A DISCRETE ANALOG OF INVERTED TOPP-LEONE DISTRIBUTION: PROPERTIES, ESTIMATION AND APPLICATIONS

A BSTRACT . In this study, a discrete inverted Topp-Leone (DITL) distribution is proposed by utilizing the survival discretization approach. The proposed distribution's mathematical features were derived. The maximum likelihood (ML), method of least squares (LS), weighted least squares (WLS), and Cramer Von-Mises (CVM) estimation techniques were used to estimate the parameter. The theoretical results of the ML, LS, WLS, and CVM estimators were demonstrated via a comprehensive simulation study. The proposed DITL distribution has been applied to analyze two count data sets number of deaths due to Covid-19 in Pakistan and India and the findings show the relevance of the proposed distribution.


INTRODUCTION
In December 2019, the first incidence of COVID-19 was reported in the Chinese city of Wuhan.COVID-19 is an extremely contagious disease.In Pakistan, the first case was reported on February 26, 2020 [1].The first death was reported in Pakistan on March 20, 2020.Many researchers make efforts to study the patterns of pandemic Covid-19 and provide models which better fit the data and can be used to have an idea about the expected number of cases to help the government to take decisions regarding precautionary measures.These efforts include the derivation of different probabilistic models and time series modeling of the data such as a new discrete Lindley [2], discrete Marshall-Olkin generalized exponential distribution to model the daily new cases in Egypt [3], a new discrete generalized distribution to analyze the count of daily cases in Hong Kong and Iran [4], a mathematical model known as SIR is used to predict the daily new cases in China [5,6], the logistic growth model is used to estimate the final size and its peak time of coronavirus epidemic [7], autoregressive time series model is used to forecast the recovered and confirmed cases [8], and discrete Marshall-Olkin Lomax distribution is used to estimate the daily new cases in Australia [9].
Practically, lifetime data sets are often recorded as whole numbers of counts.To model the count data, there are few classical distributions as geometric, Poisson, negative binomial, etc. these models sometimes do not provide a better fit due to the complex behavior of data.From the last few decades, researchers have paid attention to introduced discrete type distributions which meet the required need to model the complex behavior of data sets.Several discretization approaches are available in the literature.A detailed systematic review was conducted on discretization approaches [10].Among all approaches, one of the most important is the survival discretizing approach due to its important feature of keeping the original form of the survival function.
The inverted Topp-Leone distribution [20] was derived for the analysis of reliability observations.A comprehensive discussion about its mathematical properties, reliability characteristics, stochastic ordering, and parameter estimation via complete and censored samples, among others is also presented in the mentioned paper.The corresponding survival function is given by where  is the shape parameter.
The goal of this study is to introduce a new one-parameter discrete model called the discrete inverted Topp-Leone (DITL) distribution, which is based on the survival function approach of discretization.The DITL distribution can be used to model the over-dispersed data sets.We derive some of its properties such as, survival and hazard function, quantile function, moments, and generating function.The maximum likelihood, Cramer-von Mises, least-square, and weighted least square estimation methods are used to estimate the model's parameter.A simulation study is conducted to elaborate on the performance of these estimation methods.In the end, we will use data sets about the number of deaths due to coronavirus in Pakistan and India to illustrate the importance of the proposed distribution.
The organization of the article is as follows.Section 2 contains the derivation of the proposed distribution and its features.Section 3 addressed maximum likelihood estimation, as well as the least-squares, weighted least squares, and Cramer von Mises approach.In Section 4, a complete simulation study is used to assess the behavior of these estimators.The proposed distribution's application is discussed in Section 5.The conclusion has been presented in the final section.

THE DITL DISTRIBUTION AND PROPERTIES
Using a discretization approach based on the survival function, the discrete inverted Topp-Leone distribution has been developed.The probability mass function (pmf) of DITL distribution can be represented as The pmf plots of the DITL distribution with some selected values of parameter δ are presented in Figure 1.

