Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data

This paper addresses the estimation of the unknown parameters of the alpha power exponential distribution (Mahdavi and Kundu, 2017) using nine frequentist estimation methods. We discuss the finite sample properties of the parameter estimates of the alpha power exponential distribution via Monte Carlo simulations. The potentiality of the distribution is analyzed by means of two real data sets from the fields of engineering and medicine. Finally, we use the maximum likelihood method to derive the estimates of the distribution parameters under competing risks data and analyze one real data set.


Introduction
Several families of distributions have been constructed by extending common well-known distributions by adding one or more parameters to the baseline distribution. These generalized classes of distributions provide more flexibility in modeling data in applied areas such as economics, engineering, lifetime analysis, insurance, finance and environmental sciences. Mahdavi and Kundu (2017) introduced a new method for generating distributions by adding an extra shape parameter to a well known baseline distributions. The new method is called alpha power transformation (APT) method. Let G(x) be the cumulative distribution function (CDF) of a continuous random variable X, then the APT of G(x) is defined as and the corresponding probability density function (PDF) is Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data The rest of the paper is organized as follows. In Section 2, we discuss the APE distribution. Nine estimation methods are considered in Section 3. The performance and comparison of these methods in terms of mean relative estimates and root mean-squared error are provided in Section 4. The practical importance of the APE model is illustrated by analyzing two real data sets in Section 5. In Section 6, the APE distribution is applied to competing risks analysis and one real data set is used to show the flexibility of the APE distribution in modelling competing risks data. Finally, some concluding remarks are given in Section 7.

Alpha Power Exponential Distribution
Let X follows the exponential random variable with scale parameter λ > 0, then the CDF of the APE distribution is given by and the corresponding PDF is The survival and the hazard rate functions for the APE distribution, respectively, are in the following forms  The quantile function of the APE distribution is given by where 0 < p < 1. Also, the rth moments about zero can be expressed as follows Mahdavi and Kundu (2017) showed that the f (x) in (4) is a decreasing function of x > 0 if α ≤ e, and for α > e, f (x) is a unimodal function with mode at log(log α)/λ. They also proved that the hazard rate function in (6) is a decreasing function of x > 0 when α < 1, and for α > 1 it is an increasing function of x > 0. Figure 1 displays the various curves for the PDF and hazard rate functions of a APE distribution for λ = 1 and various values of α.
Mahdavi and Kundu (2017) studied the various properties of the APE distribution including explicit expressions of moments, moment generating function, order statistics and expression of the entropies. They used the method of maximum likelihood to estimate the unknown parameters and analyzed one real data set to show how the APE distribution works in practice.

Methods of Estimation
In this section, we consider nine methods of estimation to estimate the unknown parameters of the APE distribution. The used estimation methods are (1) maximum likelihood method, (2) methods of moments, (3) methods of L-moments, (4) ordinary least squares, (5) weighted least squares, (6) method of percentile, (7) maximum product of spacing method, (8) method of Cramér-von-Mises and (9) method of Anderson-Darling.

Maximum likelihood
Let x 1 , x 2 , · · · , x n be a random sample of size n from APE distribution, then from (4) we can write the log-likelihood function as follows Mahdavi and Kundu (2017) derived the two normal equations and obtained the the maximum likelihood estimate (MLE) of α denoted beα M LE for fixed λ aŝ and then obtained the MLE of λ denoted beλ M LE by solvinĝ upon obtainingλ M LE from (11), thenα M LE can be obtained from (10) asα M LE =α M LE (λ M LE ).

Methods of moments
The moment estimates (MEs) of α and λ, denoted byα M E andλ M E , of the APE distribution can be obtained by equating the first two sample moments with the first two theoretical moments obtained from (8). Theα M E andλ M E can be obtained by solving From (12) and for fixed α we can obtainλ M E (α) aŝ thenα M E can be obtained by solving the following nonlinear equation Onceα M E is obtained from (15),λ M E can be obtained directly from (14).

Methods of L-moments
The method of L-moments (LM) was introduced by Hosking (1990) to estimate the unknown parameters on the basis of linear combinations of order statistics. Let x 1:n < x 2:n < · · · < x n:n be the order statistics of a sample of size n from the APE distribution, then from Hosking (1990), the first two sample L-moments, respectively, are Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data The first two population L-moment for the APE distribution can be obtained as follow where Q(u|α, λ) is the quantile function of the APE distribution given by (7). The L-moment estimates of α and λ, denoted byα LM andλ LM can be obtained by equating the sample L-moments in (16) with the population L-monts in (17) and solving the following equations 1 = τ 1 (α LM ,λ LM ) and 2 = τ 2 (α LM ,λ LM ).

