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Abstract: Abstract The mathematical modeling of the coronavirus disease-19 (COVID-19) pandemic has been attempted by a large number of researchers from the very beginning of cases worldwide. The purpose of this research work is to find and classify the modelling of COVID-19 data by determining the optimal statistical modelling to evaluate the regular count of new COVID-19 fatalities, thus requiring discrete distributions. Some discrete models are checked and reviewed, such as Binomial, Poisson, Hypergeometric, discrete negative binomial, beta-binomial, Skellam, beta negative binomial, Burr, discrete Lindley, discrete alpha power inverse Lomax, discrete generalized exponential, discrete Marshall-Olkin Generalized exponential, discrete Gompertz-G-exponential, discrete Weibull, discrete inverse Weibull, exponentiated discrete Weibull, discrete Rayleigh, and new discrete Lindley. The probability mass function and the hazard rate function are addressed. Discrete models are discussed based on the maximum likelihood estimates for the parameters. A numerical analysis uses the regular count of new casualties in the countries of Angola,Ethiopia, French Guiana, El Salvador, Estonia, and Greece. The empirical findings are interpreted in-depth. PubDate: 2022-09-09

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Abstract: Abstract Inverse moments of discrete random variables are traditionally expressed by summations. But summations are often difficult to simplify. In this note, we derive an integral representation for inverse moments involving the probability generating function, making simplifications a lot easier. Two examples are provided to illustrate the result. PubDate: 2022-09-08

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Abstract: Abstract Let SPd be the class of all multivariate sign and permutation invariant densities and SPD be the corresponding class of distribution functions. The class of all distributions corresponding to a positive power of the members of SPD is called the Power Skew Symmetric class of distributions and it is denoted by PSS. We consider the problem of inference associated with the nonnegative power, in the PSS class, with a member of SPD as a nuisance parameter. As the structural forms of the members of PSS are not known, one can’t directly use the sufficiency, invariance, or likelihood function. Hence by using certain properties of the SPD and the PSS classes, we identify maximal statistics whose distributions depend only on the power but not on the members of SPD. We then use the concept of minimal dimensionality for retaining the information about the parameter of interest. We use the minimax criterion, which leads to a statistic having Binomial distribution depending only on the parameter of interest. Hence inferences about the parameter of interest can be carried out using the standard methods for the binomial model. A simulation study indicates that the proposed estimator of the power parameter is asymptotically normal and is insensitive to the nuisance parameter. The proposed method is implemented for the analysis of a data set. PubDate: 2022-08-22

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Abstract: Abstract This work is about semiparametric models assuming that the error follows a scale mixture of gaussian distributions and such that the functional relation between the response and explanatory variables is unknown. This class of distributions is particularly interesting since it includes heavy tailed distributions such as the Student’s t, symmetric stable distributions and double exponential. They are specially useful for modelling data with high incidence of extreme values. Exploring the nature of these distributions and using the concept of P-splines we obtain the complete posterior conditional distributions of all the parameters involved in the model and apply the Gibbs sampler. In this way, we show how to combine P-splines and mixture of normals under a Bayesian perspective in order to estimate such curves. We conduct some simulations in order to illustrate the proposed methodology and also analyze the case of partially linear models. PubDate: 2022-08-11

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Abstract: After publication of this paper, it was found out that there are still corrections which were not carried out in the proof. The publisher wish to apologize for this oversight. PubDate: 2022-08-01

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Abstract: Abstract A new family of multivariate distributions under elliptical models is proposed in this work. Several particular cases of this multivector variate distributions are obtained and a number of published multivariate distributions in other contexts are found as simple corollaries. An application of interest in finance is full derived and compared with the traditional methods. PubDate: 2022-08-01

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Abstract: Abstract General weighted exponential distributions including modified exponential ones are widely used with great ability in statistical applications, particularly in reliability. In this paper, we investigate full exponential weight functions and their extensions from any nonnegative continuous reference weighted distribution. Several properties and their connections with the recent variation phenomenon are then established. In particular, characterizations, weightening operations and dual distributions are set forward. Illustrative examples and concluding remarks are extensively discussed. PubDate: 2022-08-01

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Abstract: Abstract A multi-state coherent system consists of multiple components each of which passes through a sequence of states and it is of interest to estimate the distribution of time spent by the components in different states. Although not practical, for mathematical convenience, it is usually assumed that the times spent by the components in various states are independent of each other. This paper considers three-state series and parallel systems and is based on the assumption that times spent by the components in various states are positively quadrant dependent (PQD) and the corresponding dependence is modeled using a Farlie Gumbel Morgenstern (FGM) distribution. To begin with it is shown that even when marginal distributions are assumed exponential, the resulting likelihood function leads to a complicated expression making maximum likelihood (ML) based inference computationally challenging. A generalized method of moment (GMM) estimation is shown to be relatively simpler not only computationally but also the method works for arbitrary marginal distributions. The estimates obtained by GMM are shown to be uniformly consistent under some mild regularity conditions. Finite sample performances of the ML and GMM are illustrated using FGM distribution with various parametric marginal distributions. In case of exponential marginals, it is shown that GMM compares favorably to ML although the former method does not require parametric assumption for the marginals. The proposed methods are also illustrated using a real case study data of a rare type of head and neck cancer. PubDate: 2022-08-01

