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Abstract: The origins of the stick breaking construction of Sethuraman (Statistica Sinica 4 639–650. 1994) are presented here. We show the equivalence of nonparametric priors and exchangeable distributions which automatically gives the posterior distribution of nonparametric priors. The Pólya exchangeable sequence of Blackwell and MacQueen (Ann. Statist. 1 353–355. 1973) corresponds to the Dirichlet process prior which again immediately gives the posterior distribution of of the Dirichlet prior. Studying the Pólya exchangeable sequence some more shows that the random probability measure distributed as a Dirichlet process is a random discrete probability measure and has something like a stick breaking construction. Under the condition that the parameter \(\beta \) of the Dirichlet process is nonatomic, we show how to obtain the stick breaking representation of Sethuraman (Statistica Sinica 4 639–650. 1994). Noticing that this statement of the stick breaking construction puts no conditions \(\beta \) , Sethuraman (Statistica Sinica 4 639–650. 1994) used a different and a simpler method to obtain the stick breaking construction in full generality. Finally we present an example to remove a generally held misconception of Sethuraman (Statistica Sinica 4 639–650. 1994). PubDate: 2023-11-18
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Abstract: Hierarchical models enjoy great popularity due to their ability to handle heterogeneous groups of observations by leveraging on their underlying common structure. In a Bayesian nonparametric framework, the hierarchy is introduced at the level of group-specific random measures, and then translated to the observations’ level via suitable transformations. In this work, we propose a new strategy to derive closed-form expressions for the marginal and posterior distributions of each group. Indeed, by directly inserting a suitable set of latent variables into the generative model for the data, we unravel a common core shared by the different hierarchical constructions proposed in the Bayesian nonparametric literature. Specifically, we identify a key identity that underlies these models and highlight its role in the derivation of quantities of interest. PubDate: 2023-11-08
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Abstract: Failures to replicate the results of scientific studies are often attributed to misinterpretations of the p-value. The p-value may be interpreted as an approximate posterior probability, not that the null hypothesis is true but rather that it explains the data as well as the data-generating distribution. That posterior probability modifies the p-value in the following two broad areas of application, leading to new methods of hypothesis testing and effect size estimation. First, when corrected for multiple comparisons, the posterior probability that the null hypothesis adequately explains the data overcomes both the conservative bias of corrected p-values and the anti-conservative bias of commonly used false discovery rate methods. Second, the posterior probability that the null hypothesis adequately explains the data, conditional on a parameter restriction, transforms the p-value in such a way as to overcome difficulties in restricted parameter spaces. PubDate: 2023-11-01
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Abstract: Regression is one of the most fundamental statistical inference problems. A broad definition of regression problems is as estimation of the distribution of an outcome using a family of probability models indexed by covariates. Despite the ubiquitous nature of regression problems and the abundance of related methods and results there is a surprising gap in the literature. There are no well established methods for regression with a varying dimension covariate vectors, despite the common occurrence of such problems. In this paper we review some recent related papers proposing varying dimension regression by way of random partitions. PubDate: 2023-10-20
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Abstract: Survey sampling and, more generally, Official Statistics are experiencing an important renovation time. On one hand, there is the need to exploit the huge information potentiality that the digital revolution made available in terms of data. On the other hand, this process occurred simultaneously with a progressive deterioration of the quality of classical sample surveys, due to a decreasing willingness to participate and an increasing rate of missing responses. The switch from survey-based inference to a hybrid system involving register-based information has made more stringent the debate and the possible resolution of the design-based versus model-based approaches controversy. In this new framework, the use of statistical models seems unavoidable and it is today a relevant part of the official statistician toolkit. Models are important in several different contexts, from Small area estimation to non sampling error adjustment, but they are also crucial for correcting bias due to over and undercoverage of administrative data, in order to prevent potential selection bias, and to deal with different definitions and/or errors in the measurement process of the administrative sources. The progressive shift from a design-based to a model-based approach in terms of super-population is a matter of fact in the practice of the National Statistical Institutes. However, the introduction of Bayesian ideas in official statistics still encounters difficulties and resistance. In this work, we attempt a non-systematic review of the Bayesian development in this area and try to highlight the extra benefit that a Bayesian approach might provide. Our general conclusion is that, while the general picture is today clear and most of the basic topics of survey sampling can be easily rephrased and tackled from a Bayesian perspective, much work is still necessary for the availability of a ready-to-use platform of Bayesian survey sampling in the presence of complex sampling design, non-ignorable missing data patterns, and large datasets. PubDate: 2023-10-16
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Abstract: In the present paper, we are concerned with a generalized kernel estimators defined by the stochastic approximation algorithm in the case of dependent functional data. We establish the central limit theorem for the proposed estimators under some mild conditions. We then approach the distribution of the bias distribution of our estimate by the bootstrapped distribution when it is conditioned by the data using the Kolmogorov distance. PubDate: 2023-09-19
