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- Decoupling Inequalities and Decoupling Coefficients of Gaussian Processes
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Abstract: Abstract We use Brascamp-Lieb’s inequality to obtain new decoupling inequalities for general Gaussian vectors, and in particular for finite stationary Gaussian processes. In the second case, we provide an application using a version by Bump and Diaconis of the strong Szegö limit theorem. We obtain sharp estimates on the decoupling coefficient of remarkable classes of Gaussian processes.
PubDate: 2024-08-01
DOI: 10.1007/s13171-024-00347-9
- A General Formulation for the Large-Sample Behaviour of a Class of
Hypothesis Test Statistics-
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Abstract: Abstract We bring together some strands of development concerning restricted likelihood ratio estimation and testing, including boundary hypothesis testing, going back to pioneering papers of Aitchison, Silvey and Chernoff for motivation. Thus, cases where the parameters are connected by a number of functional relationships, which may involve natural restrictions on the parameters and/or restrictions imposed by a null hypothesis, as well as situations where the null and alternate hypotheses place the true parameter at the boundary of disjoint subsets of the parameter space, are considered. Our asymptotic results are proved under clearly specified and minimal assumptions, which are probably close to the weakest possible. We illustrate with an example for distributions defined on the unit sphere in \(\mathbb {R}^{\varvec{d}}\) .
PubDate: 2024-07-31
DOI: 10.1007/s13171-024-00364-8
- Multivariate Leimkuhler Curve: Properties and Applications to Analysis of
Bibliometric Data-
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Abstract: Abstract The Leimkuhler curve has established itself as an efficient tool in the analysis and comparison of concentration of bibliometric measures of productivity from different sources. Inspite of a considerable volume of literature emerging on multivariate analysis of informetric data, a multivariate version of the Leimkuhler curve does not appear to have been considered so far. In the present work we propose a multivariate Leimkuhler curve and study some of its properties. The use of our results is illustrated by analyzing multivariate informetric data.
PubDate: 2024-07-10
DOI: 10.1007/s13171-024-00363-9
- A Personal Celebration of Dr. D. Basu with Emphasis on
Examples-Counterexamples-Clarifications-
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Abstract: Abstract Preparing this centennial tribute to Dr. D. Basu (5 July, 1924 – 24 March, 2001) created an opportunity to selectively revisit a number of core notions in statistical inference. We explored intrinsic beauty, ingenuity and power of Basu’s (1955) theorem by (i) weaving through the process of Rao-Blackwellization, (ii) looking back at the uniformly minimum variance unbiased estimator (UMVUE), and (iii) proposing a randomly stopped version of Stein’s identity. In doing so, we have (i) suggested a layman’s interpretation of the notion of completeness on its own, (ii) appealed to symmetry that often remains hidden within Rao-Blackwellization, and (iii) considered notions of approximate sufficiency-ancillarity-independence via intuitive understanding. We have confronted the notion of minimal sufficiency in a common mean ( \(\mu )\) estimation problem from a bivariate normal population (with known variances and correlation coefficient), and to our surprise we find that we can do away with only one observation to construct the UMVUE or an UMP test or a likelihood ratio test for \(\mu \) in some situations. We wrap up with constructions of interesting ratios X/Z and Y/Z made up of independent or dependent random variables (X, Y, Z), where X/Z and Y/Z would be overwhelmingly favored to be dependent, but surprisingly they are not. It should not be a surprise that Basu’s theorem, and how Dr. Basu influenced statistical inference ever so gently, have acted as a common thread holding together much of the discussions included in this celebratory piece.
PubDate: 2024-07-05
DOI: 10.1007/s13171-024-00359-5
- On Integral Representations Involving the Probability Generating Function
for Inverse Moments of Positive Discrete Random Variables-
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Abstract: Abstract I simplify and note extensions of results of Shibu et al. Sankhy \(\overline{a}\) , Ser. A 85 (2023a,b) concerning integral representations involving the probability generating function for inverse moments of positive discrete random variables, both univariate and multivariate.
PubDate: 2024-06-28
DOI: 10.1007/s13171-024-00360-y
- Inferences for Fixed Effects Based Regression Parameters in a Finite
Population Setup Using Two-stage Cluster Sample-
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Abstract: Abstract In a clusters based infinite/super population (SP) setup, the inference complexity for regression parameters depend on the cluster correlations structure oriented marginal fixed or mixed effects based mean models, where as opposed to the fixed effects based models the mixed models exhibit both regression and cluster variance/correlation parameters in the mean structure especially for discrete such as clustered counts and binary data. On the other hand, as recently discussed by Sutradhar (2022, Sankhya B, 84, 259-302), in the fixed models setup the modelling of cluster correlations can be complex. In the finite population (FP) setup where FP data follow a SP model but the regression effects estimation is done using a two-stage cluster sample, the inferences become much more complex. As opposed to the recent mixed models based FP inferences (Sutradhar 2023b, Annals of the Institute of Statistical Mathematics, 75, 425-462), there does not appear any studies for fixed models based analysis for exponential family data where true correlation structures play important role in defining FP regression parameters. To resolve this inference issue, in this paper we make following specific contributions. First, the cluster correlations for linear, counts and binary data are developed in a SP setup such that the marginal mean models depend only on the fixed regression effects. Second the FP data which are hypothetical or unobserved until a sample is taken to observe a part, are utilized to develop hypothetical estimating equations for the SP regression parameters, which subsequently define the FP regression and correlation parameters. Third, the design weighted unbiased estimators are obtained for the FP regression and correlation parameters, where the specific formulas for correlation estimators depend on the nature of the response data whether linear, counts or binary. Also, as an additional application of the regression effects estimation, the FP total prediction is discussed. Next, the design consistency of the regression estimators is developed in details.
