The Exponentiated Power Akash Distribution: Properties, Regression, and Applications to Infant Mortality Rate and COVID-19 Patients’ Life Cycle

Exponentiation of power distributions has one major advantage to model data that exhibit polynomial tendencies. Distributions of lower degrees do not match data of this sort. Measures from health sciences such as the mortality rate of people living with a particular epidemic, the spread of diseases; or from engineering such as the strength of a tensile string, the life of a mechanical appliance; or from the economic sector namely income distribution of workers, inflation rate, and exchange rate of currencies of developing economies are good examples where distributions of this nature are applicable.

Relevant studies are exponentiated power Ishita by [1], exponentiated Ishita by [2], exponentiated power Lindley by [3], and exponentiated Adya by [4]. Other related studies are [5-15]. Interestingly, many of these exponentiated power transformations are on one-parameter distributions. This suggests that the Lindley class of distributions has some usefulness in modeling. Members in this class include [16-25].

Modeling some universal events such as infant mortality rate and the life cycle of COVID-19 patients is dominating the literature due to the level of concerns these events have posed to every society. Importantly, events such as these are threats to life, and hence human extinction is at the heart of modelers.

The main objective of this article is to develop a new parametric regression model that will be able to fit some skewed censored data and the rest of the article is in the following arrangement: in section 2, the new model is formulated. In section 3, some of the properties are presented. In section 4, the estimation of the uncensored data procedure is carried out. In section 5, the log-transformed regression model equivalent of the proposed distribution is derived together with the estimation. In section 6, an application to the life cycle of COVID-19 patients with a history of diabetic Mellitus with their age disparity is done. In section 7, the second application on the infant mortality rate of some countries in 2021 is also done. The paper is concluded in section 8.

The Power Akash (PA) distribution proposed by [26] with c.d.f and p.d.f given as follows;

The PA distribution is a two-component mixture that contains a Weilbull distribution (with shape parameter α and scale parameter θ ), and a generalized gamma distribution (with shape parameters 3, α and scale parameter θ ) with mixing proportion

The c.d.f and p.d.f of the exp-G distribution with power parameter c > 0 are given by

respectively, whereξ is the parameter vector. By substituting equation 1 into equation 3 and equation 1 and 2 into 4, the c.d.f and p.d.f of the random variable X  Exponentiated Power Akash EPA (c,θ ,α ) are as follows:

respectively. The hazard rate function is given as

Definition

(Linear Representation). Using the general binomial expansion, the pdf of the EPA distribution is given as follows;

The hazard function of EPA has a bathtub, increasing and decreasing shapes, see figures 3 and 4. This feature enhances the flexibility of EPA compared to the Power Akash and Akash distributions.

Definition

The maximum likelihood of EPA is given as

Equation 16 and 17 have no closed-form solutions and hence will be implemented in R using the known optim() function.

MLE of beta For Right-Censored Sample

Suppose the lifetime X of n individuals diagnosed with COVID-19 virus is EPA (℘) distributed. Let y |V ' be the response variable of a parametric regression model from the EPA(℘) distribution with pdf in eq 23. To estimate the parametersβ of the transformed model, the n individuals are quarantined and subjected to routine treatment at the same time. After time (t), (n − m) individuals recovers. If the lifetimes of the other m(> 0) individuals are denoted by 1 2 , ,...., m y y y .Then, the likelihood ofβ can be expressed as

where f ( y |V ' ) is the pdf in eq 23. It is witty to write

The unknown parametric regression coefficientsβ estimates are obtained using numerical iteration implemented in R.

The dataset comprises the lifetime (in days) of 322 individuals diagnosed with COVID-19 through RT-PCR screening in Campinas, Brazil. These data were previously studied by [1]. The response variable 1 y represents the time elapsed from the onset of symptoms until death due to COVID-19 (failure). [1] observed that about 66.45% of the observations are censored. The variables considered (f or i = 1,..., 322) include:

The Power Prakaamy (PP) distribution by [27], exponentiated Frechet (EF) distribution by [28], power Rama (PP) distribution by [29] and power Suja (PS) distribution (new) are used to compare with the proposed exponentiated power Akash (EPA) distribution. Note that the log- of each distribution is derived following the procedure in section 5 to obtain LPP, LEF, LPR, and LPS respectively.

The result from table 1 shows that the explanatory variables age and diabetes mellitus are significant at the 5% significance level. The negative signs of 1 β and 2 β mean that older individuals or those with diabetes tend to have shorter failure times. This result is in agreement with that obtained from [1] earlier study. From table 2, the LEPA regression has the lowest criterion values hence confirming that the LEPA model provides a better fit for the COVID-19 data.

Table 1: Estimates of the Regression parameters for the COVID-19 data.

Table 2: Measures of model performance.

The data on infant mortality rate per 1,000 live births for a few chosen nations in 2021, as reported by https://data.worldbank. org/indicator/SP.DYN.IMRT.IN This real data set is presented as

quick observation of the non-parametric plots in figures 8, 9, and 10 reveals that the infant mortality rate is right-skewed data. Probabil- ities taper off more slowly for higher values.

Table 3: Model Adequacy and Fitness Measures for the Infant Mortality Rate Data.

Using the following information criteria; Akaike information criterion (BIC), Corrected Akaike Information criterion (CAIC), Bayesian Infor- mation Criterion (BIC), Hannan–Quinn information criterion (HQIC), the adequacy of the model was proved since the proposed distribution has minimum value for each of the criteria. The K-S, Cramer von misses W* , Anderson Darling statistics A* , and p-value for the proposed distribution show evidence that the new distribution fits the given data more than the competitors.

Table 4: MLEs of the unknown parameters using Infant Mortality Data.

In this article, a new lifetime distribution with the potential of modeling data with inherent polynomial nature as well as skewed data. The properties of the proposed distribution were derived and the log-transformation of the proposed distribution to develop a parametric regression model was carried out. The maximum likelihood estimation aided the estimation process for uncensored samples while the procedure for the estimation of the unknown parameters when data is censored was also shown. Essentially, the censored COVID-19 data set with the age of patients and diabetic mellitus index was deployed to justify the importance of the distribution. Furthermore, the distribution was fitted to the data on infant mortality rate (below age 5 years) reported for some countries by the World Health Organization in 2021. The distribution performs pretty well in both instances of application.

The authors declare that there was no external funding received for this article.

The authors laud the comments of the reviewers and editors which have enhanced the quality of this article.

The authors declare that there is no conflict of interest.

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