Abstract
The Lindley distribution has been generalized by several researchers in recent years. In this paper, we introduce and study a new generalization of extended power Lindley distribution named alpha power transformed extended power Lindley (APTEPL) distribution that provides better fits than the extended Power Lindley distribution and existing generalizations. It includes the alpha power transformed power Lindley, alpha power transformed extended Lindley, alpha power transformed Lindley, extended power Lindley, power Lindley, extended Lindley and Lindley distribution as a special cases. In this article various properties of the APTEPL distribution such as moments, moment generating function, Characteristic function and cumulant generating function, quantiles and Order Statistics are derived. Method of maximum likelihood estimation is used to obtain the model parameters. A simulation study is performed to examine the performance of the maximum likelihood estimators of the parameters. Two data sets have been utilized to show how the APTEPL distribution works in practice.
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1 Introduction
The power Lindley (PL) distribution and two-parameter Lindley (TPL) distribution were introduced by [8] and [17] respectively, these distributions are an extension to a known Lindley distribution which was proposed by [14]. After two years [2] introduced extended power Lindley distribution whose CDF and PDF are, respectively, given by
Several methods of generating new statistical distributions were presented in the literature review such as, [16, 4], and [3] for more information can you see [12] and [11]. Another important method for generation a new distribution was proposed by [15] named alpha power transformed (APT). Its CDF and PDF are given as:
and
where F(x) and f(x) be the CDF and PDF of a random variable x and \(x>0\).
Several authors utilized APT method to re-extend Lindley distributions in particular cases; for example, [5] introduced the alpha power transformed Lindley (APTL) distribution. Also, [6] presented alpha power transformed inverse Lindley (APTIL) distribution. Whereas alpha power transformed power Lindley (APTPL) distribution was introduced by [9]. Moreover, [7] proposed alpha power transformed power inverse Lindley (APTPIL) distribution.
This study aims to introduce a new lifetime distribution, referred to alpha power transformed extended power Lindley (APTEPL) distribution using the alpha power transformed method to the extended power Lindley (EPL) distribution. This model will be more flexibility in analysing the lifetime data. We are motivated to introduce the (APTEPL) distribution because: (1) It includes a number of well-known lifetime sub-models. (2) It can simulate monotonically increasing, decreasing, constant, bathtub, upside-down bathtub, and increasing - decreasing - increasing hazard rates. and (3) It can be viewed as a suitable model for fitting skewed data that may not be properly fitted by other common distributions, and it can also be used to solve a variety of problems in various fields, such as public health, biomedical studies, and industrial reliability and survival analysis.
The paper is organized as follows: Sect. 1 is introduction. In Sect. 2, the alpha power transformed extended power Lindley (APTEPL) distribution is defined. Some properties of APTEPL distribution are derived in Sect. 3 which includes: \(r^{th}\) moment, moment generating function, Characteristic function , cumulant generating function, ordered statistics and quantiles. The estimation of the unknown parameters by using maximum likelihood estimator are studied in Sect. 4. In Sect. 5, a simulation study is conducted to evaluate the performance of the different estimators. Finally, in Sect. 6 comparison the performance of proposed distribution with other distributions is verified using two real data sets, the first data set represent the waiting time (in minutes) of 100 bank customers and second data set consists of 128 bladder cancer patients.
2 Alpha Power Transformed Extended Power Lindley Distribution
Let \(X\in R^+\) a random variable from extended power Lindley (EPL) distribution [2] with the scale parameter \(\theta >0\) and shape parameters \(\beta ,\delta >0\). By substituting the equations (2) and (1) into (3) and (4), then the CDF and PDF for APTEPL distribution given in the following expression:
and
The corresponding survival function S(x) and hazard rate function h(x) respectively, as follow:
and
A random variable X that follows alpha power transformed extended power Lindley distribution in (6) was denoted by \(X\sim APTEPL(x,\alpha ,\beta ,\delta ,\theta )\).