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Abstract: Testing a precise hypothesis can lead to substantially different results in the frequentist and Bayesian approach, a situation which is highlighted by the Jeffreys-Lindley paradox. While there exist various explanations why the paradox occurs, this article extends prior work by placing the less well-studied point-null-zero-probability paradox at the center of the analysis. The relationship between the two paradoxes is analyzed based on accepting or rejecting the existence of precise hypotheses. The perspective provided in this paper aims at demonstrating how the Bayesian and frequentist solutions can be reconciled when paying attention to the assumption of the point-null-zero-probability paradox. As a result, the Jeffreys-Lindley-paradox can be reinterpreted as a Bayes-frequentist compromise. The resolution shows that divergences between Bayesian and frequentist modes of inference stem from (a) accepting the existence of a precise hypothesis or not, (b) the assignment of positive measure to a null set and (c) the use of unstandardized p-values or p-values standardized to tail-area probabilities. PubDate: 2023-09-11
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Abstract: The classical concept of bounded completeness and its relation to sufficiency and ancillarity play a fundamental role in unbiased estimation, unbiased testing, and the validity of inference in the presence of nuisance parameters. In this short note, we provide a direct proof of a little-known result by Farrell (1962) on a characterization of bounded completeness based on an \(L^1\) denseness property of the linear span of likelihood ratios. As an application, we show that an experiment with infinite-dimensional observation space is boundedly complete iff suitably chosen restricted subexperiments with finite-dimensional observation spaces are. PubDate: 2023-09-09
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Abstract: The paper addresses asymptotic estimation of normal means under sparsity. The primary focus is estimation of multivariate normal means where we obtain exact asymptotic minimax error under global-local shrinkage prior. This extends the corresponding univariate work of Ghosh and Chakrabarti (2017). In addition, we obtain similar results for the Dirichlet-Laplace prior as considered in Bhattacharya et al. (2015). Also, following van der Pas et al. (2017), we have been able to derive credible sets for multivariate normal means under global-local priors. PubDate: 2023-09-08
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Abstract: In this article, we obtain explicit bounds on the uniform distance between the cumulative distribution function of a standardized sum \(S_n\) of \(n\) independent centered random variables with moments of order four and its first-order Edgeworth expansion. Those bounds are valid for any sample size with \(n^{-1/2}\) rate under moment conditions only and \(n^{-1}\) rate under additional regularity constraints on the tail behavior of the characteristic function of \(S_n\) . In both cases, the bounds are further sharpened if the variables involved in \(S_n\) are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. Following these theoretical results, we discuss the practical use of our bounds, which depend on possibly unknown moments of the distribution of \(S_n\) . Finally, we apply our bounds to investigate several aspects of the non-asymptotic behavior of one-sided tests: informativeness, sufficient sample size in experimental design, distortions in terms of levels and p-values. PubDate: 2023-09-08
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Abstract: Nonlinear prediction of time series can offer potential accuracy gains over linear methods when the process is nonlinear. As there are numerous examples of nonlinearity in time series data (e.g., finance, macroeconomics, image, and speech processing), there seems to be merit in developing a general theory and methodology. We explore the class of quadratic predictors, which directly generalize linear predictors, and show that they can be computed in terms of the second, third, and fourth auto-cumulant functions when the time series is stationary. The new formulas for quadratic predictors generalize the normal equations for linear prediction of stationary time series, and hence we obtain quadratic generalizations of the Yule-Walker equations; we explicitly quantify the prediction gains in quadratic over linear methods. We say a stochastic process is second order forecastable if quadratic prediction provides an advantage over linear prediction. One of the key results of the paper provides a characterization of second order forecastable processes in terms of the spectral and bi-spectral densities. We verify these conditions for some popular nonlinear time series models. PubDate: 2023-09-08
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Abstract: Basu’s via media is what he referred to as the middle road between the Bayesian and frequentist poles. He seemed skeptical that a suitable via media could be found, but I disagree. My basic claim is that the likelihood alone can’t reliably support probabilistic inference, and I justify this by considering a technical trap that Basu stepped in concerning interpretation of the likelihood. While reliable probabilistic inference is out of reach, it turns out that reliable possibilistic inference is not. I lay out my proposed possibility-theoretic solution to Basu’s via media and I investigate how the flexibility afforded by my imprecise-probabilistic solution can be leveraged to achieve the likelihood principle (or something close to it). PubDate: 2023-09-01
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Abstract: In the following study, we deal with the exponential bounds and rates for a class of sieve estimators of Grenander for Functional Autoregressive Processes when the parameter operator belongs to the parameter space of Hilbert-Schmidt operators. Two classes of parameter operators are considered where we state clearly sieve estimators formulas and derive corresponding exponential bounds. These results are applied to establish their almost sure convergence and almost complete convergence. Then, we determine rates of convergence of sieve estimators in each class. The numerical studies illustrate the performance of the sieve predictors and give comparisons with other existing prediction methods both on simulated and real functional data sets exhibiting competitive results. PubDate: 2023-08-31 DOI: 10.1007/s13171-023-00322-w
