Modelling Lifetime Data and Athletes Red Cells Counts Using the New Topp-Leone-Lomax Poisson Distribution

1. Introduction

Over the past two centuries, statistical distributions gained appreciation from many researchers as a means of describing data. A number of distributions were developed and these include the exponential distribution which was a very important lifetime distribution. This distribution was good in modelling lifetime data with constant failure rate. However, the distribution could not accommodate other failure rates that include decreasing, increasing and non-monotone. Lomax developed the shifted Pareto (Pareto II) or Lomax distribution [9] which is a mixture of the exponential and gamma distributions. This distribution applies to data with decreasing failure rate and has heavier tails compared to the Weibull [19] and gamma distributions. Statisticians came with solutions to address the short comings of classical distributions. These solutions include transformation of the classical distribution for example the T-X transformation by Alzaatreh et al. [4], gamma transformations by Zografos and Balakrishnan [20], Ristiç and Balakrishnan [15], and Torabi and Montazari [18], half logistic transformation by Cordeiro et al. [7], to mention a few. The other techniques for generalizing classical distribution includes exponentiation, power transformation, competing risks, and compounding methods.

Some generalizations of the Lomax distribution include the exponentiated Lomax (Exp-Lx) by Abdul-Moniem and Abdel-Hammed [2], exponential Lomax (E-Lx) by Bassiouny et al. [5], and the Weibull-Lomax (W-Lx) by Tahir et al. [17]. Oguntunde et al. [13] developed the Topp-Leone Lomax (TL-Lx) distribution using the generalization by Al-Shomrani et al. [3]. The TL-Lx distribution is a has a pdf that is skewed to the right. The TL-Lx distribution has cumulative distribution function (cdf) and probability density function (pdf) given by

$$G(x;\alpha ,b,\lambda )={[1-{(1+\lambda x)}^{-2b}]}^{\alpha }$$

(1)

and

$$g(x;\alpha ,b,\lambda )=2\alpha b\lambda {(1+\lambda x)}^{-(2b+1)}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1},$$

(2)

respectively, for $\alpha ,b,\lambda >0.$ Other generalizations that involves the Topp-Leone generalization include the Topp-Leone odd exponential half logistic-G family of distributions by Chipepa and Oluyede [6], The Topp-Leone-Harris-G family of distributions with applications by Oluyede et al. [12], and the Topp-Leone odd Burr III-G family of distributions by Moakofi et al. [11].

This research generalizes the TL-LxP distribution by compounding it with the power series distribution. Power series distributions includes the Poisson, logarithmic, binomial, and geometric distributions as stated by Johnson et al. [8]. The Poisson distribution is considered as a special case from the power series distributions. Given a discrete random variable, say, M, having a power series distribution with probability mass function (pmf)

$$P(M=m)={\displaystyle \frac{a_m{\theta }^m}{C\left(\theta \right)}},m=1,2,...,$$

(3)

where $C\left(\theta \right)={\sum }_{m=1}^{\infty }{a}_m{\theta }^m$ is finite, $\theta >0$ and ${\left\{a_m\right\}}_{m\ge 1}$ a sequence of positive real numbers. Taking $X_{\left(1\right)}=min(X_1,X_2,...,X_M),$ the cumulative distribution function (cdf) and probability density function (pdf) of $X_{\left(1\right)}|M=m$ are defined by

$$F_{X_{\left(1\right)}|M=m}\left(x\right)=1-{\displaystyle \frac{C\left(\theta S\right(x\left)\right)}{C\left(\theta \right)}},$$

(4)

and

$$f_{X_{\left(1\right)}|M=m}\left(x\right)={\displaystyle \frac{\theta g\left(x\right)C^{\prime }\left(\theta S\left(x\right)\right)}{C\left(\theta \right)}},$$

(5)

where $S\left(x\right)$ is the survival function given by $1-F\left(x\right)$.

The rest of the paper is organized as follows: Statistical properties are presented in Section 2. Maximum likelihood estimation technique is presented in Section 3. Simulation study and inference are presented in Section 4 and 5, followed by conclusion in Section 6.

