A novel transmuted lomax exponential distribution: Properties and applications

Abstract

Statistical lifetime distributions play a very important role in modeling data sets in various fields. Extending the existing distributions is of great interest in statistical research. The modification of the distributions provides more flexible model as compared to existing one. In this article, we propose a new probability model using Quadratic Rank Transmutation Map technique, named as Transmuted Lomax Exponential Distribution (TLED). The new distribution can model data sets with increasing, decreasing and bathtub shape hazard rates. Various statistical properties of the proposed distribution such as moments, order statistics, quantile function, mean residual life function and characteristic function are derived. Further, the parameter estimates are obtained through Maximum Likelihood method along with asymptotic confidence intervals. The utility of the new model is evaluated by analyzing two real data sets. In order to access the performance of the new model, several goodness of fit measures is used. The results indicate that the new model best fits the data as compared to the other extensions of the Lomax distribution.

Keywords

Lomax distributions maximum likelihood transmuted estimation hazard function

1 Introduction

Statistical distributions have great significance in predicting and describing the real-world phenomena in various fields such as economics, engineering, insurance and finance, reliability analysis, business and biostatistics. Determining the correct model for data under consideration is essential for reliability of the results [1]. In literature, many statistical distributions have been developed such as exponential, Rayleigh, Gamma, Log-normal and Weibull distribution for analysis of lifetime data. Due to the bulk of modern world data available for analysis, new probability distributions are required for better fit of the data sets. Over the past few years, many methodologies are developed to generate new families of distributions based on existing probability distributions in both discrete and continuous cases. Mostly, the new probability distributions are obtained by adding a new shape parameter through generators or by joining existing probability distributions or by making certain transformations in the original probability distributions. The aim of this development is to make existing distributions more flexible to adequately model the complex phenomena. Many researchers have established different procedures for adding new parameter to the classical distributions. Alpha power transformation method was suggested by Kundu and Mahdavi [2] to add an additional parameter in continuous distribution. A detailed study on techniques of developing continuous univariate distributions was carried out by Lee et al. [3]. Eugene et al. [4] introduced another method for generating new distributions named as beta family of distributions. Jones [5] extended the work of [4] using Kumaraswamy distribution instead of beta distribution. Another class of distributions known as T-X system was also proposed [6]. More recently, a new method was proposed called Quadratic Rank Transmutation Map (QRTM) for adding an extra parameter in existing distribution for making it more flexible to model several types of data more adequately [7]. This method is described as follows:

Let G(y) denote the cumulative distribution function (cdf) of the parent distribution. Then the cdf of transmuted probability distribution, denoted as F(y), is given as

$F (y) = (1 + λ) G (y) - λ (G (y))^{2}, | λ | ⩽ 1$ (1)

Then corresponding pdf of transmuted distribution is given as

$f (y) = g (y) [1 + λ - 2 λ G (y)]$ (2) where f(y) and g(y) are the pdf corresponding to F(y) and G(y) respectively. Transmuted Weibull, transmuted Gumbel and transmuted generalized extreme value distributions were proposed [8] using transmutation technique. Transmuted Pareto distribution was introduced as an extended form of the Pareto distribution [9]. Afify et al. [10] presented the transmuted complementary Weibull geometric distribution. Transmuted exponentiated Lomax and transmuted Lomax distribution were also introduced [11]. Aryal [12] introduced transmuted log logistic distribution.

Lomax or Pareto type II distribution was proposed by Lomax in 1954 to model and analyze lifetime data. The distribution is heavy tailed and also a member of family of distributions having decreasing failure rate [13]. It has applications in life testing and reliability problems in economics, engineering, business, biological sciences, actuarial modeling and survival analysis. Lomax distribution was successfully applied to model the data of business failure [14], size distribution of the computer files on servers [15] and also for wealth and income data [16]. Bryson [17] suggested the use of Lomax distribution as an alternate to exponential distribution for heavy tailed data.

Suppose random variable “Y” has Lomax Distribution having cdf

$F (y) = 1 - {[1 + (\frac{y}{β})]}^{- α}, y > 0 and α, β > 0$ (3)

The pdf of Lomax distribution is given by

$f (y) = \frac{α}{β} {[1 + (\frac{y}{β})]}^{- (α + 1)}, y > 0 and α, β > 0$ (4) where α denotes shape and β denotes scale parameter.

In the literature, researchers have introduced many extensions of Lomax distribution having different number of parameters. For example, Hassan and Abd-Allah [18] proposed exponentiated Weibull–Lomax. Kilany [19] introduced weighted Lomax distribution. Cordeiro et al. [20] proposed Gamma-Lomax. Hassan and Abd-Allah [21] studied exponentiated Lomax Geometric. Moltok et al. [22] proposed transmuted power Lomax distribution. Abdullahi and Ieren [23] introduced transmuted exponential Lomax distribution. Recently, Ijaz et al. [24] proposed a new extension of the Lomax distribution called Lomax exponential distribution by considering the transformation X = Y exp(Y). The pdf of the Lomax exponential distribution [24] is given by

$\begin{matrix} g (y) = & \frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)}, \\ y > 0 and α, β > 0 \end{matrix}$ (5)