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Abstract: Abstract In this paper, we study cliques and chromatic number in the random subgraph Gn of the complete graph Kn on n vertices, where each edge is independently open with a probability pn. Associating Gn with the probability measure ℙn, we say that the sequence {ℙn} is multiregime if the edge probability sequence {pn} is not convergent. Using a recursive method we obtain uniform bounds on the maximum clique size and chromatic number for such multiregime random graphs. PubDate: 2022-08-01

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Abstract: Abstract For a given conditional distribution for X given Y, it is important to identify the class of all conditional distributions for Y given X such that there exists at least one bivariate distribution with the given particular conditional densities. Such problems are addressed as dealing with “compatibility” of two conditional distributions. In the present note our goal is to identify all possible conditional densities for Y given X that are compatible with the given family of distributions of X given Y. PubDate: 2022-08-01

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Abstract: Abstract We study the estimation of a class of semiparametric mixture models, where the models have a symmetric nonparametric component and a parametric component of Pareto distribution with unknown parameters. We establish an estimation procedure by minimizing a criterion function after dealing with the jump point. We study the large sample properties of the proposed estimator, and prove consistency and asymptotic normality of the parameter estimation. For the nonparametric component, bias and variance are derived, and a rule-of-thumb bandwidth selection method is given. Simulation studies demonstrate good performance of the proposed methodology. PubDate: 2022-08-01

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Abstract: Abstract In a finite/survey population setup, where categorical/multinomial responses are collected from individuals belonging to a cluster, in a recent study Skinner (International Statistical Review, 87, S64-S78 2019) has modeled the means of the clustered categorical responses as a function of regression parameters, and suggested a ‘working’ correlations based GEE (generalized estimating equations) approach for the estimation of the regression parameters. However, this mean model involving only regression parameters is not justified for clustered multinomial responses because of the fact that these responses share a common cluster effect which compels the clustered correlation parameter to enter into the mean function on top of the regression parameters. Consequently, the so-called GEE approach, which requires the means to be free of correlations, is not applicable for regression analysis in the clustered multinomial setup. As a remedy, in this paper we consider a multinomial mixed model which accommodates the clustered correlation parameter in the mean functions. For inferences in the present finite population setup, as the GQL (generalized quasi-likelihood) approach is known to produce consistent and more efficient estimate than the MM (method of moments) approach in an infinite population setup, we estimate the regression parameters of primary interest by using the first order response based survey weighted GQL (WGQL) approach. For the estimation of the random effects variance (also known as clustered correlation) parameter, as it is of secondary interest, we use the second order response based survey weighted MM (WMM) approach, which is simpler than the corresponding WGQL estimation approach. The estimation steps are presented clearly for the benefit to the practitioners. Also because, in practice, survey practitioners such as statistical agencies frequently deal with a large health or socio-economic data set at national or state levels, for example, we make sure for their benefit that our proposed WGQL and WMM estimators are consistent. Thus, the asymptotic properties such as asymptotic unbiasedness and consistency for both regression and clustered correlation parameters are studied in details. The asymptotic normality property, for the benefit of constructing confidence interval for the main regression parameters, is also studied. PubDate: 2022-08-01

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Abstract: Abstract The unit Nadarajah-Haghighi (UNH) class of distributions was developed and its statistical properties investigated in this study. The generator was used to develop the UNH Weibull and UNH log-logistic distributions. For some given parameter values it was realized that the density and failure rate functions of the UNH Weibull and UNH log-logistic distributions can exhibit different kinds of shapes making the distributions suitable for modeling dataset that exhibit some of these shapes. Monte Carlo simulations were performed to examine how the maximum likelihood estimators and ordinary least squares estimators perform with regard to estimating the parameters of the distributions and the results indicated that the maximum likelihood performs better than the ordinary least squares. Applications of the UNH Weibull distribution revealed that it can provide good parametric fit to given datasets. PubDate: 2022-08-01

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Abstract: Abstract A new extension of the exponential distribution, proposed by Nadarajah and Haghighi (Statistics 45, 543–558 (2011)), is an alternative to the gamma, Weibull and generalized-exponential models, it is also known as NH distribution. The maximum likelihood and Bayes inferential approaches for estimating the unknown two-parameters and some lifetime parameters such as survival and hazard rate functions of the NH distribution in presence of progressive first-failure censored sampling are considered. Based on observed Fisher’s information matrix, the approximate confidence intervals for the two-parameters, and any function of them, are constructed. Using Lindley’s approximation and Markov chain Monte Carlo methods under the assumption of conjugate gamma priors, the Bayes estimates and associate highest posterior density credible intervals for the unknown parameters and reliability characteristics are developed based on squared error loss function. Although the proposed estimators cannot be expressed in explicit forms, these can be easily obtained through the use of appropriate numerical techniques. A Monte Carlo simulation study is carried out to examine the performance of proposed methods. Using different optimality criteria, an optimal censoring scheme has been suggested. Finally, a real data set is analyzed for illustration purposes. PubDate: 2022-08-01

