The Topp Leone-G Power Series Class of Distributions with Applications

We present a new class of distributions called the Topp-Leone-G Power Series (TL-GPS) class of distributions. This model is obtained by compounding the Topp-Leone-G distribution with the power series distribution. Statistical properties of the TL-GPS class of distributions are obtained. Maximum likelihood estimates for the proposed model were obtained. A simulation study is carried out for the special case of Topp-Leone Log-Logistic Poisson distribution to assess the performance of the maximum likelihood estimates. Finally, we apply Topp-Leone-log-logistic Poisson distribution to real data sets to illustrate the usefulness and applicability of the proposed class of distributions.


Introduction
Statistical distributions are widely used to explain different types of real life events.Because of their usefulness, statistical distribution theory is extensively researched and new distributions are being developed.The Topp Leone (TL) distribution is among the distributions used within the theory and practice of statistics.It was proposed by Topp and Leone (1955) as a lifetime model.Nadarajah and Kotz (2003) studied its properties and provided its moments and the characteristic function.Numerous authors also studied the TL distribution.Ghitany et al. (2005) provided some reliability measures of the TL distribution, while Vicaria et al. (2008) introduced a two-sided generalized version of the TL distribution and Al-Zahrani (2012) derived the goodness-of-fit test for the TL distribution.Al-Shomrani et al. (2016) proposed the Topp-Leone generated family of distributions with cumulative distribution function (cdf), probability density function (pdf), and survival function given by and respectively, for b > 0, where G(x; ψ) is the baseline cdf depending on a parameter vector ψ, g(x; ψ) = dG(x; ψ)/dx, and Ḡ(x; ψ) = 1 − G(x; ψ) is the survival function.
In this paper, we develop a new class of distributions called the Topp-Leone-G Power Series (TL-GPS) class of distributions.We are motivated by the flexibility in data fitting obtained from the TL-GPS class of distributions and the applicability of the new class of distributions to data sets that exhibit monotonic or non-monotonic hazard rate shapes.
Another motivation for developing the TL-GPS class of distributions is the applicability of the power series distributions in different fields such as finance, economics and actuarial sciences.
This paper is organized as follows.In Section 2 we introduce the new class of distributions and present its cdf and pdf.
We also discuss some sub-classes of the TL-GPS distribution and present some special cases when the baseline cdf is specified.Some statistical properties of the TL-GPS distribution including moments, conditional moments, order statistics, and Rényi entropy are presented in Section 3. Maximum likelihood estimates of the unknown parameters are presented in Section 4. Monte Carlo simulations for special cases are conducted in Section 5. Applications are given in Section 6, followed by some concluding remarks.

The Model
In this section, we develop the TL-GPS class of distributions and derive some statistical properties which include series expansion of the pdf, quantile and hazard functions, sub-classes and some special cases.
Suppose that the random variable X has the Topp-Leone-G distribution with cdf defined by equation (1).Given N, let X 1 , ..., X N be independent and identically distributed random variables from the Topp-Leone-G distribution.Let N be a discrete random variable with a power series distribution (truncated at zero) and probability mass function (pmf) where a n ≥ 0 depends only on n, C(θ ) = ∑ ∞ n=1 a n θ n and θ ∈ (0, s) (s can be ∞) is chosen such that C(θ ) is finite and its three derivatives with respect to θ are defined and given by C (.),C (.) and C (.), respectively.The power series family of distributions includes Binomial, Poisson, Geometric and Logarithmic distributions.See Johnson et al. (1994) for additional details.Let X=min(X 1 , ..., X N ), then the conditional cdf of X|N = n is given by (4) The Topp-Leone-G Power Series class of distributions is defined by the marginal cdf of X.The general form of the cdf and pdf of the Topp-Leone-G Power Series class of distributions are given by and respectively.
On the other hand, if we consider X (n) = max(X 1 , . . ., X N ) and conditioning upon N = n, then the conditional distribution of X (n) given N = n is obtained as which is also a Topp-Leone-G distribution.The marginal cdf of X (n) , say F T L−GPS , is given by The hazard rate function (hrf) is given by Similarly, the reverse hazard rate function (rhrf) becomes

Quantile function
The quantile function of the TL-GPS class of distributions is easily obtained by inverting equation ( 5), This is equivalent to Therefore, we obtain the quantile values from the TL-GPS class of distributions by solving the non-linear equation using iterative methods in R, SAS or MATLAB software.