The corresponding cdf of Lomax exponential distribution is given as

$\begin{matrix} G (y) = & 1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}, \\ y > 0 and α, β > 0 \end{matrix}$ (6)

To analyze and model real world data in several fields, various statistical distributions have been proposed but still there is a need of developing new distributions to model the real phenomena which are not best fitted by the existing classical distributions. The main aim of this article is to propose a new flexible model called as transmuted Lomax exponential distribution (TLE) using QRTM technique introduced by [7]. The proposed distribution has many tractable statistical properties and proposed model will provide an adequate fit to real life data as compared to the existing distributions. The article is organized as follows: In Section 2, we define new distribution and give its plots. In Section 3, statistical properties including quantile function, characteristic function, moment generating function and moments are derived. Section 4 provides the maximum likelihood estimates of the proposed model. In Section 5, we discussed order statistics and some entropy measures. Section 6 discusses reliability analysis of the proposed model. In section 7, the utility of the distribution is demonstrated by its application to two real data sets and a simulation study. Finally, conclusion is provided in section 8.

2 The transmuted lomax exponential distribution (TLED)

By putting equation (6) into equation (1), we get the cdf of TLED:

$\begin{matrix} F (y) = {1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}} * \\ [(1 + λ) - λ {1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}}] \end{matrix}$ (7)

And substituting Equations (5) and (6) into Equation (2), we get the pdf of the TLED:

$\begin{matrix} f (y) = \frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}] \end{matrix}$ (8) where y > 0, α>0, β>0, -1 ⩽ λ ⩽ 1. Here, λ is transmuted parameter, α and β are the shape and scale parameters respectively.

The graphs of pdf with different parameter values are displayed in Fig. 1. The plot of the pdf of TLED is positively skewed.

Fig. 1

Pdf plots of the TLED for different parameters values.

3 Statistical properties of the TLED

3.1 Quantile function and the median

Quantile function of a distribution is obtained by inverting its cumulative distribution function. If F(Y) is the cdf of Y, such that F (Y) = u then Y = F^-1 (u) is the corresponding quantile function where u ∼ U (0, 1) [25]. Quantile function for TLED is given as

$y = W [β {{(1 - \frac{(1 + λ) - \sqrt{(1 + λ)^{2} - 4 λ u}}{2 λ})}^{- (1 / α)} - 1}]$ (9)

where W(.) shows the product log function. Median of the TLED is obtained by putting u = 0.5 in Equation (9) to have

$y = W [β {{(1 - \frac{(1 + λ) - \sqrt{1 + λ^{2}}}{2 λ})}^{- (1 / α)} - 1}]$ (10)

3.2 Moments

Moments play important role in describing characteristics of the distribution such as central tendency, dispersion, skewness and kurtosis. The rth moment of the TLED is given as

$μ_{r}^{'} = E (y^{r}) = \int_{0}^{\infty} y^{r} f (y) dy, r = 1, 2, 3, 4 .$ (11)

Putting Equations (8) in (11), we have

$\begin{matrix} μ_{r}^{'} = \int_{0}^{\infty} y^{r} \frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}] dy \end{matrix}$ (12)

By using,

$\begin{matrix} {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ = \sum_{n = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) {(\frac{y exp (y)}{β})}^{n} \\ = \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) \frac{n^{k} y^{n + k}}{β^{n} k!} \end{matrix}$ (13)

In the above equation, solving the integral we get $\begin{matrix} \begin{matrix} μ_{r}^{'} = [(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) ψ_{r, n, k} \\ + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ψ_{r, n, k}] \\ [Γ (r + n + k + 2) - Γ (r + n + k + 1)] \end{matrix} \end{matrix}$ where,

$ψ_{r, n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{r + n + k} k!}$ (14)

“Coefficient of variation” (CV), “Coefficient of skewness” (CS) and “Coefficient of kurtosis” (CK) can be find using the following relationship

$C V = \sqrt{\frac{{μ^{'}}_{2}}{{μ^{'}}_{1}^{2}} - 1}$ (15)

$\begin{array}{l} C S = \frac{μ'_{3} - 3 μ'_{2} μ'_{1} + 2 μ_{1}^{' 3}}{{(μ'_{2} - μ_{1}^{' 2})}^{3 / 2}} \end{array}$ (16)

$\begin{array}{l} C K = \frac{μ'_{4} - 4 μ'_{3} μ'_{1} + 6 μ'_{2} μ_{1}^{' 2} - 3 μ_{1}^{' 4}}{{(μ'_{2} - μ_{1}^{' 2})}^{2}} \end{array}$ (17)

3.3 Moment generating function (Mgf)

The Mgf of the TLED is given

$M_{y} (t) = E (e^{ty}) = \sum_{r = 0}^{\infty} \frac{t^{r}}{r!} μ_{r}^{'}$ (18) $\begin{matrix} M_{y} (t) = \sum_{r = 0}^{\infty} \frac{t^{r}}{r!} ([(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) \\ ψ_{r, n, k} + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ψ_{r, n, k}] \\ [Γ (r + n + k + 2) - Γ (r + n + k + 1)]) \end{matrix}$ where

$ψ_{r, n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{r + n + k} k!}$ (19)

3.4 Characteristic function

The characteristic function of a random variable Y following TLE distribution is given as