2. The Model

In this paper, the TL-Lx distribution by Oguntunde et al. [13] is compounded with the Poisson power series distribution to produce a new model called the Topp-Leone-Lomax Poisson (TL-LxP) distribution. The pdf and the cdf of the TL-LxP distribution is given by

$$F(x;\alpha ,b,\lambda ,\theta )=1-{\displaystyle \frac{exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]-1}{[exp(\theta )-1]}},$$

(6)

and

$$\begin{array}{ccc}\hfill f(x;\alpha ,b,\lambda ,\theta )& =& {\displaystyle \frac{2\alpha b\lambda \theta {(1+\lambda x)}^{-(2b+1)}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1}}{exp\left(\theta \right)-1}}\hfill \\ & \times & exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right],\hfill \end{array}$$

(7)

for $\alpha ,b,\lambda ,\theta >0.$ The corresponding hazard rate function is given by

$$\begin{array}{ccc}\hfill h(x;\alpha ,b,\lambda ,\theta )& =& {\displaystyle \frac{2\alpha b\lambda \theta {(1+\lambda x)}^{-(2b+1)}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1}}{exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]-1}}\hfill \\ & \times & exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right].\hfill \end{array}$$

(8)

The pdf of the TL-LxP takes reverse-J shape, right or left-skewed shapes as shown in Figure 1.

3. Some Statistical Properties

We present in this section quantile function, moments, moment generation function, and Rényi Entropy.

3.1. Quantile Function

The $u^{th}$ quantile function is defined by $F\left(x\right)=u,$$0<x<1.$ That is

$$1-{\displaystyle \frac{exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]-1}{[exp(\theta )-1]}}=u$$

which simplifies to

$$\begin{array}{c}\hfill x={\displaystyle \frac{1}{\lambda }}\left\{{\left[1-{\left\{{\displaystyle \frac{\theta -ln[1+(1-u)(e^{\theta }-1)]}{\theta }}\right\}}^{{\displaystyle \frac{1}{\alpha }}}\right]}^{-{\displaystyle \frac{1}{2b}}}-1\right\}\end{array}$$

(9)

The first, second and third quantiles for the TL-LxP distribution are obtained by substituting in equation (9) $u=0.25,0.5and0.75,$ respectively. Table 1 show quantile values for the TL-LxP distribution for selected parameter values ($\alpha ,b,\lambda ,and\theta$).

3.2. Moments and Moment Generating Function

The $r^{th}$ raw moment (${\mu }_r$) of the TL-LxP distribution is given by

$${\mu }_r=E\left(X^r\right)={\int }_{-\infty }^{\infty }{x}^{r}f(x;\alpha ,b,\lambda ,\theta )dx,$$

(10)

with $f(x;\alpha ,b,\lambda ,\theta )$ defined in Equation (7). Using the generalized binomial series expansion ${(1-z)}^a={\sum }_{j=0}^{\infty }{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{a}{j}}\right)z^j,$ we have

$$\begin{array}{ccc}\hfill E\left(X^r\right)& =& \sum _{i,j=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{i}{j}}\right){\displaystyle \frac{{(-1)}^{j}2\alpha b\lambda {\theta }^{i+1}}{(e^{\theta }-1)i!}}{\int }_0^{\infty }{x}^r{(1+\lambda x)}^{-(2b+1)}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha +j-1}dx.\hfill \end{array}$$

Let $u=1+\lambda x,$ implying $x={\displaystyle \frac{1}{\lambda }}(u-1)$ and $du=\lambda dx,$ we get

$$\begin{array}{ccc}\hfill {\int }_1^{\infty }{x}^r{(1+\lambda x)}^{-(2b+1)}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha +j-1}dx& =& \sum _{m=0}^{\infty }{(-1)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +j-1}{m}}\right)\hfill \\ & \times & {\int }_1^{\infty }{\displaystyle \frac{1}{{\lambda }^{r+1}}}{(u-1)}^r{u}^{-2mb-2b-1}du.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill {\int }_1^{\infty }{\displaystyle \frac{1}{{\lambda }^{r+1}}}{(u-1)}^r{u}^{-2mb-2b-1}du& =& {\int }_0^{\infty }{\displaystyle \frac{{(u-1)}^{(r+1)-1}}{{\lambda }^{r+1}{u}^{2mb+2b+1}}}du\hfill \\ & =& {\displaystyle \frac{1}{{\lambda }^{r+1}}}{\int }_0^{\infty }{\displaystyle \frac{{(u-1)}^{(r+1)-1}}{u^{2mb+2b+1}}}du\hfill \\ & =& {\displaystyle \frac{1}{{\lambda }^{r+1}}}B(r+1,2mb+2b+1-(r+1))\hfill \\ & =& {\displaystyle \frac{1}{{\lambda }^{r+1}}}B(r+1,2mb+2b-r)\hfill \\ & =& {\displaystyle \frac{\Gamma (r+1)\Gamma (2mb+2b-r)}{{\lambda }^{r+1}\Gamma (2mb+2b+1)}}\hfill \end{array}$$