Expansion of the Density Function
Expansion of the density function of the TL-GPS class of distributions is presented in this sub-section.Equation ( 6) can be rewritten as Using the generalized binomial expansion the pdf of the TL-GPS class of distribution is given by Also, applying the generalized binomial expansion we get Furthermore, using the binomial expansion where g k+1 (x; ψ) = (k + 1)g(x; ψ)G(x; ψ) k is the exponentiated-G (Exp-G) distribution with power parameter (k + 1), and It follows that the TL-GPS distribution can be expressed as an infinite linear combination of Exp-G densities.

Sub-classes of the TL-GPS Distribution
We derive expressions for cdfs of sub-classes of the TL-GPS class of distributions and these are presented in Table 1.

Some Special Cases of the TL-GPS Class of Distributions
In this section, we present some special cases of the TL-GPS class of distributions.We consider cases when the baseline distribution are Weibull and log-logistic distributions.

Topp-Leone-Weibull-Poisson Distribution
The cdf and pdf of the Topp-Leone-Weibull Poisson (TL-WP) distribution are given by respectively, for α, β , b, θ > 0 and x > 0. The hrf and rhrf are given by , respectively.Figure 1 shows the plots of the pdfs and hrfs for the TL-WP distribution for selected parameter values.
As such, random numbers can be generated from the TL-WP distribution by numerically solving the non-linear equation ( 12).Quantile values of the TL-WP distribution are given in Table 2.  (0.8,2.1,0.9,1.0) (1.5,1.2,1.8,2.1) (3.5,1.2,5.5,0.6) (0.5,1.0,0.3,0.1) (2.0,3.0,0.4,1.5)  0 The cdf and pdf of the Topp-Leone-Weibull Binomial (TL-WB) distribution are given by respectively for α, β , b, θ > 0 and x > 0. The hrf and rhrf are given by respectively.Figure 2 shows the plots of the pdfs and hrfs for the TL-WB distribution for selected parameters values.Plots of the TL-WB pdf exhibit different shapes including symmetric, skewed to the right, skewed to the left and reverse-J shapes.Plots of the hrf of the TL-WB distribution shows different shapes including increasing, decreasing, bathtub and uni-modal shapes.
The quantile function of the TL-WB distribution can be obtained by solving the non-linear equation The Topp-Leone-G Power Series Class of Distributions with Applications As such, random numbers can be generated from the TL-WB power series distribution by numerically solving the non-linear equation ( 13).Quantile values of the TL-WB distribution are given in Table 3.
Therefore, random numbers can be generated from the TL-LLP distribution by numerically solving the non-linear equation ( 14).Quantile values of the TL-LLP distribution are given in Table 4.The cdf and the pdf of the Topp-Leone-Log-Logistic Binomial (TL-LLB) distribution are given by respectively for θ , b, c > 0 and x > 0. The hrf and rhrf are given by respectively.Figure 4 shows the plots of the pdfs and hrfs for the TL-WP distribution for selected parameters values.The quantile function obtained by solving the non-linear equation Therefore, random numbers can be generated from the TL-LLB distribution by numerically solving the non-linear equation ( 15).Quantile values of the TL-LLB distribution are given in Table 5.

Moments, Conditional Moments and Mean Deviations
In this section, the r th moment, conditional moments, mean deviations, Lorenz and Bonferroni curves of the TL-GPS class of distributions are presented.

Moments and Generating Function
If X follows the TL-GPS distribution and Y ∼ Exp-G(k + 1), then using equation ( 10), the r th moment of the TL-GPS class of distributions is obtained as follows where E[Y r ] is the r th moment of the Exp-G distribution with power parameter (k + 1) and η k+1 is given by equation ( 11).The moment generating function (mgf) of the TL-GPS class of distributions is given by where M Y (t) is the mgf of the Exp-G distribution and η k+1 is given by equation ( 11).