$φ_{y} (t) = E (e^{ι ty})$ (20)

$φ_{y} (t) = E [cos (ty)] + iE [sin (ty)]$ (21)

By employing simple algebra and power series expansion, we get

$φ_{y} (t) = \sum_{r = 0}^{\infty} \frac{(- 1)^{r} (t)^{2 r}}{(2 r)!} μ_{2 r}^{'} + \sum_{r = 0}^{\infty} \frac{(- 1)^{r} (t)^{2 r + 1}}{(2 r + 1)!} μ_{2 r + 1}^{'}$ (22) where $\begin{matrix} μ_{2 r}^{'} = ([(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) \\ ψ_{2 r, n, k} + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ψ_{2 r, n, k}] \\ [Γ (2 r + n + k + 2) - Γ (2 r + n + k + 1)]) \end{matrix}$

$ψ_{2 r, n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{2 r + n + k} k!}$ (23) $\begin{matrix} μ_{2 r + 1}^{'} = ([(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) \\ ψ_{2 r + 1, n, k} + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ψ_{2 r + 1, n, k}] \\ [Γ (2 r + n + k + 3) - Γ (2 r + n + k + 2)]) \end{matrix}$

$ψ_{2 r + 1, n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{2 r + 1 + n + k} k!}$ (24)

4 Estimation of the parameters

In this section, the method of MLE will be used to estimate the parameters of the Transmuted Lomax Exponential distribution.

4.1 Maximum likelihood estimation

Suppose a sample Y₁, Y₂, Y₃, ... .,Yn of size ‘n’ is taken randomly from Transmuted Lomax Exponential distribution with parameters α, β and λ. The likelihood function is given as

$\begin{matrix} L = \prod_{i = 1}^{n} (\frac{α}{β} (y_{i} + 1) exp (y_{i}) {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α}]) \end{matrix}$ (25)

Taking natural logarithm of the equation (25), we get log likelihood function

$l (α, β, λ) = log L (\underline{y}; α, β, λ)$ (26)

$\begin{matrix} l (α, β, λ) = n log α - n log β + \sum_{i = 1}^{n} log (y_{i} + 1) \\ + \sum_{i = 1}^{n} y_{i} - (α + 1) \sum_{i = 1}^{n} log [1 + (\frac{y_{i} exp (y_{i})}{β})] \\ + \sum_{i = 1}^{n} log [1 - λ + 2 λ {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α}] \end{matrix}$ (27)

Computing partial derivatives of Equation (27) with respect toα, β and λ respectively, we obtain the following equations

$\begin{matrix} \frac{\partial l (α, β, λ)}{\partial α} = \frac{n}{α} - \sum_{i = 1}^{n} log [1 + (\frac{y_{i} exp (y_{i})}{β})] + 2 λ \\ \sum_{i = 1}^{n} (\frac{{[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α} log [1 + (\frac{y_{i} exp (y_{i})}{β})]}{[1 - λ + 2 λ {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α}]}) \end{matrix}$ (28)

$\begin{matrix} \frac{\partial l (α, β, λ)}{\partial β} = - \frac{n}{β} + \frac{(α + 1)}{β} \sum_{i = 1}^{n} (\frac{y_{i} exp (y_{i})}{β + (y_{i} exp (y_{i}))}) \\ + \frac{2 λ α}{β^{2}} \sum_{i = 1}^{n} (\frac{y_{i} exp (y_{i}) {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- (α + 1)}}{[1 - λ + 2 λ {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α}]}) \end{matrix}$ (29)

$\begin{matrix} \frac{\partial l (α, β, λ)}{\partial λ} \\ = \sum_{i = 1}^{n} (\frac{2 {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α} - 1}{[1 - λ + 2 λ {[1 + (\frac{y_{i} exp (y_{i})}{β})]}^{- α}]}) \end{matrix}$ (30)

Equating the result of (28–30) to zero, i.e $\frac{\partial l (α, β, λ)}{\partial α} = 0$ , $\frac{\partial l (α, β, λ)}{\partial β} = 0$ , $\frac{\partial l (α, β, λ)}{\partial λ} = 0$ , and solving equations simultaneously give the MLE of α, β and λ. Since the expressions do not have closed form, so therefore, cannot be solved analytically. Above equations can be solved numerically by employing iterative techniques like Bisection method, Newton-Raphson method or any other through statistical software.

4.2 Asymptotic confidence bounds

The exact distribution of the MLE cannot be derived due to explicit expressions of above equations. However, asymptotic confidence bounds for unknown parameters of Transmuted Lomax Exponential distribution can be obtained on the basis of asymptotic distribution of MLE. We assume that the MLE’s ( $\hat{α}$ , $\hat{β}$ , $\hat{λ}$ ) have approximately normal distribution with mean (α, β,λ) and covariance matrix Σ. All second order partial derivatives exist for TLED therefore $(\begin{matrix} \hat{α} \\ \hat{β} \\ \hat{λ} \end{matrix}) \sim N [(\begin{matrix} α \\ β \\ λ \end{matrix}), Σ]$