which is a beta function over a half line. Therefore, the $r^{th}$ moment is given by

$$\begin{array}{ccc}\hfill E\left(X^r\right)& =& \sum _{i,j,m=0}^{\infty }{\displaystyle \frac{{(-1)}^{j+m}2\alpha b{\theta }^{i+1}}{(e^{\theta }-1)i!{\lambda }^r}}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +j-1}{m}}\right){\displaystyle \frac{\Gamma (r+1)\Gamma (2mb+2b-r)}{{\lambda }^{r+1}\Gamma (2mb+2b+1)}}.\hfill \end{array}$$

This result is important in determining the moments and measures of central tendency and dispersion (standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS), and coefficient of kurtosis (CK)). Table 2 presents the first five moments and the measures of dispersion for selected parameter values ($\alpha ,b,\lambda ,and\theta$).

The moment generating function of the TL-LxP is given by

$$\begin{array}{ccc}\hfill M_X\left(t\right)=E\left[e^{tX}\right]& =& \sum _{p=0}^{\infty }{\displaystyle \frac{t^{p}E\left[X^p\right]}{p!}}\hfill \\ & =& \sum _{i,j,m,p=0}{\displaystyle \frac{{(-1)}^{j+m}2\alpha b{\theta }^{i+1}}{(e^{\theta }-1)i!p!{\lambda }^p}}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{j}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +j-1}{m}}\right){\displaystyle \frac{\Gamma (p+1)\Gamma (2mb+2b-p)}{{\lambda }^{p+1}\Gamma (2mb+2b+1)}}\hfill \end{array}$$

3.3. Distribution of Order Statistics

The pdf of the $i^{th}$ order statistics is given by

$$\begin{array}{ccc}\hfill f_{i:n}\left(x\right)& =& {\displaystyle \frac{n!f\left(x\right)}{(i-1)!(n-i)!}}\sum _{q=0}^{\infty }{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{q}}\right){\left[F\left(x\right)\right]}^{q+i-1}\hfill \\ & =& {\displaystyle \frac{n!2\alpha b\lambda \theta {(1+\lambda x)}^{-(2b+1)}}{(i-1)!(n-i)!(e^{\theta }-1)}}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1}exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]\hfill \\ & \times & \sum _{q=0}^{\infty }{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{q}}\right){\left\{1-{\displaystyle \frac{exp\left[\theta (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]-1}{e^{\theta }-1}}\right\}}^{q+i-1}\hfill \\ & =& {\displaystyle \frac{n!2\alpha b\lambda \theta }{(i-1)!(n-i)!}}\sum _{q,s,t=0}^{\infty }{\displaystyle \frac{{(-1)}^{q+s+t}(1+t)}{(1+t){(e^{\theta }-1)}^{s+1}}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{q}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{q+i-1}{s}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{s}{t}}\right){(1+\lambda x)}^{-(2b+1)}\hfill \\ & \times & {[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1}exp\left[\theta (1+t)(1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]\hfill \end{array}$$

(11)

which is a linear combination of exponentiated TL-Lx distribution.

3.4. Rényi Entropy

The two commonly used measures of uncertainty are Shannon [16] and Rényi [14] entropies. Rényi entropy is a generalization of the Shannon entropy, hence, we will derive only the Rényi entropy in this paper. Rényi entropy for the TL-LxP distribution is derived as follows:

$$I_R\left(\nu \right)={\displaystyle \frac{1}{1-\nu }}log\left({\int }_0^{\infty }{\left[f(x;\alpha ,b,\lambda ,\theta )\right]}^{\nu }dx\right).$$