Conditional Moments
It is also of interest to obtain the r th conditional moments.The conditional r th moment of the TL-GPS distribution is given by • g k+1 (y; ψ)dy.

Mean Deviations, Lorenz and Bonferroni Curves
The mean deviation about the mean and mean deviation about the median as well as Lorenz and Bonferroni curves for the TL-GPS class of distributions are presented in this subsection.

Mean Deviations
If X has the TL-GPS distribution, then we can derive the mean deviation about the mean D(µ) and the median deviation about the median D(M) as follows and respectively, where µ = E(X) and M = Median(X) is the median of F T L−GPS (x; θ , b, ψ).Note that The Topp-Leone-G Power Series Class of Distributions with Applications and

Bonferroni and Lorenz Curves
In this subsection, we present Bonferroni and Lorenz curves for TL-GPS class of distributions.The Bonferroni and Lorenz curves are given by respectively, where q 0 x • g k (x; ψ)dx is the first incomplete moment of the Exp-G distribution with power parameter (k + 1) and η k+1 is given by equation (11).

Order Statistics and Rényi Entropy
In this section, we present the distribution of the order statistic and Rényi entropy of the TL-GPS class of distributions.

Distribution of Order Statistics
Let X 1 , X 2 , ..., X n be a random sample from the TL-GPS distribution and let X i:n be the corresponding i th order statistics.The pdf of the i th order statistic, X i:n is given by where B(.,.) is the beta function.Substituting the pdf and cdf of the TL-GPS family of distributions, we write Using the generalized binomial expansion and applying the result on power series raised to a positive integer, we get where b m, j = (ma 0 ) −1 ∑ m l=1 (l( j + 1) − m)a l b m−l and b 0, j = a j 0 (Gradshetyn (2000)).Also, using the following generalized binomial expansion Furthermore, applying the generalized binomial expansion Also, applying the binomial expansion where g r+1 (x; ψ) = (r + 1)g(x; ψ)G(x; ψ) r is the Exp-G distribution with power parameter (r + 1), and Therefore, substituting equation ( 17) in ( 16) we obtain It follows that the distribution of the i th order statistic from the TL-GPS class of distributions can be obtained directly from the distribution of the i th order statistic from the Exp-G distribution.

Rényi Entropy
In this subsection, Rényi entropy of the TL-GPS class of distributions is derived.Entropy measures the uncertainty or variation of a random variable.Rényi entropy (Rényi (1960)) is a generalization of Shannon entropy (Shannon (1951)).Rényi entropy of the TL-GPS class of distributions is defined as Note that f ν T L−GPS (x; θ , b, ψ) can be written as Considering the following series expansions we get Therefore, the Rényi entropy of the TL-GPS class of distributions is given by where and is the Rényi entropy of the Exp-G distribution with parameter q υ + 1 .As such, we can directly derive the Rényi entropy of the TL-GPS family of distributions from the Rényi entropy of the Exp-G distribution.

Estimation
In this section, we derive the maximum likelihood estimates of the parameter vector (θ , b, ψ) T of the TL-GPS class of distributions.Let X i ∼ TL-GPS(θ , b, ψ) and ∆ = (θ , b, ψ) T be the parameter vector.The log-likelihood = (∆) based on a random sample of size n is given by The elements of the score vector are given by and The equations obtained by setting the partial derivatives equal to zero are not in closed form.The maximum likelihood estimates of the parameters denoted by ∆ are obtained by solving the non-linear equation numerical methods such as the Newton-Raphson procedure.The multivariate normal distribution N(0, J −1 ( ∆)), where the mean vector 0 = (0, 0, 0) T and J −1 ( ∆) is the observed Fisher information matrix evaluated at ∆ can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions.