Ordinary least squares
Consider a random sample of size n from the APE distribution, and let x 1:n < x 2:n < · · · < x n:n be the order observations, then we can obtained the ordinary lease squares (OLS) estimates of α and λ, denoted byα OLS and λ OLS by minimizing with respect to α and λ. These estimates can be obtained by solving the following nonlinear equations and ψ 2 (x i:n |α, λ) = log(α) α − 1 x i:n e −λxi:n α (1−e −λx i:n ) .
3.5. Weighted least squares Using the same notation as in the previous subsection, the weighted least squares (WLS) estimates of α and λ, denoted byα W LS andλ W LS can be obtained for the APE distribution by minimizing the following function with respect to α and λ, or equivalently by solving the following two equations (n + 1) 2 (n + 2) i(n − i + 1) where ψ j (x i:n |α, λ), j = 1, 2 are defined by (18) and (19).

Method of percentile estimation
The percentile (PE) method can be used to obtain the PE estimates of α and λ which denoted byα P E andλ P E by equating the sample percentile points with the corresponding population percentile points. For the APE distribution and based on the quantile function (7), we can obtain the PE estimates by minimizing the following function with respect to α and λ Also, we can obtain these estimates by solving

Maximum product of spacing method
The maximum product of spacing (MPS) method was introduced by Cheng and Amin (1979,1983) as an alternative to the maximum likelihood method. They proved that the MPS estimates are efficient as the MLEs. For the APE distribution, we define the uniform spacings of a random sample of size n as where F (x 0:n |α, λ) = 0, F (x n+1:n |α, λ) = 1. Now, the MPS estimates of α and λ are denoted byα M P S andλ M P S can be obtained by maximizing the following function with respect to α and λ Theα M P S andλ M P S can also be obtained by solving where ψ j (x i:n |α, λ), j = 1, 2 are give by (18) and (19).
Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data

Minimum distance methods
In this subsection, we use two estimation methods based on the minimization of two well-known goodness-of-fit statistics. The two methods are Cramér-von-Mises method and method of Anderson-Darling which based on the difference between the estimates of the CDF and the empirical distribution function, see D'Agostino (1986).

Cramér-von-Mises
The Cramér-von-Mises (CM) estimatesα CM andλ CM of the APE distribution parameters α and λ are obtained by minimizing the following function with respect to α and λ These estimates also can be obtained by solving where ψ j (x i:n |α, λ), j = 1, 2 are given by (18) and (19).

Method of Anderson-Darling
The Anderson-Darling (AD) estimates of the APE distribution parameters α and λ denoted byα AD andλ AD can be obtained by minimizing the following function with respect to α and λ, or by solving the following two equations where ψ j (.|α, λ), j = 1, 2 are defined by (18) and (19).

Simulation Study
Here, we conduct Monte Carlo simulation studies in order to assess the performance among of the different estimation methods introduced and discussed in this article. The application of simulation studies involve the following steps: 1. Generate random samples of size n from ALE distribution By using this approach, the most efficient estimation method is expected to have MREs approaching to 1 and RMSEs decaying to 0, as the sample size getting large. The results of the Monte Carlo simulation procedures are obtained from simulating 10, 000 samples from APE(α, λ) distribution of sizes n = 10, 20, 30, . . . , 250 and (α, λ)=(0.5, 0.5), (3,3), (10, 0.3), (0.5, 3), (5, 1), (5,15). The estimates of the parameters are obtained by executing the nls function implemented in the software R. It should be noted that the estimation methods are put in the same conditions, i.e., using the same sample sizes and same initial values. However, some estimation methods namely method of moments and L-moments do not preform well in finding the parameter estimates and therefore they are excluded from the simulation study. Additionally, a rate convergence study is performed to assess the behavior of the convergence rate of the numerical solution. The rate study is carried out by calculating the rate of times that each estimation fails in finding the numerical solution. Figure  It is evident that for large sample size, the calculated RMSEs using these estimators tend to zero and the values of MREs tend to one. This implies that these estimators are consistent and asymptotically unbiased. Indeed, consistent estimates using these estimators can be achieved for n > 50.