With, $\begin{matrix} Σ = - E {(\begin{matrix} V_{α α} & V_{α β} & V_{α λ} \\ V_{β α} & V_{β β} & V_{β λ} \\ V_{λ α} & V_{λ β} & V_{λ λ} \end{matrix})}^{- 1} \end{matrix}$

Where,

$\begin{matrix} \begin{matrix} V_{α α} = \frac{\partial^{2} l (α, β, λ)}{\partial α^{2}}, V_{α β} = \frac{\partial^{2} (α, β, λ)}{\partial α \partial β}, V_{α λ} = \frac{\partial^{2} (α, β, λ)}{\partial α \partial λ} \\ V_{β α} = \frac{\partial^{2} l (α, β, λ)}{\partial β \partial α}, V_{β β} = \frac{\partial^{2} l (α, β, λ)}{\partial β^{2}}, V_{β λ} = \frac{\partial^{2} l (α, β, λ)}{\partial β \partial λ} \\ V_{λ α} = \frac{\partial^{2} l (α, β, λ)}{\partial λ \partial α}, V_{λ β} = \frac{\partial^{2} l (α, β, λ)}{\partial λ \partial β}, V_{λ λ} = \frac{\partial^{2} l (α, β, λ)}{\partial λ^{2}} \end{matrix} \end{matrix}$

The estimate of Σ is obtained by replacing all parameters by their MLE’s. We get

$\hat{Σ} = - E {(\begin{matrix} {\hat{V}}_{α α} & {\hat{V}}_{α β} & {\hat{V}}_{α λ} \\ {\hat{V}}_{β α} & {\hat{V}}_{β β} & {\hat{V}}_{β λ} \\ {\hat{V}}_{λ α} & {\hat{V}}_{λ β} & {\hat{V}}_{λ λ} \end{matrix})}^{- 1}$ (31)

Using (31), approximately (1 - ξ) 100% confidence interval can be derived for parameters α, β, λ as $\hat{α} \pm Z_{ξ / - 2} \sqrt{{\hat{V}}_{α α}}, \hat{β} \pm Z_{ξ / - 2} \sqrt{{\hat{V}}_{β β}}, \hat{λ} \pm Z_{ξ / - 2} \sqrt{{\hat{V}}_{λ λ}}$

Where Z_ξ/2 denote the upper (ξ/2)^th percentile of standard normal distribution.

5 Order statistics and entropy measures

In this section, densities for order statistics and entropy measures such as Renyi entropy and q- entropy are considered.

5.1 Order statistics

Let a random sample Y1, Y2, ... . Yn is drawn from the TLE distribution and Y_1:n, Y_2:n, …, Y_n:n be corresponding order statistics of this sample in such a way that Y_1:n ⩽ Y_2:n ⩽ , … . , ⩽ Y_n:n. Then the pdf of jth order statistics, (j = 1,2,3, ... .,n) is given by

$f_{j : n} (y) = \frac{{[F (y)]}^{j - 1} {[1 - F (y)]}^{n - j} f (y)}{B (j, n - j + 1)}$ (32)

Where B(.,.) is beta function [26].

$\begin{matrix} f_{j : n} (y) = & \frac{1}{B (j, n - j + 1)} \sum_{k = 0}^{n - j} (\begin{matrix} n - j \\ k \end{matrix}) (- 1)^{k} \\ {[F (y)]}^{j + k - 1} f (y) \end{matrix}$ (33)

Substituting Equations (7) and (8) in (33), we get

$\begin{matrix} f_{j : n} (y) = n (\begin{matrix} n - 1 \\ j - 1 \end{matrix}) \sum_{k = 0}^{n - j} (\begin{matrix} n - j \\ k \end{matrix}) (- 1)^{k} \\ [{1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}} * \\ {{1 + λ {[1 + (\frac{y exp (y)}{β})]}^{- α}}]}^{j + k - 1} \\ [\frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}]] \end{matrix}$ (34)

By putting j = 1 in (34), we get pdf of minimum order statistics Y_1:n

$\begin{matrix} f_{1 : n} (y) = n \sum_{k = 0}^{n - 1} (\begin{matrix} n - 1 \\ k \end{matrix}) (- 1)^{k} \\ [{1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}} * \\ {{1 + λ {[1 + (\frac{y exp (y)}{β})]}^{- α}}]}^{k} \\ [\frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}]] \end{matrix}$ (35)

By putting j = n in (34), we get pdf of maximum order statistics Y_n:n

$\begin{matrix} f_{n : n} (y) = n [{1 - {[1 + (\frac{y exp (y)}{β})]}^{- α}} * \\ {{1 + λ {[1 + (\frac{y exp (y)}{β})]}^{- α}}]}^{n - 1} \\ [\frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} \\ [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}]] \end{matrix}$ (36)

5.2 Renyi entropy

The Renyi entropy for TLED is given as

${RE}_{y} (v) = \frac{1}{1 - v} log [\int_{0}^{\infty} (f (y))^{v} dy]$ (37) $\begin{matrix} \begin{matrix} {RE}_{y} (v) = \frac{v}{1 - v} log α - \frac{v}{1 - v} \\ log β + \frac{v}{1 - v} log (1 - λ) + \frac{1}{1 - v} \\ log [\sum_{k, l, m, n = 0}^{\infty} \frac{ω_{k, l, m, n}}{β^{m + k}} \frac{Γ (l + k + m + 1)}{(- (v + m + k))^{l + k + m + 1}}] \end{matrix} \end{matrix}$