Simulation Study
In this section, a simulation study was conducted to assess consistency of the maximum likelihood estimators.We considered a special case of the TL-LLP distribution.We simulated for the sample sizes n= 25,50, 100, 200, 400, 800, and 1000, for N=1000 for each sample.We estimate the mean, root mean square error (RMSE), and average bias.The bias and RMSE for the estimated parameter, say, ∆ , are given by respectively.We consider simulations for the following sets of initial parameters values (I: θ = 0.5, b = 1.5, c = 1.0), (II: θ = 1.5, b = 1.5, c = 0.5), (III: θ = 0.5, b = 1.0, c = 1.5), and (IV: θ = 1.0, b = 1.5, c = 0.5).If the model performs better, we except the mean to approximate the true parameter values, the RMSE, and bias to decay toward zero for an increase in sample size.From the results in Table 6, the mean values approximate the true parameter values, RMSE and bias decay towards zero for all the parameter values.

Applications
In this section, we present examples to illustrate the usefulness and applicability of the TL-GPS class of distributions.This is achieved by applying the special case of Topp-Leone-Log-Logistic Poisson to two real data sets and comparing it to several equal-parameter non-nested models.Model parameters were estimated via the maximum likelihood estimation technique using the R software.The performance of the models were assessed using the following several goodness-of-fit statistics; -2loglikelihood (-2 log L), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Cramer von Mises (W * ) and Andersen-Darling (A * ) (as described by Chen and Balakrishnan (1995)), Kolmogorov-Smirnov (K-S) statistic and its p-value.The model that has smaller values of these above mentioned goodness-of-fit statistics and larger p-value of the K-S statistics is deemed as the best model.
Tables 7 and 8 show the model parameters estimates (standard errors in parenthesis) and the goodness-of-fit-statistics for the two data sets considered.Plots of the fitted densities, the histogram of the data and probability plots (Chambers et al. (1983)) are also presented to show how well our model fits the observed data set compared to the selected non-nested models.

Growth Hormone Data
The data consists of the estimated time since growth hormone medication until the children reached the targeted age.This data was used by Alizadeh et al. (2017) and are as follows: 2. 15, 2.20, 2.55, 2.56, 2.63, 2.74, 2.81, 2.90, 3.05, 3.41, 3.43, 3.43, 3.84, 4.16, 4.18, 4.36, 4.42, 4.51, 4.60, 4.61, 4.75, 5.03, 5.10, 5.44, 5.90, 5.96, 6.77, 7.82, 8.00, 8.16, 8.21, 8.72, 10.40, 13.20, 13.70.The estimated variance-covariance matrix for the TL-LLP model on growth hormone  has the smallest values of all the goodness-of-fit statistics and bigger value for the K-S p-value.We therefore conclude that the TL-LLP distribution performs better than the several models considered in this paper.The fitted densities and probability plots in Figure 5 also shows that the TL-LLP model fit the growth hormone data set better than the selected non-nested models.

Concluding Remarks
We developed a new class of distributions, called the Topp-Leone-G Power Series (TL-GPS) class of distributions.We presented some sub-classes and some special cases of the new proposed distribution.Structural properties were also derived including moments, mean deviations, distribution of order statistics, Rényi entropy, and maximum likelihood estimates.We also presented two real data examples to show the usefulness of the new class of distributions.The proposed model performs better than the several models on the selected data sets.

Figure 1 :
Figure 1: Pdfs and hrfs plots for the TL-WP distribution Plots of the TL-WP pdf exhibit different shapes including almost symmetric, left-skewed, right-skewed, and reverse-J shapes.Plots of the hrf of the TL-WP distribution shows different shapes including increasing, decreasing, upsidedown bathtub followed by bathtub and uni-modal shapes.

Figure 2 :
Figure 2: Pdfs and hrfs plots for the TL-WB distribution

Figure 4 :
Figure 4: Pdfs and hrfs plots for the TL-LLB distribution Plots of the TL-LLB pdf exhibit different shapes skewed to the right, skewed to the left, reverse-J and almost symmetric shapes.Plots of the hrf of the TL-LLB distribution shows different shapes including reverse-J, decreasing, bathtub followed by an upside-down bathtub and uni-modal shapes.

Table 4 :
Table of Quantiles for the TL-LLP Distribution

Table 5 :
Table of Quantiles for the TL-LLB Distribution

Table 6 :
Monte Carlo Simulation Results for TL-LLP Distribution: Mean, RMSE and Average Bias

Table 7 :
Parameter estimates and goodness-of-fit statistics for various fitted models for growth hormone data set