(12)

$$\begin{array}{ccc}\hfill {\left[f(x;\alpha ,b,\lambda ,\theta )\right]}^{\nu }& =& {\left[{\displaystyle \frac{2\alpha b\lambda \theta }{e^{\theta }-1}}\right]}^{\nu }{(1+\lambda x)}^{-(2b\nu +\nu )}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha \nu -\nu }\hfill \\ & \times & exp\left[\theta \nu (1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha })\right]\hfill \\ & =& {\left[{\displaystyle \frac{2\alpha b\lambda \theta }{e^{\theta }-1}}\right]}^{\nu }\sum _{t,s,u=0}^{\infty }{\displaystyle \frac{{(-1)}^{t+u}{\left(\theta \nu \right)}^s}{s!}}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{t}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha t+\alpha \nu -\nu }{u}}\right){(1+\lambda x)}^{-2bu-2b\nu -\nu }.\hfill \end{array}$$

Now

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }{(1+\lambda x)}^{-2bu-2b\nu -\nu }dx& =& {\int }_0^{\infty }{\displaystyle \frac{dx}{{(1+\lambda x)}^{2bu+2b\nu +\nu }}}.\hfill \end{array}$$

Let $u=1+\lambda x$⇒$du=\lambda dx$⇒$dx={\displaystyle \frac{1}{\lambda }}du.$ When $x=0,u=1$ and when $x=\infty ,u=\infty .$ Note

$${\int }_1^{\infty }{\displaystyle \frac{{(t-1)}^a}{t^b}}=B(a+1,b-a-1)={\displaystyle \frac{\Gamma (a+1)\Gamma (b-a-1)}{\Gamma \left(b\right)}}.$$

Hence,

$$\begin{array}{ccc}\hfill {\int }_1^{\infty }{\displaystyle \frac{dx}{{(1+\lambda x)}^{2bu+2b\nu +\nu }}}& =& {\displaystyle \frac{1}{\lambda }}{\int }_1^{\infty }{\displaystyle \frac{{(u-1)}^{1-1}}{{(1+\lambda x)}^{2bu+2b\nu +\nu }}}\hfill \\ & =& {\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{\Gamma \left(1\right)\Gamma 2bu+2b\nu +\nu -1}{\Gamma (2bu+2b\nu +\nu )}}\hfill \\ & =& {\displaystyle \frac{1}{\lambda (2bu+2b\nu +nu-1)}}.\hfill \end{array}$$

Therefore, we define Rényi entropy of the TL-LxP distribution by

$$\begin{array}{ccc}\hfill I_R\left(\nu \right)& =& {\displaystyle \frac{1}{1-\nu }}log\left[{\left[{\displaystyle \frac{2\alpha b\lambda \theta }{e^{\theta }-1}}\right]}^{\nu }\sum _{t,s,u=0}^{\infty }{\displaystyle \frac{{(-1)}^{t+u}{\left(\theta \nu \right)}^s}{s!}}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{t}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha t+\alpha \nu -\nu }{u}}\right){\displaystyle \frac{1}{\lambda (2bu+2b\nu +nu-1)}}\right].\hfill \end{array}$$

4. Estimation

If $X_i\sim TL-LxP(\alpha ,b,\lambda ,\theta )$ with the parameter vector $\Delta ={(\alpha ,b,\lambda ,\theta )}^T$. The total log-likelihood $\ell =\ell \left(\Delta \right)$ from a random sample of size n is given by

$$\begin{array}{ccc}\hfill \ell & =& nln\left[2\alpha b\lambda \theta \right]-(2b+1)\sum _{i=1}^{n}ln[1+\lambda x]+(\alpha -1)\sum _{i=1}^{n}ln[1-{(1+\lambda x)}^{-2b}]\hfill \\ & +& \sum _{i=1}^n\theta [1-{(1-{[1+\lambda x]}^{-2b})}^{\alpha }]-\sum _{i=1}^n(e^{\theta }-1).\hfill \end{array}$$

The elements of the score vector $U=\left({\displaystyle \frac{\partial \ell }{\partial \alpha }},{\displaystyle \frac{\partial \ell }{\partial b}},{\displaystyle \frac{\partial \ell }{\partial \lambda }},{\displaystyle \frac{\partial \ell }{\partial \theta }}\right)$ are given below

$${\displaystyle \frac{\partial \ell }{\partial \alpha }}={\displaystyle \frac{n}{\alpha }}+\sum _{i=1}^{n}ln[1-{(1+\lambda x)}^{-2b}]-\sum _{i=1}^n\theta {[1-{(1+\lambda x)}^{-2b}]}^{\alpha }ln[1-{(1+\lambda x)}^{-2b}],$$

