Research Article Alpha Power Moment Exponential Model with Applications to Biomedical Science

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.


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. e APT-G family's cumulative distribution function (CDF) is defined as follows: e 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], APTpower 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. e 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. is 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 υ is the scale parameter. Different values of the shape parameter lead to different shapes of the density function. e 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. e APTME model's structural characteristics and parameter estimators are derived for a number of APTME models. e 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.

Description of the APTME Model
e APTME distribution, based on the APT method, is introduced in this section. e 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 (α, υ), with shape parameter α and scale parameter υ, if the PDF and CDF of X for x ≥ 0 are given by Some plots of the density (4) for different values of α and υ are illustrated in Figure 1. ey 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). ese functions are especially important in analyzing the survival and are included in the definition below.

Definition 2.
e survival function (SF) and the HRF of the APTME distribution are given, respectively, by Some 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.
e 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, x q � Q (u). Solving (6) numerically, we easily simulate data from the APTME. Quantities of interest are obtained from (6) where Q (.) is the APITL quantile function. Moor's kurtosis is given as Skewness and kurtosis plots of the APITL model, based on quantile, are exhibited in Figure 3.

Distributional Properties
In this section, we derive the expressions for some essential properties of the APTME model.

Important Expansion.
Here, we obtain a simple expression for PDF (4) to aid the derivation of some structural properties for the APTME model. e 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:

Ordinary and Incomplete
Moments. If X has the PDF (12), then its kth moment can be obtained as follows: where Γ(.) stands for the gamma function. e central moments (μ k ) of the APTME distribution can be obtained from e skewness and kurtosis measures can be evaluated from the ordinary moments using the well-known relationships. e moment and cumulant generating functions are obtained as follows: Numerical values of the μ 1 ′ , μ 2 ′ , μ 3 ′ , μ 4 ′ , variance (σ 2 ), 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 (μ 1 ′ ), variance (σ 2 ), 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 m th upper incomplete (UI) moment, say η m (t) of the APTME distribution, is given by where Similarly, the m th lower incomplete (LI) moment is given by where c(m, t) � t 0 x m− 1 e − x dx is the LI gamma function.

Information
Measures. e Rényi entropy (RE), presented by Rényi [14], is defined by (20) Using expansion (9) in (20), we get Employing the binomial expansion in (21), we have e Havrda and Charvat entropy (HCE) measure (see [15]) is defined by Hence, the HCE of the APTME distribution is given by e Arimoto entropy (AE) measure (see Arimoto [16]) of the APTME is defined by Hence, the AE of the APTME distribution is given by e 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 α and ε increases, the measures values of the RE, HCE, AE, and TE also increase. For the same value of ε and ε, the measures values of the RE, HCE, AE, and TE increase with α. As the value of α and ε increases, for fixed value of ε, the measures values of the RE, HCE, AE, and TE increase.

Parameter Estimation
is section deals with the parameter estimation for the APTME distribution based on six frequentist estimation procedures under complete and RC samples.

M1 Estimators under the Complete Sample.
Let X 1 , X 2 , . . ., X n be the observed values from the APTME with parameters α and υ; then, from (4), we can write the loglikelihood function (LLFu) under the complete sample as follows: e partial derivatives of ln ℓ for α and υ are given by and (31) e nonlinear equations z ln ℓ/zα � 0 and z ln ℓ/zυ � 0 are solved numerically; we obtain the ML estimators of α and υ.

M1 Estimators under the RC Sample.
Let X 1 , X 2 , . . . , X r be a RC sample of size r observed from the lifetime testing experiment on n items whose lifetime has the PDF for APTME. e LLFu of the RC sample is e partial derivatives of ln ℓ for α and υ are given by and e nonlinear equations z ln ℓ/zα � 0 and z ln ℓ/zυ � 0 are solved numerically, and we obtain the ML estimators of α and υ.

e 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 α and υ are obtained by minimizing the following: with respect to α and υ. e M3 estimator of α and υ is derived by minimizing the following: with respect to α and υ.

e M5 and M6 Estimators.
e 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. e M5 estimator of the APTME parameters is derived by minimizing the following: with respect to α and υ. e M5 estimator of the APTME distribution parameters is obtained by minimizing the following: with respect to α and υ.

e M4 Estimators.
For estimating the population parameters of continuous distributions, the M4 method is a powerful alternative to the M1 method.
. . , n + 1, be the uniform spacings of a random sample drawn from the APTME distribution, e M6 estimator is obtained by maximizing the geometric mean (GEOM) of the spacings: e M6 estimator of α and υ can be obtained by maximizing the logarithm of the GEOM of sample spacing's (39).

Applications to Real Data
In this section, we present two applications to real-world datasets to demonstrate the empirical significance of the APTME distribution. e 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. e M1 estimates (M1Es) of the parameters and their standard errors (SEs) for all models are computed for both datasets (see Tables 9 and 10 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.

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 α � 0.15, 0.75, 1.5, 3 and υ � 0.6, 1.6, 3, 5. 10000 random sample of sizes n � 50, 100, 150 are generated from the APTME distribution using (6).  (1) For a fixed value of α and υ, 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.

Conclusions
We developed a generalized version of the moment exponential as a member of the alpha power-transformed family. e alpha power-converted moment exponential distribution has two parameters. We investigated some of its most significant statistical features. e model parameters are estimated using six different estimation methods. e 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
e authors declare no conflicts of interest.