The PDF and HRF plots of \(X\sim APTEPL(x,\alpha ,\beta ,\delta ,\theta )\) are presented in Figs. 1 and 2 respectively,
Also, the reversed hazard rate function r(x) and the cumulative hazard rate function H(x) of the APTEPL distribution are, respectively, given as follows:
and
2.1 Special Cases
Some well-known distributions are special cases of the APTEPL distribution. We present these cases for selected values of parameters.
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If \(\alpha =1\) the APTEPL distribution reduces to extended power Lindley (EPL) distribution [2].
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If \(\delta = 1\) the APTEPL distribution reduces to alpha power transformed extended Lindley (APTEL) distribution. (new)
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If \(\beta = 1\) the APTEPL distribution reduces to alpha power transformed power Lindley (APTPL) distribution [9].
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If \(\beta = \delta = 1\) the APTEPL distribution reduces to alpha power transformed Lindley (APTL) distribution [5].
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If \(\alpha = \beta = 1\) the APTEPL distribution reduces to power Lindley (PL) distribution [8].
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If \(\alpha = \delta = 1\) the APTEPL distribution reduces to two-parameter Lindley (TPL) distribution [17]
-
If \(\alpha = \beta = \delta = 1\) the APTEPL distribution reduces to the well known Lindley (L) distribution [14].
Table 1 shows the specific values of the parameters used to generate the special cases of APTEPL distribution.
Note: Some spacial cases for the hazard rate function arising from the APTEPL distribution by assigning relevant values of the parameters.
3 Some Mathematical Properties of APTEPL Distribution
This section includes some properties of the APTEPL distribution like, \(r^{th}\) moment, moment generating function, characteristic function, cumulant generating function, and order statistics. and obtain the mean, standard deviation, skewness, kurtosis, and coefficients of variation.
3.1 \(r^{th}\) Moment
The \(r^{th}\) moment of APTEPL distribution is
Using power series ,
Therefore, (11) can be written as follows:
Also, by using series of Taylor,
Hence (12) can be expressed as follows:
After some steps we found that
where :
Remark: the mean of APTEPL distribution is \(E(X)=\mu ={\mu _{1}^{'}}\) also, The variance, skewness, kurtosis and coefficient of variation can be obtained by: \(variance= \sigma ^2=\mu _{2}^{'}-\mu ^{2}\) ; \(\,\,\, skewness=\frac{\mu _{3}}{\sigma ^{3}} =\frac{\mu _{3}^{'}-3\mu _{2}^{'} \mu +2 \mu ^{3}}{(\mu _{2}^{'}- \mu ^{2} )^{(3/2)}}\) ; \(kurtosis=\frac{\mu _{4}}{\sigma ^{2}} =\frac{\mu _{4}^{'}-4\mu _{3}^{'} \mu +6\mu _{2}^{'}\mu ^{2}-2\mu ^{3}}{(\mu _{2}^{'}-\mu ^{2} )^{2}}\) and \(CV= \frac{\sigma }{\mu }=\sqrt{\frac{\mu _{2}^{'}}{\mu ^{2}}-1}\)
3.2 Moment Generating Function, Characteristic Function and Cumulant Generating Function
-
The moment generating function is given by
$$\begin{aligned} M_{X}(t)&=E(e^{tx})\nonumber \\ {}&=\int _{0}^{\infty }e^{tx}f(x)\,\,df\nonumber \\&=\sum _{r=0}^{\infty }\frac{t^{r}}{r!}\int _{0}^{\infty }x^{r} f(x,\alpha ,\beta ,\theta ) \,\, dx\nonumber \\&=\sum _{r=0}^{\infty }\frac{t^{r}}{r!}\mu _{r}^{'} \end{aligned}$$(14)In case of using the equations (13) and (14) we found that moment generating function of APTEPL distribution can be expressed as follows:
