Four statistical models for wet-spell analysis

Wet-spell analysis is an important part of rainfall analysis. The distribution of the length of wet-spells provides useful information on the temporal distribution of rainfall. This distribution has traditionally been modelled through different probability distributions. Here we compare four such models, namely, Cochran's model, truncated Poisson distribution, truncated negative binomial distribution, and logarithmic series distribution. These comparisons are accomplished with help of application to five rainguage stations in India.

Negative Binomial Distribution to Explain the Domestic Fire Incidence in Nepal

Background: Fire disaster is one of the most destructive disasters. According to global dataset of Sendai Framework, domestic fire incidence was 9.9% up to 2019. In Nepal, 62% fire incidence was reported during 2017 and 2018. However, many studies have been conducted on fire incidence, few of them are based on domestic fire incidence. Objective: To find the descriptive statistics of fire occurrences and fire fatalities, and to identify the probability distributions that best fit the data of fire occurrences observed in three ecological regions as well as overall in Nepal. Material and Methods: The data of fire incidences from May 2011 to April 2021 were retrieved from Nepal Disaster Risk Reduction Portal, Government of Nepal. At first, a statistical software "Mathwave EasyFit" of 30 days trial version was used to identify the candidate probability models. Further, the best probability model was determined after testing the goodness of fit of the candidate models by using graphical tools-histogram and theoretical densities, empirical and theoretical CDFs, Q-Q plot and P-P plot; and mathematical tools-maximum likelihood, Akaike Information Criteria and Bayesian Information Criteria by using the package “fitdistrplus” of software R version 4.1.1. Results: On an average, 135 fire incidences per month were occurred in Nepal. However, the Terai faced the highest monthly fire incidences compared to the Hill and the Mountain, it has less fatality per 100 fire incidence followed by the Hill and the Mountain. Descriptive statistics reveals that fire occurrences are moderate during November to February and high in March and April. The fire incidences were reported high during spring and winter and low during summer and autumn season which reveals that fire incidence might be related with the precipitation and temperature. The sample data was run in "Mathwave EasyFit" software which suggested Poisson, geometric and negative binomial distribution as candidate probability models. The goodness of fit of these models were further tested by graphical as well as mathematical tools where negative binomial distribution was found to be best among the candidate models for the data set. Conclusion: Incidence of fire disasters varies by ecological regions as well as by seasons. It is low in the Mountain region and during Monsoon/rainy season. Negative binomial distribution fits the best to monthly data of fire incidence in Nepal.

A New Neutrosophic Negative Binomial Distribution: Properties and Applications

Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.

Local limit theorems for generalized scheme of allocation of particles into ordered cells

A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.

On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.