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Abstract: Abstract In this paper, the general proportional mean past lifetime frailty model is considered, from which the unconditional cumulative distribution and density functions of the lifetime variable are derived. Dependency between the two variables are studied. Stochastic comparisons are made through which it is shown that some well-known stochastic orderings between two frailty variables carry over to the corresponding lifetime variables. The effects of baseline variable and the frailty variable on the proposed frailty model are studied. The relative mean past lifetime ordering is introduced and some relative ordering between two lifetime random variables with different frailty variables are studied. Also a simulation study is given to illustrate some results. PubDate: 2022-08-01

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Abstract: Abstract The Generalized Chinese Restaurant Process (GCRP) describes a sequence of exchangeable random partitions of the numbers \(\{1,\dots ,n\}\) . This process is related to the Ewens sampling model in Genetics and to Bayesian nonparametric methods such as topic models. In this paper, we study the GCRP in a regime where the number of parts grows like nα with α > 0. We prove a non-asymptotic concentration result for the number of parts of size \(k=o(n^{\alpha /(2\alpha +4)}/(\log n)^{1/(2+\alpha )})\) . In particular, we show that these random variables concentrate around ckV∗nα where V∗nα is the asymptotic number of parts and ck ≈ k−(1+α) is a positive value depending on k. We also obtain finite-n bounds for the total number of parts. Our theorems complement asymptotic statements by Pitman and more recent results on large and moderate deviations by Favaro, Feng and Gao. PubDate: 2022-08-01

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Abstract: Abstract We prove two estimates of the rate of convergence in the Lindeberg theorem, involving algebraic truncated third-order moments and the classical Lindeberg fraction, which generalize a series of inequalities due to Esseen (Arkiv För Matematik 8, 1, 7–15, 1969), Rozovskii (Bulletin of Leningrad University (in Russian), (1):70–75, 1974), & Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016), some of our recent results in Gabdullin, Makarenko, Shevtsova, (J. Math Sci. 234, 6, 847–885, 2018, J. Math Sci. 237, 5, 646–651, 2019b) and, up to constant factors, also Katz (Ann. Math. Statist. 34, 1107–1108, 1963), Petrov (Soviet Math. Dokl. 6, 5, 242–244, 1965), & Ibragimov and Osipov (Theory Probab. Appl. 11, 1, 141–143, 1966b). The technique used in the proof is completely different from that in Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016) and is based on some extremal properties of introduced fractions which has not been noted in Katz (Ann. Math. Statist. 34, 1107–1108, 1963), Petrov (Soviet Math. Dokl. 6, 5, 242–244, 1965), & Wang and Ahmad (Sankhya A: Indian J.Stat. 78, 2, 180–187, 2016). PubDate: 2022-08-01

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Abstract: Abstract Let X be a real random variable having f as density function. Let F be its cumulative distribution function and Q its quantile function. For h > 0, let Fh and Qh denote respectively the Nadaraya kernel estimator of F and Q. In the first part of this paper the almost sure convergence of the conventional L1 distance between Qh and Q is established. In the second part, the L1 right inversion distance is introduced. The representation of this L1 right inversion distance in terms of Fh and F is given. This representation allows us to suggest ways to choose a global bandwidth for the estimator Qh. PubDate: 2022-08-01

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Abstract: Abstract Spatio-temporal data indexed by sampling locations and sampling time points are encountered in many scientific disciplines such as climatology, environmental sciences, and public health. Here, we propose a novel spatio-temporal expanding distance (STED) asymptotic framework for studying the properties of statistical inference for nonstationary spatio-temporal models. In particular, to model spatio-temporal dependence, we develop a new class of locally stationary spatio-temporal covariance functions. The STED asymptotic framework has a fixed spatio-temporal domain for spatio-temporal processes that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illustrated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions. A simulation study suggests sound finite-sample properties and the method is applied to a sea-surface temperature dataset. PubDate: 2022-08-01

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Abstract: Abstract We obtain simple non-parametric estimators of Gini-based covariance, correlation and regression coefficients. We then establish the consistency and asymptotic normality of the proposed estimators. We provide an explicit formula for finding the asymptotic variance of the estimators. We also discuss jackknifed versions of the proposed estimators for reducing the bias of the estimators in case of small sample sizes. Finally, we evaluate the finite-sample performance of these estimators through on Monte Carlo simulations from a bivariate Pareto distribution. PubDate: 2022-08-01