$$\begin{aligned} M_{X}(t)=\frac{\alpha }{\alpha -1}\sum _{k=0}^{\infty }\sum _{l=0}^{k} C_{k,l}\frac{t^{r}}{r!}\left[ \frac{\Gamma (\frac{r}{\delta }+l+1)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+1}}+\beta \frac{\Gamma (\frac{r}{\delta }+l+2)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+2}}\right] \end{aligned}$$(15) -
The Characteristic function is given by
$$\begin{aligned} \phi _{X}(t)=E(e^{itx})=\int _{0}^{\infty }e^{itx}f(x)\,\,df \end{aligned}$$similarly, after some steps the Characteristic function can be expressed:
$$\begin{aligned} \phi _{X}(t)=\frac{\alpha }{\alpha -1}\sum _{k=0}^{\infty }\sum _{l=0}^{k} C_{k,l}\frac{(it)^{r}}{r!}\left[ \frac{\Gamma (\frac{r}{\delta }+l+1)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+1}}+\beta \frac{\Gamma (\frac{r}{\delta }+l+2)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+2}}\right] \end{aligned}$$(16)where \(i=\sqrt{-1}\)
-
The Cumulant generating function is given by:
$$\begin{aligned} K(t)=\log \phi _{X}(t) \end{aligned}$$(17)In case of using the equations (16) and (17) we found that cumulant generating function of APTEPL distribution can be expressed:
$$\begin{aligned} K(t)=\log \left( \frac{\alpha }{\alpha -1}\right) +\log C \end{aligned}$$(18)where:
$$\begin{aligned} C=\sum _{k=0}^{\infty }\sum _{l=0}^{k} C_{k,l}\frac{(it)^{r}}{r!}\left[ \frac{\Gamma (\frac{r}{\delta }+l+1)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+1}}+\beta \frac{\Gamma (\frac{r}{\delta }+l+2)}{\left[ \theta (1+k)\right] ^{\frac{r}{\delta }+l+2}}\right] \end{aligned}$$
3.3 Quantile Function
The quantile function of the APTEPL distribution random variable X is \(Q_{X}(u)=G_{X}^{-1}(u)\) , \(0<u<1\), and for any \(\alpha ,\beta ,\delta ,\theta >0\) and \(\alpha \ne 1\). In the following steps, we find out the expression of \(Q_{X}\).
By considering the \(p=1-\left( 1+\frac{\beta \theta x^{\delta }}{\theta +\beta }\right) e^{-\theta x^{\delta }}\), the CDF of APTEPL distribution (5) can be written as
By solving \(G_{X}(x)=u\) for p, we get
By solving \(p=1-\left( 1+\frac{\beta \theta x^{\delta }}{\theta +\beta }\right) e^{-\theta x^{\delta }}\) for x, we obtain
By using negative Lambert \(W_{-1}\) function, from (21) we get
From (22) and put \(Q_{X}(u)=x\), we obtain
By substituting (20) into (23), then the quantile function \(Q_{X}(u)\) is given as
whereas \(W_{-1}(.)\) is the negative Lambert W function and \(u\in (0,1)\). Further, the first quantile Q1 obtained by substituting \(u=0.25\) in (24). The median Q2 and third quantile Q3 obtained in a similar manner. Tables 2 and 3 show behavior the first three quartiles of APTEPL distribution with the different values of Its parameters. We observed that if value of \(\alpha ,\,\beta , \, \text {and} \, \delta\) are constant,while value of \(\theta\) increases, value of each of Q1, Q2 and Q3 decrease. But in the case of values of \(\beta ,\,\delta ,\, \text {and} \,\theta\) are fixed, values of Q1, Q2 and Q3 increase when value of \(\alpha\) increases. Also, in the case of value of \(\beta\) increases, whereas values of \(\alpha ,\,\delta ,\, \text {and} \, \theta\) are fixed , values of each of Q1, Q2 and Q3 increase too. Finally, values of each of Q1, Q2 and Q3 increase when value of \(\delta\) increases, while value of \(\alpha ,\,\beta , \,\text {and}\, \theta\) are constant.