Where, $ω_{k, l, m, n} = (\begin{matrix} v \\ l \end{matrix}) (\begin{matrix} - v (α + 1) \\ m \end{matrix}) (\begin{matrix} v \\ n \end{matrix}) (\begin{matrix} - α n \\ k \end{matrix}) {(\frac{2 λ}{1 - λ})}^{n}$

5.3 q- entropy

Havrda and Charvat [27] defined q- entropy as

$I_{H} (q) = \frac{1}{q - 1} {1 - \int_{0}^{\infty} (f (y))^{q} dy}, q > 0 and q \neq 1$ (38)

The q- entropy for a random variable Y from TLED is given as follows

$\begin{matrix} I_{H} (q) = & \frac{1}{q - 1} {1 - \sum_{k, l, m, n = 0}^{\infty} \frac{γ_{k, l, m, n} α^{q}}{β^{m + k + q}} \\ (\frac{Γ (l + k + m + 1)}{(- (q + m + k))^{l + k + m + 1}})} \end{matrix}$ (39)

Where, $γ_{k, l, m, n} = (\begin{matrix} q \\ l \end{matrix}) (\begin{matrix} - q (α + 1) \\ m \end{matrix}) (\begin{matrix} q \\ n \end{matrix}) (\begin{matrix} - α n \\ k \end{matrix}) (1 - λ)^{q} {(\frac{2 λ}{1 - λ})}^{n}$

6 Reliability analysis of TLED

6.1 Survival function

This function gives the likelihood of an object, individual or a system living longer than a given time. The survival function is given as

$\begin{matrix} S (y) = & {[1 + (\frac{y exp (y)}{β})]}^{- α} \\ [1 - λ + λ {[1 + (\frac{y exp (y)}{β})]}^{- α}] \end{matrix}$ (40)

6.2 Hazard rate function

The hazard rate function plays a fundamental role in lifetime modeling. It is also known as force of mortality or failure rate.

$h (y) = \frac{α (y + 1) exp (y) [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}]}{β [1 + (\frac{y exp (y)}{β})] [1 - λ + λ {[1 + (\frac{y exp (y)}{β})]}^{- α}]}$ (41)

If λ = 0, then the failure rate is same as Lomax exponential distribution. $\begin{matrix} h (y) = \frac{α (y + 1) exp (y)}{β [1 + (\frac{y exp (y)}{β})]} \end{matrix}$

Figure 2 represents different shapes of hazard rate function of TLED for several values of the parameter. The following figure shows that the monotonic as well as non-monotonic hazard rate shapes can be modeled through TLED. Therefore, the suggested distribution has more flexibility for modeling different lifetime datasets.

Fig. 2

Plots of hazard rate function of Transmuted Lomax Exponential Distribution.

6.3 Mean residual life function (MRL)

This function determines the expected additional lifetime of a unit given that the unit has lasted up to a fixed time t. The MRL say u (t) is given as

$\begin{matrix} u (t) = E (Y - t / - Y > t) \\ u (t) = \frac{1}{S (t)} [\int_{t}^{\infty} yf (y) dy] - t \end{matrix}$ (42)

$u (t) = \frac{\int_{t}^{\infty} y \frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}] dy}{{[1 + (\frac{t exp (t)}{β})]}^{- α} [1 - λ + λ {[1 + (\frac{t exp (t)}{β})]}^{- α}]} - t$ (43)

Using (13) and solving integral of above equation using upper incomplete gamma function $\frac{Γ (a, cx)}{c^{a}} = \int_{x}^{\infty} t^{a - 1} exp (- ct) dt$ and simplifying the result we get

$\begin{matrix} \begin{matrix} u (t) \frac{[(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) ς_{n, k} + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ς_{n, k}] [Γ (n + k + 2, - t) - Γ (n + k + 3, - t)]}{[(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α \\ n \end{matrix}) + λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α \\ n \end{matrix})] \frac{t^{n + k} n^{k}}{β^{n} k!}} - t \\ where ς_{n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{2 + n + k} k!} \end{matrix} \end{matrix}$ (44)

6.4 Mean waiting time

The mean waiting time reflects the waiting period that has elapsed after a unit failed on the condition that the failure happened in [0, t] interval. The mean waiting time, denoted by $\bar{u} (t)$ is given by

$\begin{matrix} \bar{u} (t) = E [t - Y / - Y ⩽ t] \\ \bar{u} (t) = t - \frac{1}{F (t)} \int_{0}^{t} yf (y) dy \end{matrix}$ (45)

$\bar{u} (t) = t - \frac{\int_{0}^{t} y \frac{α}{β} (y + 1) exp (y) {[1 + (\frac{y exp (y)}{β})]}^{- (α + 1)} [1 - λ + 2 λ {[1 + (\frac{y exp (y)}{β})]}^{- α}] dy}{{1 - {[1 + (\frac{t exp (t)}{β})]}^{- α}} * [(1 + λ) - λ {1 - {[1 + (\frac{t exp (t)}{β})]}^{- α}}]}$ (46)