Real Data Analysis
In this section, we illustrate the practical importance of the APE model by analyzing two real data sets. The first data set represents failures times of 50 items from Murthy et al. (2004). The second data represent the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938 (Lee, 1992).
The Kolmogorov-Smirnov (KS) test and its p-value (PV) are considered to check the goodness of fit. Tables 1 and 2 provide the KS test and PV for all fitted distributions. Both criteria provide evidence in favor of the APE distribution for both data sets. Hence, the APE distribution can be seen as a competitive distribution of practical interest for different fields. Figures 9 and 10 show the fitted PDF, CDF, survival function (SF) and PP plots of the APE distribution for both data sets, respectively.      Tables 3 and 4 provide the estimates of the unknown parameters of the APE distribution, for both the data sets, using seven methods of estimation and the values of − , KS and corresponding P-values. The WLS method can be used to estimate the parameters of the APE distribution for data I, whereas, the MLE method can be used to estimate the parameters of the APE distribution for data II. However, all estimation methods perform well.

Competing Risks Analysis
In reliability and medical studies, the cause of death of units can be due to more than one source. This problem is known in the literature as the analysis of competing risks model. In this section, we consider the APE distribution to the competing risks data. We derive the MLEs of the APE distribution parameters with different parameters. We also analyze one real data set to show the flexibility of the APE distribution in modelling competing risks data.

Estimation under competing risks data
Consider a lifetime experiment with n identical objects. Without loss of generality, we assume that the failure of the objects due to only two causes of failure. Let X 1i and X 2i are the latent failure time of the i − th object, where i = 1, 2, ..., n, then X i = min{X 1i , X 2i }. Based on the competing risks data we have the following observations where δ i ∈ (1, 2) is an indicator to determine the cause of failure of the ith unit. Furthermore, let I(δ i = 1) = 1, δ i = 1 0 else and I(δ i = 2) = 1, δ i = 2 0 else, then, the random variables n 1 = n i=1 I(δ i = 1) and n 2 = n i=1 I(δ i = 2) are the number of failures due to cause one and cause two, respectively, and n = n 1 + n 2 . Now, for a random sample of size n we can write the likelihood function of the observed data (x 1 , δ 1 ), , ..., (x n , δ n ) as follows Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data whereF k = 1 − F k (x), k = 1, 2. Now, let X ki s, k = 1, 2 are APE random variables with parameters α k and λ k with the following PDF and survival function in the formF Then from (20), (21) and (22) we can obtain the likelihood function as follows The log-likelihood function of (23) can be written in the following form From (24), the MLEs of α k and λ k , k = 1, 2 can be obtained by solving the following likelihood equations and where k = 1, 2. As it seems, equations (25) and (26) have no closed form solution in α 1 , α 2 , λ 1 and λ 2 , so a numerical technique is used to solve these equation.

Analysis of competing risks data set
In this subsection, we analyze one data set which was originally analyzed by Hoel (1972), and later by many authors, see for example Kundu et al. (2004), Cramer and Schmiedt (2011) and Nassar (2014, 2017). The data was obtained from a laboratory experiment in which male mice received a radiation dose of 300 roentgens at 35 days of age. The death cause for each mouse was classified by autopsy to be thymic lymphoma, reticulum cell sarcoma, or other causes. Here, for the analysis purpose we consider reticulum cell sarcoma as cause 1 and the other causes of death as cause 2.
From the original data we have n = 77 observations, where n 1 = 38 deaths due to cause one and n 2 = 39 deaths due to cause two. The MLEs of the unknown parameters are obtained and presented in Table 5 with the corresponding SEs. We compare the fitting of the APE distribution to the Hoel (1972) data with the exponential, Weibull and exponentiated exponential distributions. The estimates of the parameters of these distributions are given in Table 5. We use the Akaike information criterion (AIC), and Bayesian information criterion (BIC) to compare the competitive models. The values of AIC and BIC are also displayed in Table 5. From Table 5, we conclude that the APE distribution provides better fits than the exponential, Weibull and exponentiated exponential distributions.

Conclusion
We discuss the estimation of the parameters of the alpha power exponential (APE) distribution using nine methods, namely, maximum likelihood estimation, moments estimation, L-moments estimation, ordinary least squares estimation, weighted least squares estimation, percentile estimation, maximum product of spacing estimation, Cramér-von Mises estimation and Anderson-Darling estimation. An extensive simulation study is performed to compare the performance of these methods. Overall, the simulation results reveal that the MPS method perform better than other methods in terms of MREs and RMSEs. Also, the results of moments and L-moments are unsatisfactory, therefore their results are excluded from the simulation study. We have also compared estimators by two real data applications. Using two real data applications, we show that the APE provides better fits than other revial models. Furthermore, real data set is used to show the flexibility of the APE distribution in modelling competing risks data.