3.4 Order Statistics
Let \(X_{(i)}, i =1,2,......., n\) denoted to n independent random variables from any distribution with CDF \(F_{X} (x)\) and PDF \(f_{X} (x)\), then the PDF of \(X_{(i)}\) is given by :
In case of substituting (6) and (5) into equation (25), the PDF of \(X_{i}\) according to APTEPL distribution given as the following:
Now, the PDF of \(X_{(n)}\) and \(X_{(1)}\) respectively, given by :
and
4 Simulation Study
The simulation study for MLEs of APTEPL distribution is performed by generating \(N=1000\) samples of sizes \(n=20, 40,\,\,80,\,\, and \,\, 100\) from APTEPL distribution and studied the behavior of estimates, based on certain measures, which are mean square errors (MSEs) and absolute biases (ABs). Considering CDF of APTEPL distribution and after performing mathematical calculations it was found that if u is a random number from U(0, 1), then
of the APTEPL distribution with parameters \(\alpha ,\beta ,\delta ,\) and \(\theta\). where \(W_{-1}(.)\) is the negative Lambert W function.
Tables 4 and 5 show the empirical results of a simulation study using MLEs and the goodness.fit() function in the \('AdequacyModel'\) package via the R program. The MSEs and ABs decrease as the sample size increases.
5 Maximum Likelihood Estimator
Let \(X= ( x_{1}, x_{2}, ........, x_{n} )\) be the random variables of size n belong to \(APTEPL( x,\alpha ,\beta ,\delta , \theta )\), then the Likelihood function can be written as,
Substituting (6) into (30) and taking the logarithm function for two side we found that the \(Loglikelihood\,\, function\) is given by:
By taking the partial derivative for Log L in (31) with respected to \(\alpha ,\theta , \beta \, \text {and} \, \delta\) respectively , we got that
The maximum likelihood estimator of \(\alpha\) ,\(\theta\) , \(\beta\) and \(\delta\) can be obtained by solving the above four nonlinear equations when \(\frac{\partial \log L}{\partial \alpha }=0\), \(\frac{\partial \log L}{\partial \theta }=0\) , \(\frac{\partial \log L}{\partial \beta }=0\) and \(\frac{\partial \log L}{\partial \delta }=0\) with using the numerical approach.
6 Application
In this section, two real data sets are used to compare the performance of proposed APTEPL distribution with three existing models: alpha power transformed power Lindley (APTPL) distribution [9] , alpha power transformed extend Lindley (APTEL) which is as spacial case of our proposed distribution and alpha power transformed Lindley (APTL) distribution [5]. To compare the performance of our model with the others models the following criterions are used: Akaike information criterion (AIC), Bayesian information criterion (BIC), Corrected Akaike information criterion (AICc), and Kolmogorov-Smirnov goodness of fit test (KS). The distribution with the smallest values of \(AIC, BIC,\,\, \text {and} \,\, AICc\) or maximum \(p-value\) for (KS) test is considering as the best model for the given data. In this section the numerical results are obtained by using of R software.
First data set taken from Ijaz Muhammad et al. [10]. The data set represent the waiting time (in minutes) of 100 bank customers. The MLE of the parameters and standard errors and goodness of fit statistics of the model parameters are provided in Tables 6 and 7. It is clear from Table 7 that the APTEPL distribution gives better fit than the other distributions. Figure 3 shows the theoretical and empirical \(PDF\,\, \text {and} \,\,CDF\) of the \(APTEPL, APTPL, APTEL,\,\, \text {and} \,\, APTL\) distributions using Bank customer data.
