Abstract
The moment exponential distribution has recently been generalized by a number of authors. The two-parameter alpha power-transformed moment exponential (APTME) distribution is introduced. In terms of fit, the APTME distribution outperforms the moment exponential distribution. Exact expressions for ordinary moments, incomplete and conditional moments, the moment generating function, the cumulant generating function, and information measures are obtained for some APTME distribution features. To estimate the model parameters, six well-known frequentist techniques were applied. The behavior of the various estimators was investigated using a simulated exercise. To examine the practical significance of the APTME distribution, real-world datasets were used. In terms of performance, we show that the APTME distribution beats other models.
1. Introduction
Several univariate continuous distributions have been widely used for modelling lifetime data in environmental, engineering, financial, and biomedical sciences, among other fields. However, there is still a strong need for significant improvement of the classical distributions via various techniques for modelling various data lifetimes. In this regard, Mahdavi and Kundu [1] introduced the alpha power transformation (APT) method, which is based on adding a parameter to a family of distributions to improve their flexibility. The APT- G family’s cumulative distribution function (CDF) is defined as follows:
The probability density function related to (1) is
In literature, many probability distributions have been prepared by several researches using this approach; for instance, APT Weibull (APTW) distribution [2], APT inverse exponential [3], APT extended exponential distribution [4], APT Lindley model [5], APT inverse Lindley model [6], APT power Lindley [7], APT Kumaraswamy distribution [8], APT Pareto model [9], APT inverse Lomax [10], Marshall–Olkin APT Weibull distribution with different estimation methods based on Type-I and Type-II censored samples [11], and APT Lomax distribution [12], among others.
The moment exponential (ME) distribution (also known as the length biased exponential (LBE) distribution) is regarded as one of the most important univariate and parametric models. It is commonly used in the analysis of lifetime data as well as problems involving the modelling of failure processes. This distribution is also a flexible lifetime distribution model that may fit some failure datasets well. Dara and Ahmed [13] proposed the ME with the PDF and CDF files shown as follows:where is the scale parameter. Different values of the shape parameter lead to different shapes of the density function.
The following notable contributions to the associated literature are made by this study: As a generalization of the ME distribution, the APT moment exponential (APTME) distribution is presented; it is a more flexible model than the ME distribution. Two real-world datasets are used to assess the model’s applicability. Maximum likelihood (M1) under complete and right-censored (RC) samples, least squares (M2), weighted least squares (M3), maximum product of spacing (M4), Cramer von Mises (M5), and Anderson Darling (M6) methods are used to estimate the unknown parameters of the APTME distribution. The APTME model’s structural characteristics and parameter estimators are derived for a number of APTME models. The following is a presentation of the study’s content. In Section 2, we present the APTME model’s PDF, CDF, and the hazard rate function (HRF). Section 3 contains some structural properties. Section 4 discusses parameter estimation using the methods of M1, M2, M3, M4, M5, and M6. In Section 5, two applications to real-world datasets demonstrate the APTME model’s empirical importance. Section 6 provides a comparison and precision of these methods versus a simulation study. Conclusions are provided at the end of the paper.
2. Description of the APTME Model
The APTME distribution, based on the APT method, is introduced in this section. The PDF, CDF, survival function, and the HRF of the APTME distribution are defined.
Definition 1. A random variable X is said to have APTME distribution, denoted by APTME , with shape parameter and scale parameter , if the PDF and CDF of X for are given bySome plots of the density (4) for different values of and are illustrated in Figure 1. They reveal that the PDF of X is quite flexible and can take asymmetric forms, among others.
Other important functions, such as the survival function and the HRF, can be used to highlight the APTME model in addition to the density function (4) and distribution function (4). These functions are especially important in analyzing the survival and are included in the definition below.
Definition 2. The survival function (SF) and the HRF of the APTME distribution are given, respectively, bySome plots of the HRF are displayed in Figure 2. An increasing or decreasing hazard rate is frequently used to model survival and failure time data. As seen from Figure 2, the HRF of the APTME takes an upside-down bathtub shape, i.e., it is increasing and then decreasing, with a single maximum.