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \ell }{\partial b}}& =& {\displaystyle \frac{n}{b}}-2\sum _{i=1}^{n}ln[1+\lambda x]+2(\alpha -1)\sum _{i=1}^n{\displaystyle \frac{ln[1+\lambda x]{(1+\lambda x)}^{-2b}}{1-{(1+\lambda x)}^{-2b}}}\hfill \\ & -& 2\sum _{i=1}^n\theta \alpha ln[1+\lambda x]{(1+\lambda x)}^{-2b}{[1-{(1+\lambda x)}^{-2b}]}^{\alpha -1},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \ell }{\partial \lambda }}& =& {\displaystyle \frac{n}{\lambda }}-(2b+1)\sum _{i=1}^n{\displaystyle \frac{\lambda }{1+\lambda x}}+2(\alpha -1)\sum _{i=1}^n{\displaystyle \frac{bx{(1+\lambda x)}^{-2b-1}}{1-{(1+\lambda x)}^{-2b}}}\hfill \\ & -& 2\sum _{i=1}^n\theta b\alpha x{(1+\lambda x)}^{-2b}{(1-{[1+\lambda x]}^{-2b})}^{\alpha -1},\hfill \end{array}$$

. and

$${\displaystyle \frac{\partial \ell }{\partial \theta }}={\displaystyle \frac{n}{\theta }}+\sum _{i=1}^{n}1-{[1-{(1+\lambda x)}^{-2b}]}^{\alpha }-n,$$

respectively. The solutions to the nonlinear equation ${\left({\displaystyle \frac{\partial \ell }{\partial \alpha }},{\displaystyle \frac{\partial \ell }{\partial b}},{\displaystyle \frac{\partial \ell }{\partial \lambda }},{\displaystyle \frac{\partial \ell }{\partial \theta }}\right)}^T=\mathbf{0},$ gives the maximum likelihood estimates of the model parameters.

5. Simulations

To evaluate the consistency of the model parameter estimation technique. A Monte Carlo simulation study was conducted. Results for the simulation study are summarised in Table Table 3. A simulation study was conducted for sample sizes n= 20, 40, 60,120, 240 and 480 for N=3000 repetitions from the TLL-LxP distribution. The average bias (AB) and root mean square error (RMSE) for an estimated parameter, say ($\widehat{\Phi }$), is calculated using the following formulae

$$ABIAS\left(\widehat{\Phi }\right)={\displaystyle \frac{{\sum }_{i=1}^N{\widehat{\Phi }}_i}{N}}-\Phi ,\mathrm{and}RMSE\left(\widehat{\Phi }\right)=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{({\widehat{\Phi }}_i-\Phi )}^2}{N}}},$$

respectively. The simulation study results demonstrates that the parameter estimates from the TL-LxP distribution are consistent since the mean values approaches the true parameter values for increasing sample size. Furthermore, RMSE and ABias decays as the sample size increases demonstrating consistence in the parameter estimation process. The results are presented in Table Table 3.

6. Inference

To assess the utility of the new compounded model. Three real data sets are used for demonstration purposes. The TL-LxP distribution was compared to the TL-Lx by Oguntunde et al. [13], exponentiated Lomax (Exp-Lx) by Abdul-Moniem and Abdel-Hammed [2], exponential Lomax (E-Lx) by Bassiouny et al. [5], and the Weibull-Lomax (W-Lx) by Tahir et al. [17]. Various goodness-of-fit (GoF) statistics were used to assess model performance. The GoF statistics used are the -2log-likelihood statistic $(-2ln(L\left)\right)$, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Consistent Akaike Information Criterion (AICC), Cramér-von Mises ($W^{*}$) and Anderson-Darling ($A^{*}$). Generally, a good fitting model is expect to have smaller values for all the GoF statistics. Graphical displays were further used to demonstrate how the proposed model fit the data sets. The graphs considered are the histogram of data and fitted pdf, probability plot, fitted cdf, Kaplan-Meier curves, scaled total time on test (TTT) transform proposed by Aarset [1] and the fitted hazard rate function. Data analysis results are presented in Table 4 and Table 5

