θ = 0 and solving them simultaneously yields the MLEs α ^ and λ ^ of α and λ, respectively.The observed information matrix for the parameter vector θ = (α, λ) ┬ is given in the Appendix.The estimates of α ^ and λ ^ , come from the simultaneous solution of the non linear equations The simulation of the M-O-EE independent deviates can be performed using where U ~ U (0,1).The evaluation of point estimation was performed based on the following quantities for each sample size: (i) mean; (ii) relative bias (the relative bias of an estimator (θ ^) of a parameter θ is defined as {E(θ ^θ ^)} / θ, its estimate being obtained by estimating E(θ ^) by Monte Carlo); (iii) root mean squared error, MSE √ , where MSE is the mean squared error estimated from R Monte Carlo replications.The sample sizes are taken as n = 50 and 150.The values of the parameters were set at λ = 0.5 and α = 0.2, 0.6, 1.5, 1.8, 2.0, 2.5, 3.0, 4.0, 5.0, 7.0, 10 and 20.
The point estimates are presented in Tables I and II for n = 50 and n = 150, respectively.From these tables, note that the root mean squared errors of α ^ and α ^ increase with α whereas the root mean squared error of λ ^ decreases with α.The relative bias of λ ^ also decreases as α increases.Additionally, in most of the cases, the relative bias of α ^ is smaller than the relative bias (in absolute value) of α ^ .Also, the relative bias and the root mean squared error of λ ^ is smaller than the relative bias (in absolute value) and the root mean squared error of λ ^ in most of the cases.Also noteworthy is that as the sample size increases, the root mean squared error decreases.In general, the maximum likelihood method yields better estimates of α and λ than the estimation-type method of moments.
Tables III and IV evaluate the overall performance of each of the two different estimators, for each value of n.Each entry in Table III corresponds to a specific estimator and a specific value of n.We obtain what we call the integrated relative bias squared norm (Cribari-Neto and Vasconcellos 2002).This is computed as where the r h 's (h = 1, ..., 12) correspond to the twelve different values of the relative bias of each estimator.Similarly, Table IV III provides a measure of the overall performance of the estimators regarding the bias, and Table IV gives a measure of their overall performance regarding the root mean squared error.In short, the figures in these tables reveal that the maximum likelihood method should be preferred than the estimation-type method of moments in order to estimate the model parameters.

APPLICATION
As Marshall and Olkin (1997) pointed out the M-O-EE distribution considered in the previous section may be an alternative to the Weibull and gamma distributions.In what follows, we shall present an empirical application to a real data set in which the M-O-EE distribution may be preferred than the Weibull and gamma models.We consider the active repair times (hours) for an airborne communication transceiver given in Jørgensen (1982).All the computations were done using the Ox matrix programming language (Doornik 2006).  of the estimated density of all fitted models are given in Figure 1.Note that the M-O-EE model provides a better fit than the other models.In Figure 2, we plot the estimated hazard rate for the M-O-EE, Weibull and gamma distributions.Notice that the estimated hazard ratio of the M-O-EE distribution is decreasing and belongs to the interval [λ ^; λ ^/ α ^] = [0.1618;0.3935],expected.Now, we shall apply formal goodness-of-fit tests in order to verify which distribution fits better to these data.We apply the Cramér-von Mises (W*) and Anderson-Darling (A*) statistics.The statistics W* and A* are described in details in Chen and Balakrishnan (1995).In general, the smaller the values of the statistics W* and A*, the better the fit to the data.Let H(x; θ) be the cdf, where the form of H is known but θ (a k-dimensional parameter vector, say) is unknown.

CONCLUDING REMARKS
We study some mathematical properties of the Marshall and Olkin's family of distributions (Marshall and Olkin 1997).This family is defined by adding a parameter to a baseline distribution, giving more flexibility to model various type of data.In the last few years, several authors proposed Marshall-Olkin models to extend well-known distributions (see, for example, Ghitany 2005, Zhang and Xie 2007, Ristic´ et al. 2007, Ghitany et al. 2007, Ghitany and Kotz 2007, Gómez-Déniz 2010).In this article, we derive various structural properties of the Marshall-Olkin extended-F distribution not explored before including simple expansions for the density function and explicit expressions for the moments, moments of the order statistics and Rénvy entropy.Our formulas related with this class of distributions are manageable, and with the use of modern computer resources with analytic and numerical capabilities, may turn into adequate tools comprising the arsenal of applied statisticians.Several Marshall-Olkin extended-F distributions are proposed and some of their mathematical properties are given.We discuss maximum likelihood estimation of the model parameters and propose an alternative estimation method.Monte Carlo simulation experiments are also considered.Finally, an empirical application to a real data set is presented.

Figure 1 -
Figure 1 -Estimated pdf of the Marshall-Olkin extended exponential (M-O-EE), Weibull and gamma models for a real data set.
provides the average root mean squared error expressed as √of each estimator.For each value of n, Table Table V lists the MLEs of the parameters (standard errors in parentheses) and the values of the loglikelihood functions.The M-O-EE distribution yields the highest value of the log-likelihood function.Plots MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS

TABLE I Point estimates for n=50 and different values of α.
MSE: mean squared error; Rel.Bias: relative bias.
To obtain the statistics W* and A*, one can proceed as BARRETO-SOUZA, ARTUR J. LEMONTE and GAUSS M. CORDEIRO

TABLE III Integrated relative bias squared norm.
MLE: maximum likelihood estimate.MME: modified moment estimate.

TABLE II Point estimates for n=50 and different values of α.
MSE: mean squared error; Rel.Bias: relative bias.MARSHALL AND OLKIN`S FAMILY OF DISTRIBUTIONS

TABLE IV Average root mean squared error.
MLEs: maximum likelihood estimate.M-O-EE: Marshall-Orkin extended exponential.