Definition 3. The APTME quantile function, say Q (u) = F−1 (u), is straightforward given by means of the inverse transformation method inverting (4) as follows:where G (.) is the CDF of the ME distribution, xq = Q (u). Solving (6) numerically, we easily simulate data from the APTME. Quantities of interest are obtained from (6) by inserting appropriate values for q. Furthermore, Bowley’s skewness that depends on quartiles is as follows:where Q (.) is the APITL quantile function. Moor’s kurtosis is given asSkewness and kurtosis plots of the APITL model, based on quantile, are exhibited in Figure 3.
3. Distributional Properties
In this section, we derive the expressions for some essential properties of the APTME model.
3.1. Important Expansion
Here, we obtain a simple expression for PDF (4) to aid the derivation of some structural properties for the APTME model.
The power series can be represented as follows:
Hence, inserting (9) in PDF (4), then
Using the binomial expansion in (10),
Hence, (11) is expressed as follows:
3.2. Ordinary and Incomplete Moments
If X has the PDF (12), then its kth moment can be obtained as follows:where stands for the gamma function. The central moments of the APTME distribution can be obtained from
The skewness and kurtosis measures can be evaluated from the ordinary moments using the well-known relationships. The moment and cumulant generating functions are obtained as follows:
Numerical values of the , variance , coefficient of variation (CV), coefficient of skewness (CS), and coefficient of kurtosis (CK) of the APTEL distribution for certain values of parameters are obtained and recorded in Table 1.
Numerical outputs for the mean , variance , skewness (S), kurtosis (K), and coefficient of variation (CoV) of the APTME model for some values of parameters are mentioned in Table 1.
Furthermore, the upper incomplete (UI) moment, say of the APTME distribution, is given bywhere is the UI gamma function. Similarly, the lower incomplete (LI) moment is given bywhere is the LI gamma function.
3.3. Information Measures
The Rényi entropy (RE), presented by Rényi [14], is defined by
Substituting (4) in (19), we have
Using expansion (9) in (20), we get
Employing the binomial expansion in (21), we have
The Havrda and Charvat entropy (HCE) measure (see [15]) is defined by
Hence, the HCE of the APTME distribution is given by
The Arimoto entropy (AE) measure (see Arimoto [16]) of the APTME is defined by
Hence, the AE of the APTME distribution is given by
The Tsallis entropy (TE) measure (see [17]) is defined by
Hence, the TE of the APTME distribution is obtained as follows:
Some of the numerical values of the RE, HCE, AE, and TE for some selected values of parameters are given in Tables 2–7.
As the value of and increases, the measures values of the RE, HCE, AE, and TE also increase. For the same value of and , the measures values of the RE, HCE, AE, and TE increase with As the value of and increases, for fixed value of , the measures values of the RE, HCE, AE, and TE increase.
4. Parameter Estimation
This section deals with the parameter estimation for the APTME distribution based on six frequentist estimation procedures under complete and RC samples.
4.1. M1 Estimators under the Complete Sample
Let X1, X2, …, Xn be the observed values from the APTME with parameters and ; then, from (4), we can write the log-likelihood function (LLFu) under the complete sample as follows:
The partial derivatives of for and are given byand
The nonlinear equations and are solved numerically; we obtain the ML estimators of and
4.2. M1 Estimators under the RC Sample
Let be a RC sample of size r observed from the lifetime testing experiment on n items whose lifetime has the PDF for APTME. The LLFu of the RC sample is
The partial derivatives of for and are given byand
The nonlinear equations and are solved numerically, and we obtain the ML estimators of and .
4.3. The M2 and M3 Estimators
Consider a random sample of size n from the APTME distribution, and let x(1)< x(2)<…< x(n) be the associated order random observations; then, the M2 estimators of and are obtained by minimizing the following:with respect to and
The M3 estimator of and is derived by minimizing the following:with respect to and
4.4. The M5 and M6 Estimators
The M5 method is regarded as a class of minimum distance estimators based on the difference between the estimate of the CDF and the empirical CDF. The M5 estimator of the APTME parameters is derived by minimizing the following:with respect to and .