Second data set consists of 128 bladder cancer patients taken from Lee Elisa et al. [13] and used by Ahmad et al. [1]. The MLEs of the parameters and standard errors and goodness of fit statistics of the model parameters are provided in Table 8 and 9, from Table 9 it can be that the APTEPL distribution gives better fit than the other distributions. Figure 4 shows the theoretical and empirical PDF and CDF of the \(APTEPL, APTPL, APTEL,\,\, \text {and} \,\, APTL\) distributions regarding data set 2.
7 Conclusion
In this article, we have proposed alpha power transformation extended power Lindley distribution which is extension of extended power Lindley distribution using alpha power transformation. Some properties of the proposed distribution are derived such as moments, moment generating function, Characteristic function , cumulant generating function, quantiles and order statistics. From the proposed distribution various distributions related to Lindley distribution can be obtained as a special cases of it. If \(\beta =1\) alpha power transformed power Lindley is obtained, similarly, if \(\delta =1\) alpha power transformed extended Lindley, if \(\beta =1\) and \(\delta =1\) alpha power transformed Lindley, if \(\alpha =1\) extended power Lindley, if \(\alpha =1\) and \(\beta =1\) power Lindley, if \(\alpha =1\) and \(\delta =1\) extended Lindley and if \(\alpha =1\), \(\beta =1\) and \(\delta =1\) Lindley distribution are obtained. The performance of the proposed distribution is verified by using real data sets and performing simulation study. It was observed that the proposed distribution is good w.r.t the comparing criterions \(AIC, BIC \,\, \text {and}\,\, AICc\). Therefore it was better than other models considered in this article. It works well for the real life data to model the waiting time (in minutes) of 100 bank customers data and the data of 128 bladder cancer patients.
Availability of data and material
The real data sets used in Applications: Data set 1 taken from Ijaz et al. (2021). Data set 2 taken from Lee and Wang (2003) and used by Ahmad et al. (2019).
Abbreviations
- CDF :
-
The cumulative distribution function
- PDF :
-
The probability density function
- L :
-
Lindley distribution
- PL :
-
Power Lindley distribution
- EPL :
-
Extended power Lindley distribution
- TPL :
-
Two parameter Lindley distribution
- APT :
-
Alpha power transformed
- APTL :
-
Alpha power transformed Lindley distribution
- APTEL :
-
Alpha power transformed extended Lindley distribution
- APTPL :
-
Alpha power transformed power Lindley distribution
- APTEPL :
-
Alpha power transformed extended power Lindley distribution
- MLEs :
-
Maximum likelihood estimators
- AIC :
-
Akaike information criterion
- BIC :
-
Bayesian information criterion
- AICc :
-
Corrected Akaike information criterion
- KS :
-
Kolmogorov-Smirnov goodness of fit test
- f(x):
-
Probability density function
- F(x):
-
Cumulative distribution function
- S(x):
-
Survival function
- h(x):
-
Hazard function
- H(x):
-
Cumulative hazard function
- r(x):
-
Reverse hazard rate function
- \(E(x^r)\) :
-
The rth moment of distribution.
- \(M_{x}(t)\) :
-
Moment generating function
- \(\phi _{x}(t)\) :
-
Characteristic function
- K(t):
-
Cumulant generating function
- Q(u):
-
Quantile function
- \(W_{-1}(.)\) :
-
Negative Lambert function
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The authors would like to express their heartfelt appreciation to the Editors of the Journal of Statistical Theory and Applications for waiving the APC payment. The authors would also like to thank the editor and anonymous referees for their insightful comments and suggestions, which helped to improve the paper’s quality.
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Conceptualization, F. E.; Writing - original draft preparation, F. E. ; review and editing, C. D. S.
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Eissa, F.Y., Sonar, C.D. Alpha Power Transformed Extended power Lindley Distribution. J Stat Theory Appl 22, 1–18 (2023). https://doi.org/10.1007/s44199-022-00051-3
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DOI: https://doi.org/10.1007/s44199-022-00051-3