Sankhya A
Journal Prestige (SJR): 0.106 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 0976836X  ISSN (Online) 09768378 Published by SpringerVerlag [2467 journals] 
 Families of Discrete Circular Distributions with Some Novel Applications

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Abstract: Abstract We give a unified treatment of constructing families of circular discrete distributions. Some of these families are deduced from established distributions such as von Mises and wrapped Cauchy. Some others are derived directly such as a flexible family based on trigonometric sums and the circular location family. Results interrelating these families are discussed. These distributions have been motivated by two examples of discrete circular data: casino roulette spins and smart health acrophase monitoring, and these data are analyzed using our proposed models. We discuss how using continuous circular models for circular discrete data can be misleading.
PubDate: 20221122

 Adaptive Estimation of a Function from its Exponential Radon Transform in
Presence of Noise
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Abstract: Abstract In this article we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without prior knowledge of the smoothness of functions that are to be estimated. We build a nonparametric kernel type estimator and show that for a class of functions comprising a wide Sobolev regularity scale, our proposed strategy follows the minimax optimal rate up to a \(\log {n}\) factor. We also show that there does not exist an optimal adaptive estimator on the Sobolev scale when the pointwise risk is used and in fact the rate achieved by the proposed estimator is the adaptive rate of convergence.
PubDate: 20221103

 JonesBalakrishnan Property for Matrix Variate Beta Distributions

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Abstract: Abstract Let X and Y be independent m × m symmetric positive definite random matrices. Assume that X follows a matrix variate beta distribution with parameters a and b and that Y has a matrix variate beta distribution with parameters a + b and c. Define \(\boldsymbol {R}= \left (\boldsymbol {I}_{m}  \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{1/2} \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\) \( \left (\boldsymbol {I}_{m}  \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{1/2} \) and \(\boldsymbol {S}= \boldsymbol {I}_{m}  \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\) , where Im is an identity matrix and A1/2 is the unique symmetric positive definite square root of A. In this paper, we have shown that random matrices R and S are independent and follow matrix variate beta distributions generalizing an independence property established by Jones and Balakrishnan (Statistics and Probability Letters, 170 (2021), article id 109011) in the univariate case.
PubDate: 20221024

 Prediction Theory for Multinomial Proportions Using Twostage Cluster
Samples
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Abstract: Abstract In a twostage clusters sampling setup for categorical data, it is well known that the socalled best prediction of the category based proportions involves computing the conditional means of the nonsampled multinomial variables conditional on the sampled multinomial responses. This computation is however not easy mainly due to the complex cluster correlations among multinomial responses within a cluster. The independence assumption based approach or any linear model approach for cluster correlated data those used so far in the existing studies are not valid for the computation of such conditional means in the prediction function for multinomial data. As opposed to these ‘working’ independence or linear models based approaches, in this paper we first develop a cluster correlation structure for multinomial data and exploit this structure to compute theoretically valid formulas for the conditional means of nonsampled hypothetical responses. Next because these conditional means or equivalently the prediction function contains the regression and clustered variance/correlation parameters, we estimate these parameters using the survey sampling weights based conditional likelihood approach, whereas the existing studies mostly use the independence assumption based likelihood or moment approaches which are invalid or inadequate in a correlation setup. The proposed conditional likelihood estimators are shown to be consistent for their respective parameters leading to the consistent estimation of the prediction function for the multinomial proportions.
PubDate: 20221024

 On Resolving Problems with Conditionality and Its Implications for
Characterizing Statistical Evidence
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Abstract: Abstract The conditionality principle C plays a key role in attempts to characterize the concept of statistical evidence. The standard version of C considers a model and a derived conditional model, formed by conditioning on an ancillary statistic for the model, together with the data, to be equivalent with respect to their statistical evidence content. This equivalence is considered to hold for any ancillary statistic for the model but creates two problems. First, there can be more than one maximal ancillary in a given context and this leads to C not being an equivalence relation and, as such, calls into question whether C is a proper characterization of statistical evidence. Second, a statistic A can change from ancillary to informative (in its marginal distribution) when another ancillary B changes, from having one known distribution PB, to having another known distribution QB. This means that the stability of ancillarity differs across ancillary statistics and raises the issue of when a statistic can be said to be truly ancillary. It is therefore natural, and practically important, to limit conditioning to the set of ancillaries whose distribution is irrelevant to the ancillary status of any other ancillary statistic. This results in a family of ancillaries for which there is a unique maximal member. This also gives a new principle for inference, the stable conditionality principle, that satisfies the criteria required for any principle whose aim is to characterize statistical evidence.
PubDate: 20221018

 Correction to: Inferences and Optimal Censoring Schemes for Progressively
FirstFailure Censored NadarajahHaghighi Distribution
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PubDate: 20221006

 Identifiability of Asymmetric Circular and Cylindrical Distributions

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Abstract: Abstract Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sineskewed circular distributions, including the sineskewed von Mises and sineskewed wrapped Cauchy distributions, and that of a Möbius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sineskewed wrapped Cauchy distribution, and conditional distributions of a random variable on the nonnegative real line given the circular random variable are a Weibull distribution and a generalized Paretotype distribution, respectively.
PubDate: 20220927

 An Overview of Discrete Distributions in Modelling COVID19 Data Sets

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Abstract: Abstract The mathematical modeling of the coronavirus disease19 (COVID19) 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 COVID19 data by determining the optimal statistical modelling to evaluate the regular count of new COVID19 fatalities, thus requiring discrete distributions. Some discrete models are checked and reviewed, such as Binomial, Poisson, Hypergeometric, discrete negative binomial, betabinomial, Skellam, beta negative binomial, Burr, discrete Lindley, discrete alpha power inverse Lomax, discrete generalized exponential, discrete MarshallOlkin Generalized exponential, discrete GompertzGexponential, 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 indepth.
PubDate: 20220909
DOI: 10.1007/s13171022002916

 An Integral Representation for Inverse Moments

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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: 20220908
DOI: 10.1007/s13171022002934

 A Class of Multivariate Power Skew Symmetric Distributions: Properties and
Inference for the PowerParameter
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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: 20220822
DOI: 10.1007/s13171022002925

 Bayesian PSplines Applied to Semiparametric Models with Errors Following
a Scale Mixture of Normals
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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 Psplines 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 Psplines 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: 20220811
DOI: 10.1007/s13171022002907

 Correction to: Formulas of Absolute Moments

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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: 20220801

 Multivector Variate Distributions

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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: 20220801

 On General Exponential Weight Functions and Variation Phenomenon

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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: 20220801

 Parameter Estimation for Multistate Coherent Series and Parallel Systems
with Positively Quadrant Dependent Models
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Abstract: Abstract A multistate 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 threestate 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: 20220801

 Cliques and Chromatic Number in Multiregime Random Graphs

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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: 20220801

 All Conditional Distributions for Y Given X that are Compatible with a
Given Conditional Distribution for X Given Y
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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: 20220801

 Estimation for a Class of Semiparametric Pareto Mixture Densities

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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 ruleofthumb bandwidth selection method is given. Simulation studies demonstrate good performance of the proposed methodology.
PubDate: 20220801

 Multinomial Logistic Mixed Models for Clustered Categorical Data in a
Complex Survey Sampling Setup
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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, S64S78 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 socalled 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 quasilikelihood) 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 socioeconomic 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: 20220801

 Unit NadarajahHaghighi Generated Family of Distributions: Properties and
Applications
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Abstract: Abstract The unit NadarajahHaghighi (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 loglogistic distributions. For some given parameter values it was realized that the density and failure rate functions of the UNH Weibull and UNH loglogistic 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: 20220801