6.1. Failure Times Data

The first dataset represents time to failure of 59 test conductors of 400 micrometer length. The specimens were subjected to high temperature and current density. The times to failure in hours for the 59 test conductors are shown below. 6.545, 9.289, 7.543, 6.956, 6.492, 5.459, 8.120, 4.706, 8.687,2.997, 8.591, 6.129, 11.038, 5.381, 6.958, 4.288, 6.522, 4.137, 7.459, 7.495, 6.573, 6.538, 5.589, 6.087, 5.807, 6.725, 8.532, 9.663,6.369, 7.024, 8.336, 9.218, 7.945, 6.869, 6.352, 4.700, 6.948, 9.254, 5.009, 7.489, 7.398, 6.033, 10.092, 7.496, 4.531, 7.974, 8.799, 7.683, 7.224, 7.365, 6.923, 5.640, 5.434, 7.937, 6.515, 6.476, 6.071, 10.491, 5.923.

Table 4 shows the model parameters and GoF statistics for the new model and some selected competing models. The standard errors for the parameter estimates are given in parenthesis. The estimated variance-covariance matrix for the TL-LxP model on the failure times dataset is given by

$$\left(\begin{array}{cccc}2.5478& 1.7123& -0.0118& 0.3297\\ 1.7123& 1.1507& -0.0079& 0.2216\\ -0.0118& -0.0079& 5.5589\times {10}^{-5}& -0.0015\\ 0.3297& 0.2216& -0.0015& 0.0426\end{array}\right)$$

with the approximate asymptotic $95\%$ two-sided confidence intervals for the parameter estimates $\alpha ,b,\lambda$ and $\theta$ given by $10.6068\pm 3.1286,$ $3.2757\pm 2.1026,$$0.0291\pm 0.0146$ and $30.7911\pm 0.4049,$ respectively.

The results presented in Table 4 shows that the TL-LxP distribution performs better than the other selected generalizations of the Lomax distribution available in the literature. This is supported by the evidence that the TL-LxP distribution has lower values for all the GoF statistics compared to the selected competing models. Graphical plots from the fitted model also show how good the TL-LxP distribution performs on the failure times dataset.

6.2. Red Cells Data

The second dataset represents red cell counts of 202 athletes from Australia. The data are was obtained from the sn package of the R software. The observations are as follows: 3.80, 3.90, 3.90, 3.91, 3.95, 3.95, 3.96, 3.96, 4.00, 4.02, 4.03, 4.06, 4.07, 4.08, 4.09, 4.09, 4.10, 4.11, 4.11, 4.12, 4.13, 4.13, 4.14, 4.15, 4.16, 4.16, 4.17, 4.17, 4.19, 4.20, 4.20, 4.21, 4.23, 4.23, 4.24, 4.24, 4.25, 4.26, 4.26, 4.27, 4.27, 4.30, 4.31, 4.31, 4.32, 4.32, 4.32, 4.35, 4.36, 4.36, 4.37, 4.38, 4.38, 4.39, 4.40, 4.40, 4.40, 4.41, 4.41, 4.41, 4.42, 4.42, 4.44, 4.44, 4.44, 4.45, 4.45, 4.46, 4.46, 4.46, 4.46, 4.46, 4.48, 4.49, 4.50, 4.50, 4.51, 4.51, 4.51, 4.51, 4.52, 4.53, 4.54, 4.55, 4.56, 4.57, 4.58, 4.62, 4.63, 4.63, 4.63, 4.64, 4.66, 4.68, 4.71, 4.71, 4.71, 4.71, 4.73, 4.75, 4.75, 4.76, 4.77, 4.77, 4.78, 4.81, 4.81, 4.82, 4.82, 4.83, 4.83, 4.83, 4.83, 4.84, 4.86, 4.86, 4.87, 4.87, 4.87, 4.87, 4.87, 4.87, 4.88, 4.89, 4.89, 4.90, 4.90, 4.91, 4.91, 4.92, 4.93, 4.93, 4.94, 4.95, 4.95, 4.96, 4.97, 4.97, 4.98, 4.99, 5.00, 5.00, 5.00, 5.01, 5.01, 5.01, 5.02, 5.02, 5.03, 5.03, 5.03, 5.03, 5.04, 5.04, 5.08, 5.09, 5.09, 5.09, 5.10, 5.11, 5.11, 5.11, 5.11, 5.11, 5.13, 5.13, 5.13, 5.13, 5.16, 5.16, 5.16, 5.16, 5.17, 5.17, 5.18, 5.21, 5.21, 5.22, 5.22, 5.24, 5.24, 5.25, 5.29, 5.31, 5.32, 5.33, 5.33, 5.34, 5.34, 5.34, 5.34, 5.38, 5.40, 5.48, 5.48, 5.49, 5.50, 5.59, 5.66, 5.69, 5.93, 6.72.

