A New Three-Parameter Discrete Distribution With Associated INAR(1) Process and Applications

Abstract

The aim of this article is to propose a new three-parameter discrete Lindley distribution. A wide range of its structural properties are investigated. This includes the shape of the probability mass function, hazard rate function, moments, skewness, kurtosis, index of dispersion, mean residual life, mean past life and stress-strength reliability. These properties are expressed in explicit forms. The maximum likelihood approach is used to estimate the model parameters. A detailed simulation study is carried out to examine the bias and mean square error of the estimators. Using the proposed distribution, a new first-order integervalued autoregressive process is introduced for the over-dispersed, equi-dispersed and under-dispersed time series of counts. To demonstrate the importance of the proposed distribution, three data sets on coronavirus, length of stay at psychiatric ward and monthly counts of larceny calls are analyzed. INDEX TERMS Survival discretization method, over-dispersion, INAR(1) process, simulation.

Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process

“…These data were based on the discretization concept. Adopting a data analysis setting, we compared the DPsL, discrete three-parameter Lindley (DTPL) (see [9]), discrete log-logistic (DLL) (see [23]), discrete inverse Weibull (DIW) (see [2]), discrete Burr-Hutke (DBH) (see [6]), discrete Pareto (DP) (see [3]), Poisson (P) and geometric (G) distributions. The MLEs with standard errors (SEs) and confidence intervals (CIs) for the parameter(s), estimated −log Likelihood (−L), Akaike information criterion (AIC), Bayesian information criterion (BIC) and goodness of fit statistic (Kolmogorov statistic (K-S) and p-value) of these distributions for this dataset are given in Table 9.…”

“…(1) Some of the recently introduced discrete distributions based on this survival discretization method are as follows: Discrete Lindley distribution by [1], discrete inverse Weibull distribution by [2], discrete Pareto distribution by [3], discrete Rayleigh distribution by [4], two-parameter discrete Lindley distribution by [5], exponentiated discrete Lindley distribution by [6], discrete Burr-Hatke distribution by [7], discrete Bilal distribution [8], discrete three-parameter Lindley distribution by [9], etc. Recently, Ref.…”

“…Hence, researchers have assembled many approaches concerning innovations in modelling over-dispersed time series count datasets. The INAR (1) process with geometric innovations (INAR(1)G) by [14], INAR (1) process with Poisson-Lindley innovations (INAR(1)PL) by [15], INAR (1) process with a new Poisson weighted exponential innovation ((INAR(1)NPWE)) by [16], INAR(1) process with discrete three-parameter Lindley as innovation by [9], INAR (1) process with discrete Bilal as innovation by [8], INAR (1) process with Poisson quasi Gamma innovations (INAR(1)PQX) by [17] and the INAR(1) process with Bell innovations (INAR(1)BL) by [18] are some of the recently developed over-dispersed INAR(1) processes.…”

“…Even though these processes provide better solutions to over-dispersed time series count datasets, they have some limitations that can sometimes cause computing difficulties. Even if a model has one parameter, the inclusion of special functions in the pmf, cumulative distribution function (cdf) and other statistical properties makes it difficult to obtain explicit expressions and, hence, for estimation procedures to generate them (see, e.g., [9,19]).…”

A New Skewed Discrete Model: Properties, Inference, and Applications

“…The first data set contains 64 observations, and it refers to numbers of daily cases in Egypt due to COVID-19 infections from 8 March to 10 May, 2020. The data are: 12,33,7,4,8,13,13,17,16,40,30,14,46,29,9,33,39,36,54,39,41,40,33,47,54,69,86,120,85,103,149,128,110,139,95,145,126,125,160,155,168,171,188,112,189,157,169,232,201,227,215,248,260,226,269,358,298,272,348,388...…”