Figure 1: Behavior of pmf of DITLD for different parameter values
The cumulative distribution and survival functions of DITL are given, respectively as and, The hazard function of the DITL distribution is obtained using Eq. ( 3) and Eq. ( 5).The behavior of the hazard function is illustrated in Figure 2.
where  > 0 &  = 0,1,2, ….Note that, for  → 0 the HRF turn into  The reverse hazard function of DITL is given as The second rate of failure of DITL is The recurrence relation for generating the probabilities of discrete DITL distribution is given by or

Quantile Function
The p th quantile function of DITL distribution is given by

The moments of DITL distribution
The non-central moments of DITL distribution can be obtained using Eq. ( 3) as follows: In particular, the mean of DITL distribution is The central moments of DITL distribution can be obtained using the following relation The variance of DITL distribution is given as The dispersion index can be calculated using the expression

𝐷𝐼 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝐷𝐼𝑇𝐷 𝑀𝑒𝑎𝑛 𝑜𝑓 𝐷𝐼𝑇𝐿𝐷
Since the above equation cannot be solved, so the descriptive measures, i.e., mean and variance are computed numerically.These measures are presented in Figure 3. From Figure 3, it is apparent that the mean, variance, and DI of the DITL distribution have decreasing behavior with an increase in parameter δ.

PARAMETER ESTIMATION OF DITL DISTRIBUTION
This section describes the parameter estimation of DITL distribution using four different estimation methods.These methods are Maximum Likelihood Estimator (MLE), Cramer Von-Mises Estimator (CVM), Least Square Estimator (LS), and Weighted Least Square Estimator (WLSE).

Method of Maximum Likelihood
Let  1 , The MLE of DITL distribution parameter δ can be obtained from the above Equation (11).The exact solution is not possible, so we can be obtained numerically.

Method of Cramer von-Mises
The Cramèr-von-Mises estimators (CVME), can be determined depending on the difference between both the estimated and exact distributions.The CVME estimator ( ̂ ) can be obtained by minimizing

Method of Least Squares and Weighted Least Squares
The least-square estimator can be obtained by minimizing the sum of the square of residuals.

SIMULATION
In this section, a simulation analysis evaluates the output of four different estimators of the DITL for different values of parameter δ.We consider the different sample sizes n = 10, 20, 50, and 100 for the different values of parameter = (0.8, 1, 1.5, 2, 3, 5, 10).From DITL distribution, we generate 10000 iterations of random samples.For each computation, we get the average estimations (AEs) and mean square error (MSE).The output of considered estimators is compared in terms of MSE, with the lowest MSE values indicating the best successful technique of estimation.The R program is used to obtain simulation results.The AE and MSE values for the MLE, CVM, LS, and WLS approaches are shown in Table 1.Methods tend towards the true parameter values, suggesting that all estimators are asymptotically unbiased.

APPLICATION
In this section, we illustrate the importance of the proposed distribution using two data sets.Both data sets are counts.Five one-parameter competitive distributions of the DITL distribution are Poison distribution, discrete Pareto distribution [13], discrete Rayleigh distribution [12], discrete inverse Rayleigh distribution [21], and discrete Burr-Hutke distribution [19].      is expected to make a significant contribution to the field of count data modeling.
X a continuous random variable.If X has a survival function   (), then the discrete random Variable  = [], where [] indicates the smallest integer part or equal to , have probability mass function (PMF) written as":

Figure 2 :
Figure 2: Behavior of HRF of DITLD for different parameter values

Figure 3 :
Figure 3: Plots of Mean, Variance and Dispersion Index for DITL distribution Figure 4.

Figure 4 .
Figure 4. Plots for COVID-19 daily deaths in Pakistan and India The results presented in Tables 2-3 and Figures 5-6 demonstrate the sufficiency and superiority of the proposed distribution in modeling the data sets when compared to the competitive distributions.

Figure 5 :
Figure 5: Density plot for the deaths due to COVID-19 in Pakistan

Figure 6 :
Figure 6: Density plot for the deaths due to COVID-19 in India

Table 1 :
The simulation results for the parameter δ If =0.8, 1, 1.5, and 2, the MLE is the best estimation method in all sample sizes.If  =3 with sample size n=10, the LSE method is the best estimation method while in other sample sizes the MLE is the best method of estimation.If =5 with sample sizes n=10 and 20 the LS is the best estimation method while the MLE is the best method in sample sizes n= 50 and 100.If = 10 the CVME is the best estimation method in all sample sizes.

Table 2 :
The MLEs, standard errors of the competing models for number of deaths in Pakistan

Table 3 :
The MLEs, standard errors of the competing models for number of deaths in India