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Abstract: A variety of parametric models are specified by a mix of discrete parameters, which take values from a countable set, and continuous parameters, which take values from a continuous space. However, the asymptotic properties of the parameter estimators are not well understood in the literature. In this paper, we consider the general framework of M-estimation and derive the asymptotic properties of the M-estimators of both discrete and continuous parameters. In particular, we show that the M-estimators are consistent and the continuous parameters are asymptotically normal. We also extend a large deviation principle from models with only discrete parameters to models with discrete and continuous parameters. The finite-sample properties are assessed by a simulation study, and for illustration, we perform a break-point analysis for the clinical outcomes of COVID-19 patients. PubDate: 2023-08-30 DOI: 10.1007/s13171-023-00317-7
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Abstract: This article presents a bivariate extension of the Teissier distribution, whose univariate marginal distributions belong to the exponentiated Teissier family. Analytic expressions for the different statistical quantities such as conditional distribution, joint moments, and quantile function are explicitly derived. For the proposed distribution, the concepts of reliability and dependence measures are also explored in details. Both the maximum likelihood technique and the Bayesian approach are utilised in the process of parameter estimation for the proposed distribution with unknown parameters. Several numerical experiments are reported to study the performance of the classical and Bayes estimators for varying sample size. Finally, a bivariate data is fitted using the proposed distribution to show its applicability over the bivariate exponential, Rayleigh, and linear exponential distributions in real-life situations. PubDate: 2023-08-29 DOI: 10.1007/s13171-023-00314-w
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Abstract: In this paper we have considered the chirp like model which has been recently introduced, and it has a very close resemblance with a chirp model. We consider the weighted least squares estimators of the parameters of a chirp like model in presence of an additive stationary error, and study their properties. It is observed that although the least squares method seems to be a natural choice to estimate the unknown parameters of a chirp like model, the least squares estimators are very sensitive to the outliers. It is observed that the weighted least squares estimators are quite robust in this respect. The weighted least squares estimators are consistent and they have the same rate of convergence as the least squares estimators. We have further extended the results in case of multicomponent chirp like model. Some simulations have been performed to show the effectiveness of the proposed method. In simulation studies, weighted least squares estimators have been compared with the least absolute deviation estimators which, in general, are known to work well in presence of outliers. One EEG data set has been analyzed and the results are quite satisfactory. PubDate: 2023-08-22 DOI: 10.1007/s13171-023-00313-x
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Abstract: Recently, a modification of fractional entropy based on the inverse Mittag-Leffler function (MLF) was proposed by Zhang and Shang (2021). In this paper, we present an extension of the fractional cumulative entropy (FCE) and obtain some further results about this measure. We study new equivalent expressions, bounds, stochastic ordering, and properties of dynamic generalized FCE. By using the empirical approach, we give an estimator of this measure and study large sample properties of it. In addition, the validity of this new measure is supported by numerical simulations on logistic map equations. Finally, an application of this measure is proposed in the evaluation of MRI scans for brain cancer. PubDate: 2023-08-14 DOI: 10.1007/s13171-023-00316-8
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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: 2023-08-01 DOI: 10.1007/s13171-022-00293-4
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Abstract: The discrete time insurance risk process with a constant interest force is an interesting stochastic model in risk theory. This paper considers the issue of ruin probabilities of an insurer whose portfolio is exposed to insurance risk arising from a discrete time risk model under the constant initial capital, capital injections and insurer’s aggregate claim size. Some expressions are obtained for the ruin probabilities within finite and infinite time. We compute the exact and approximation to the density and cumulative distribution of the time to ruin in the continuous risk model with capital injections for some light and heavy-tailed distributions. Additionally, we illustrate our results and try to minimize the infinite time ruin probability for different values of initial capital and level of capital injection. PubDate: 2023-08-01 DOI: 10.1007/s13171-022-00305-3
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Abstract: Methods of testing for the equality of two distributions on a manifold are unveiled in this paper. One defines the extrinsic energy distance associated with two probability measures on a complete metric space embedded in a numerical space. One derives the extrinsic energy statistic test for homogeneity of such distributions. This test is validated via a simulation example on the Kendall space of planar k-ads with a Veronese-Whitney (VW) embedding. Imaging data driven examples are also considered here. In one application, central to the paper, one tests for homogeneity the distributions of planar Kendall shapes of midsections of the Corpus Callosum in a clinically normal population vs a population of ADHD diagnosed individuals; these distributions are not significantly different, although they are known to have highly significant VW-means. On the other hand, in 3D, the reflection shapes of configurations of Acrosterigma Magnum shells are not significantly different, and do not have significantly similar different 3D Schoenberg means. PubDate: 2023-05-15 DOI: 10.1007/s13171-023-00310-0
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