Simplifying (46) and then solving integral using lower incomplete gamma function, $\frac{γ (a, cx)}{c^{a}} = \int_{0}^{x} t^{a - 1} exp (- ct) dt$ , $\bar{u} (t)$ can be written as

$\begin{matrix} \bar{u} (t) = t - \frac{[(1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α - 1 \\ n \end{matrix}) ɛ_{r, n, k} + 2 λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α - 1 \\ n \end{matrix}) ɛ_{r, n, k}] [γ (n + k + 2, - t) - γ (n + k + 3, - t)]}{1 - (1 - λ) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α \\ n \end{matrix}) ν_{n, k} - λ \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α \\ n \end{matrix}) ν_{n, k}} \end{matrix}$ (47) where, $\begin{matrix} \begin{matrix} ν_{n, k} = \frac{t^{n + k} n^{k}}{β^{n} k!} \\ ɛ_{r, n, k} = \frac{α n^{k}}{β^{n + 1} (- 1)^{n + k + 2} k!} \end{matrix} \end{matrix}$

6.5 Stress strength parameter

Let Y1 and Y2 be the independent and continuous random variables from TLE distribution such that Y₁ ∼ TLE (α₁, β, λ₁) and Y₂ ∼ TLE (α₂, β, λ₂), then the stress strength parameter, say S is given as

$S = \int_{- \infty}^{+ \infty} f_{1} (y) F_{2} (y) dy$ (48)

$\begin{matrix} S = (\begin{matrix} (1 - λ_{1}) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α_{1} - 1 \\ n \end{matrix}) - (1 - λ_{1}) (1 - λ_{2}) \sum_{n = k = 0}^{\infty} (\begin{matrix} - α_{1} - α_{2} - 1 \\ n \end{matrix}) - λ_{2} (1 - λ_{1}) \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α_{2} - α_{1} - 1 \\ n \end{matrix}) \\ + 2 λ_{1} \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α_{1} - 1 \\ n \end{matrix}) - 2 λ_{1} (1 - λ_{2}) \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α_{1} - α_{2} - 1 \\ n \end{matrix}) - 2 λ_{1} λ_{2} \sum_{n = k = 0}^{\infty} (\begin{matrix} - 2 α_{2} - 2 α_{1} - 1 \\ n \end{matrix}) \end{matrix}) \\ ω_{(n, k)} [Γ (n + k + 2) - Γ (n + k + 1)] \end{matrix}$ (49)

where $ω_{(n, k)} = \frac{α_{1} n^{k}}{β^{n + 1} (- 1)^{n + k} k!}$

7 Applications and simulation

7.1 Simulation study

Simulation study has been carried out by generating random data from TLED using equation (10). The simulation experiment was performed W = 100 times at different sample sizes n with different parameter combinations. Mean Square Error (MSE) and bias are calculated using the following formula $\begin{matrix} \begin{matrix} MSE = \frac{1}{W} \sum_{i = 1}^{W} ({\hat{a}}_{i} - a)^{2} \\ Bias = \frac{1}{W} \sum_{i = 1}^{W} ({\hat{a}}_{i} - a) \end{matrix} \end{matrix}$

where, a = (α, β, λ). Table 1 shows the values for average bias and MSE. From Table 1, it can be clearly seen that the values of bias and MSE decrease by increasing the sample size.

Table 1
Average MSE and Bias

Parameter N MSE ( $\hat{α}$ ) MSE ( $\hat{β}$ ) MSE ( $\hat{λ}$ ) Bias ( $\hat{α}$ ) Bias ( $\hat{β}$ ) Bias ( $\hat{λ}$ )

∝=2, β = 1.5, λ = 1 30 11.9932 6.056116 2.126028 1.198104 1.278938 1.262871

∝=2, β = 1.5, λ = 1 100 4.394712 1.589837 1.897881 0.6514929 0.5991997 0.9075802

∝=2, β = 1.5, λ = 1 200 3.998618 1.366789 1.67525 0.2497315 0.4851594 0.4936793

∝=5, β = 1.5, λ = 0.5 30 29.52892 1.494484 3.502715 0.4212225 0.2233092 1.711524

∝=5, β = 1.5, λ = 0.5 100 14.23456 1.334339 2.411169 –2.184051 –0.375053 1.03124

∝=5, β = 1.5, λ = 0.5 200 13.01173 1.285163 1.9542 –2.913859 –0.4015392 0.839173

∝=2.5, β = 1.5, λ = 1 30 4.56359 2.088077 2.205388 –2.029838 0.4505282 1.252266

∝=2.5, β = 1.5, λ = 1 100 4.426673 1.343842 1.985506 –2.099751 –0.6181949 0.9098496