It can be observed from the results shwon in Table 5 that the TL-LxP distribution performs better on red cell counts data compared to the other selected models since it has the smallest values for the goodness-of-fit statistics.

The estimated variance-covariance matrix for the TL-LxP model for Red cells dataset is given by

$$\left(\begin{array}{cccc}2.2023\times {10}^{-17}& -1.5450\times {10}^{-15}& -9.1006\times {10}^{-13}& -1.8641\times {10}^{-16}\\ -1.5450\times {10}^{-15}& 1.0839\times {10}^{-13}& 6.3844\times {10}^{-11}& 1.3077\times {10}^{-14}\\ -9.1006\times {10}^{-13}& 6.3844\times {10}^{-11}& 3.7606\times {10}^{-8}& 7.7029\times {10}^{-12}\\ -1.8641\times {10}^{-16}& 1.3077\times {10}^{-14}& 7.7029\times {10}^{-12}& 1.5778\times {10}^{-15}\end{array}\right)$$

and the approximate $95\%$ two-sided confidence intervals for the parameter estimates are $\alpha ,b,\lambda$ and $\theta$ are given by $275.1900\pm 9.1981\times {10}^{-9},$ $18.0300\pm 6.4529\times {10}^{-7},$$0.0287\pm 3.8009\times {10}^{-4}$ and $12.1300\pm 7.7855\times {10}^{-8},$ respectively.

6.3. Acute Bone Cancer Data

The third dataset represents the survival times (in days) of 73 patients diagnosed with acute bone cancer as reported by Mansour et al. [10]. The data are as follows: 0.09, 0.76, 1.81, 1.10, 3.72, 0.72, 2.49, 1.00, 0.53, 0.66, 31.61, 0.60, 0.20, 1.61, 1.88, 0.70, 1.36, 0.43, 3.16, 1.57, 4.93, 11.07, 1.63, 1.39, 4.54, 3.12, 86.01, 1.92, 0.92, 4.04, 1.16, 2.26, 0.20, 0.94, 1.82, 3.99, 1.46, 2.75, 1.38, 2.76, 1.86, 2.68, 1.76, 0.67, 1.29, 1.56, 2.83, 0.71, 1.48, 2.41, 0.66, 0.65, 2.36, 1.29, 13.75, 0.67, 3.70, 0.76, 3.63, 0.68, 2.65, 0.95, 2.30, 2.57, 0.61, 3.93, 1.56, 1.29, 9.94, 1.67, 1.42, 4.18, 1.37.

Furthermore, results presented in Table 6 supports the superiority of the TL-LxP distribution compared to the other selected models. From the plots of the fitted densities (Figure 2, Figure 5 and Figure 8), we observe that the model is applicable to data containing some outlying values and that the model is applicable to data with increasing or non monotonic failure rates as shown in Figure 4, Figure 7 and Figure 10.

The estimated variance-covariance matrix for the TL-LxP model for acute bone cancer dataset is given by

$$\left(\begin{array}{cccc}0.5494& -0.0559& 0.2477& -0.1902\\ -0.0559& 0.0978& -0.0999& -0.4936\\ 0.2477& -0.0999& 0.1845& 0.2232\\ -0.1902& -0.4936& 0.2232& 4.1950\end{array}\right)$$

and the approximate $95\%$ two-sided confidence intervals for the parameter estimates are $\alpha ,b,\lambda$ and $\theta$ are given by $2.5917\pm 1.4528,$ $0.5558\pm 0.6130,$$0.5351\pm 0.8418$ and $4.1833\pm 4.0144,$ respectively.

7. Concluding Remarks

A new compounded distribution was developed. The new distribution is a generalization of the Lomax distribution via the Topp-Leone generalization and compounding using the Poisson distribution. Statistical properties for this distribution were derived. Results from applications to real datasets of the new model shows the utility of the proposed model in comparison to some selected models.