The M5 estimator of the APTME distribution parameters is obtained by minimizing the following:with respect to and .
4.5. The M4 Estimators
For estimating the population parameters of continuous distributions, the M4 method is a powerful alternative to the M1 method. Let , be the uniform spacings of a random sample drawn from the APTME distribution, and .
The M6 estimator is obtained by maximizing the geometric mean (GEOM) of the spacings:
The M6 estimator of and can be obtained by maximizing the logarithm of the GEOM of sample spacing’s (39).
5. Applications to Real Data
In this section, we present two applications to real-world datasets to demonstrate the empirical significance of the APTME distribution. The APTME model is compared to the Marshall–Olkin E (MOE), beta exponential (BE), ME, and E models. Data I shows the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli (see Bjerkedal [18]). Data II, which consists of 20 patients, refers to the lifetimes of patients who received an analgesic and their relief times (in minutes) (see Gross [19]). Table 8 shows some descriptive statistics for both datasets.
The M1 estimates (M1Es) of the parameters and their standard errors (SEs) for all models are computed for both datasets (see Tables 9 and 10). We use some model selections like Akaike information criterion (B1), Bayesian information criterion (B2), consistent AIC (B3), and Hannan–Quinn information criterion (B4). In addition, we calculate the Kolmogorov–Smirnov (B5) along with their P–values (B6), Anderson–Darling (B7), and Cram`er–von Mises (B8) test statistics. The values of the criteria measures and the values of the test statistics are listed in Tables 11 and 12 for both datasets.
Plots of the estimated PDFs, CDFs, SF, and probability–probability (PP) of the APTME model for the two considered data are displayed in Figures 4 and 5. According to the numerical values provided in Tables 11 and 12 along with Figures 4 and 5, the APTME model is much better than the above-mentioned extensions of the exponential model; thus, the APTME model is a good alternative to these models in both datasets.
6. Simulation Study
In this portion, a simulation analysis evaluates the output of six different estimates of the APTME parameters. We chose different parameter values as and . 10000 random sample of sizes n = 50, 100, 150 are generated from the APTME distribution using (6). The average values of all estimates (mean) and their corresponding mean square error (MSE) are calculated. The results of the different estimates are evaluated via their mean and MSE. Simulation outputs are obtained via the R program. Tables 13–16 list the mean and MSE for M1, M2, M3, M4, M5, and M5 estimate values.
6.1. Final Thoughts on the Simulation Results
For the same value of and increased values of , the MSE of both the parameters for the APTME distribution increases in most cases except when = 0.15:(1)For a fixed value of and , the mean values of the estimated parameters tend to the true values with increased sample size. Also, the MSE of the estimated parameters decreases with increased sample size.(2)In most cases, the M2 method was the best method for estimating the parameter referring to the mean values of the parameter estimates and MSE as in Tables 13–16.(3)All estimates perform very well and provide very small MSE and the average (Av) values of all estimates tend to the true value of the parameters.(4)The differences between all estimates’ values are very small, referring to the MSE values and the average value of the estimated parameters.
7. Conclusions
We developed a generalized version of the moment exponential as a member of the alpha power-transformed family. The alpha power-converted moment exponential distribution has two parameters. We investigated some of its most significant statistical features. The model parameters are estimated using six different estimation methods. The estimation methods given are M1, M2, M3, M4, M5, and M6. A simulation research was carried out in order to assess the accuracy and behavior of several estimators. According to the numerical results, even with small sample numbers, the estimates offer desirable qualities such as minimal biases and variances. Finally, we demonstrate empirically that the APTME distribution better matches a real dataset than alternative models.
Data Availability
If you would like to get the numerical dataset used to conduct the study reported in the publication, please contact the appropriate author.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was funded by the Researchers Supporting Project (RSP2022R464), King Saud University, Riyadh, Saudi Arabia.