∝=2.5, β = 1.5, λ = 1 200 4.39859 1.282503 1.4358 –2.994607 –0.9439927 0.406014

∝=3, β = 1.5, λ = 1 30 41.18275 7.625437 14.70299 2.224058 1.725246 3.576002

∝=3, β = 1.5, λ = 1 100 11.83388 2.552058 12.47451 0.8139115 0.3173025 2.950032

∝=3, β = 1.5, λ = 1 200 4.59965 1.796907 8.913386 0.08966367 0.2922758 1.914666

Parameter	N	MSE ( $\hat{α}$ )	MSE ( $\hat{β}$ )	MSE ( $\hat{λ}$ )	Bias ( $\hat{α}$ )	Bias ( $\hat{β}$ )	Bias ( $\hat{λ}$ )
∝=2, β = 1.5, λ = 1	30	11.9932	6.056116	2.126028	1.198104	1.278938	1.262871
∝=2, β = 1.5, λ = 1	100	4.394712	1.589837	1.897881	0.6514929	0.5991997	0.9075802
∝=2, β = 1.5, λ = 1	200	3.998618	1.366789	1.67525	0.2497315	0.4851594	0.4936793
∝=5, β = 1.5, λ = 0.5	30	29.52892	1.494484	3.502715	0.4212225	0.2233092	1.711524
∝=5, β = 1.5, λ = 0.5	100	14.23456	1.334339	2.411169	–2.184051	–0.375053	1.03124
∝=5, β = 1.5, λ = 0.5	200	13.01173	1.285163	1.9542	–2.913859	–0.4015392	0.839173
∝=2.5, β = 1.5, λ = 1	30	4.56359	2.088077	2.205388	–2.029838	0.4505282	1.252266
∝=2.5, β = 1.5, λ = 1	100	4.426673	1.343842	1.985506	–2.099751	–0.6181949	0.9098496
∝=2.5, β = 1.5, λ = 1	200	4.39859	1.282503	1.4358	–2.994607	–0.9439927	0.406014
∝=3, β = 1.5, λ = 1	30	41.18275	7.625437	14.70299	2.224058	1.725246	3.576002
∝=3, β = 1.5, λ = 1	100	11.83388	2.552058	12.47451	0.8139115	0.3173025	2.950032
∝=3, β = 1.5, λ = 1	200	4.59965	1.796907	8.913386	0.08966367	0.2922758	1.914666

7.2 Applications

The flexibility of the proposed distribution is demonstrated through its application to two real data sets.

Data set 1: Losses from wind disaster: The data set comprises 40 observations of losses resulting from wind disaster reported in 1977 [28]. The values (in millions) of the data set are given as

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 9, 15, 17, 22, 23, 24, 24, 25, 27, 32, 43.

Data set 2: Breaking stress of carbon fibres: The second set of the data representing breaking stress of carbon fibres (in Gba), consists of 100 values given by [29]

3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.

A comparison is made between the proposed distribution and various extensions of Lomax distribution such as power Lomax [30], exponential Lomax [31], transmuted power Lomax [22], weibull Lomax [32], transmuted exponential Lomax [23], Kumaraswamy-Generalized Lomax [33], Lomax exponential [24]. The usefulness of the proposed model is demonstrated by using various goodness of fit criteria such as “Akaike Information Criterion” (AIC), “Hannan-Quin Information Criterion” (HQIC), “Consistent Akaike Information Criterion” (CAIC), “Bayesian Information Criterion” (BIC), Anderson–Darling (AD), Kolmogorov-Smirnov (K-S) statistics and its p-value. Generally, model having minimum value of all these tests and maximum p-value would be considered best model to fit the given set of the data. Tables 2 and 4 represent the maximum likelihood estimates while Tables 3 and 5 show the corresponding goodness of fit measures, i.e., AIC, BIC, HQIC, CAIC, K-S, AD for the data sets 1 and 2. Tables 3 and 5 show that TLED adequately fits the data in comparison with the other fitted models. Figures 3 and 4 represent the Q-Q and P-P plot of the TLED for data sets 1 and 2 respectively.

Table 2
MLEs with standard errors (in brackets) for data set 1

Distribution M L E

E LD(α, β, λ) 1.049 (0.118) 0.075 (0.058) 0.007 (0.005)

K G L D(a, b, α, λ) 1.354 (0.209) 12.204 (6.750) 0.3559 (0.186) 14.708 (8.342)

W L D(a, b,α, β) 4.435 (2.821) 1.987 (0.346) 0.179 (0.21) 0.99 (0.671)

T E LD(α, β, θ, λ) 1.551 (0.369) 25.537 (12.337) 0.005 (0.002) 0.268 (0.308)

L E D(α, β) 0.107 (0.018) 6.365 (3.878)

T L E D(α, β, λ) 0.091 (0.023) 6.874 (3.829) 0.486 (0.312)

Distribution	M L E
K G L D(a, b, α, λ)	1.354 (0.209)	12.204 (6.750)	0.3559 (0.186)	14.708 (8.342)
W L D(a, b,α, β)	4.435 (2.821)	1.987 (0.346)	0.179 (0.21)	0.99 (0.671)
T E LD(α, β, θ, λ)	1.551 (0.369)	25.537 (12.337)	0.005 (0.002)	0.268 (0.308)
L E D(α, β)	0.107 (0.018)	6.365 (3.878)
T L E D(α, β, λ)	0.091 (0.023)	6.874 (3.829)	0.486 (0.312)

Table 3

Goodness of fit measures for fitted distributions for data set 1

Distribution	AIC	CAIC	BIC	HQIC	$- ln l (\overset{\land}{ψ})$	AD	K-S	p-value
E LD	254.810	255.496	259.802	256.602	124.405	2.505	0.202	0.08
K G L D	262.158	263.300	268.913	264.600	127.08	2.109	0.188	0.121
W LD	249.175	250.351	255.829	251.562	120.587	1.922	0.196	0.100
T E LD	253.402	254.579	260.056	255.789	122.701	2.216	0.210	0.065
L ED	253.723	254.047	257.101	254.944	124.86	1.904	0.201	0.07
T L ED	243.646	244.332	248.637	245.437	118.823	1.739	0.165	0.238

Table 4

MLEs with standard errors (in brackets) for data set 2

Dist.	M L E
P LD(α, β,λ)	1.567 (0.497)	3.328 (0.351)	33.901 (10.33)
E LD(α, β,λ)	9.726(2.534)	9.420(4.702)	0.136(0.088)
T P L D (α, β,θ,λ)	3.528 (2.849)	2.409 (0.619)	22.512 (9.739)	–0.852 (0.271)
W L D(a, b,α, β)	0.007 (0.005)	0.969 (0.567)	4.467 (2.615)	1.448 (1.094)
L E D(α, β)	0.945 (0.159)	26.137 (6.916)
T L E D(α, β,λ)	1.087 (0.201)	15.652 (6.853)	–0.894 (0.209)

Table 5

Goodness of fit measures for fitted distributions for data set 2

Dist.	AIC	CAIC	BIC	HQIC	$- ln l (\overset{\land}{ψ})$	AD	K-S	p-value
P LD	295.392	295.641	303.207	298.554	144.695	0.889	0.114	0.147
E LD	305.336	305.586	313.512	308.499	149.668	1.615	0.115	0.140
T P LD	294.067	294.885	304.488	298.284	143.033	0.723	0.105	0.217
W LD	295.868	296.289	306.289	300.086	143.934	0.664	0.077	0.291
L ED	295.384	295.508	300.595	297.493	145.692	0.641	0.101	0.254
T L ED	291.095	291.345	298.911	294.258	142.547	0.617	0.052	0.358

Fig. 3

Theoretical and empirical pdf and cdf along with Q-Q and P-P plot for data set 1 (losses from wind disaster).

Fig. 4

Theoretical and empirical pdf and cdf along with Q-Q and P-P plot for data set 2 (breaking stress of carbon fibers).

TTT Plot: This plot is a graphical tool to identify different shapes of hazard rate for a given data set. The hazard rate is constant of a data set, if the plot shows the straight line. The plot is concave for increasing hazard rate and convex for decreasing hazard rate. If the plot is first convex, then concave it indicates the bath-tub hazard rate shape [24]. The formula for determining TTT plot is given as

$\begin{matrix} G (\frac{r}{n}) = \frac{\sum_{i = 1}^{r} y_{i : n} + (n - r) y_{r : n}}{\sum_{i = 1}^{n} y_{i : n}}, r = y_{i : n} = 1, 2, 3, . . ., n \end{matrix}$

Where y_i:n shows order statistics.

The TTT plot for data sets are given in the Fig. 5. From the Fig. 5 (a and b), it is clear that both monotonic and non-monotonic shapes of hazard function can be modeled by the new distribution.

Fig. 5

(a) TTT plot for 1st data set (b) TTT plot for 2nd data set.

8 Conclusion

A new three parameter distribution called, Transmuted Lomax Exponential Distribution (TLED) was proposed and studied in this article. The suggested distribution is the extension of Lomax exponential distribution. The expressions of characteristic function, moments, quantile function, mgf were derived for TLED. The pdfs of order statistics and some entropy measures such as Renyi entropy and q-entropy were obtained. For estimation purpose, MLE method was used and also asymptotic confidence bounds based on asymptotic distribution of MLE were obtained. Reliability measures including survival function, stress strength parameter, hazard rate function and mean residual life function were derived for TLED. Plots of the pdf, cdf and hazard rate function, TTT plot were made to visualize the performance of the new model. TTT plot demonstrated that TLED can model data sets with different hazard rate shapes. Simulation results were conducted to examine the performance of the MLE. Finally, applications of the real data sets clearly showed the potentiality of the model. Goodness of fit criteria were used and results reveal that TLED is best fitted model with minimum AIC, BIC, CAIC, HQIC, K-S and AD statistics.

Conflict of interest

The authors have no conflict of interest.

Footnotes

Acknowledgment

This project was supported by the deanship of scienti?c research at Prince Sattam bin Abdulaziz University, Al-Kharj, Saudi Arabia.

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Distribution	M L E
E LD(α, β, λ)	1.049 (0.118)	0.075 (0.058)	0.007 (0.005)
K G L D(a, b, α, λ)	1.354 (0.209)	12.204 (6.750)	0.3559 (0.186)	14.708 (8.342)
W L D(a, b,α, β)	4.435 (2.821)	1.987 (0.346)	0.179 (0.21)	0.99 (0.671)
T E LD(α, β, θ, λ)	1.551 (0.369)	25.537 (12.337)	0.005 (0.002)	0.268 (0.308)
L E D(α, β)	0.107 (0.018)	6.365 (3.878)
T L E D(α, β, λ)	0.091 (0.023)	6.874 (3.829)	0.486 (0.312)