PubDate: 2024-06-25
DOI: 10.1007/s13171-024-00362-w
- Valid Edgeworth Expansion of the Bootstrap t-statistic of the Whittle MLE
for Linear Regression Models with Long-Memory Residuals-
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Abstract: Abstract In this paper we provide a valid Edgeworth expansion of the parametric bootstrap t-statistic for the Whittle maximum likelihood estimator of a linear regression time series model whose residuals are stationary, Gaussian, and long-memory. Under some sets of conditions on the spectral density function and the parametric values, an Edgeworth expansion of the bootstrap t-statistic of arbitrarily large order of the model is established to have an error of \(o(n^{1-s/2})\) , where \(s \ge 3\) is a positive integer. The result is obtained by extending the Edgeworth expansion obtained by Andrew et al. (2006), which was established for the parametric bootstrap t-statistic of the same model without the linear regression component.
PubDate: 2024-06-21
DOI: 10.1007/s13171-024-00361-x
- How Certain are You of Your Minimum AIC or BIC Values'
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Abstract: Abstract In choosing a candidate model in likelihood-based inference by minimizing an information criterion, the practitioner is often faced with the difficult task of deciding how far up the ranked list to look. Motivated by this pragmatic necessity, we derive an approximation to the quantiles of a generalized (model selection) information criterion (ZIC), defined as a criterion for which the limit in probability is identical to that of the normalized log-likelihood, and which includes common special cases such as AIC and BIC. The method starts from the joint asymptotic normality of the ZIC values, and proceeds by deriving the (asymptotically) exact distribution of the minimum, which can be efficiently (numerically) computed. High quantiles can then be obtained by inverting this distribution function, resulting in what we call a certainty envelope (CE) of plausible models, intended to provide a heuristic upper bound on the location of the actual minimum. The theory is established for three data settings of perennial classical interest: (i) independent and identically distributed, (ii) regression, and (iii) time series. The development in the latter two cases invokes Lindeberg-Feller type conditions for, respectively, normalized: sums of conditional distributions and quadratic forms, in the observations. The performance of the methodology is examined on simulated data by assessing CE nominal coverage probabilities, and comparing them to the bootstrap. Both approaches give coverages close to nominal for large samples, but the bootstrap is on average two orders of magnitude slower. Finally, we hint at the possibility of producing confidence intervals for individual parameters by pivoting the distribution of the minimum ZIC, thus naturally accounting for post-model selection uncertainty.
PubDate: 2024-06-05
DOI: 10.1007/s13171-024-00344-y
- On Robust Change Point Detection and Estimation in Multisubject Studies
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Abstract: Abstract A variety of change point estimation and detection algorithms have been developed for random variables observed over time. The acquisition of data in current practice often results in multiple subjects studied. The traditional treatment of such observations involves the assumption of their independence. In practice, however, this assumption is often inadequate or unrealistic. We propose an effective and modern computerized approach to estimating and detecting change points in linear model time series processes in the situation when the assumption of independent observations is not feasible. The developed methodology relies on the multivariate transformation and matrix normal distribution. The latter is used for separating the sources of variability. The application of the back-transform of the exponential transformation leads to a flexible distribution that effectively accounts for deviations from normality. The developed procedure has been successfully tested in various settings and applied to a crime rate data set.
PubDate: 2024-05-29
DOI: 10.1007/s13171-024-00355-9
- Valid Confidence Intervals for $$\mu , \sigma $$ When There Is Only One
Observation Available-
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Abstract: Abstract Portnoy (The American Statistician, 73:1, 10–15, 2019) considered the problem of constructing an optimal confidence interval for the mean based on a single observation \(\, X \sim \mathcal{{N}}(\mu , \, \sigma ^2) \,\) . Here we extend this result to obtaining 1-sample confidence intervals for \(\, \sigma \,\) and to cases of symmetric unimodal distributions and of distributions with compact support. Finally, we extend the multivariate result in Portnoy (The American Statistician, 73:1, 10–15, 2019) to allow a sample of size \(\, m \,\) from a multivariate normal distribution where m may be less than the dimension.
PubDate: 2024-05-24
DOI: 10.1007/s13171-023-00338-2
- Finite Sample Rates for Logistic Regression with Small Noise or Few
Samples-
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Abstract: Abstract The logistic regression estimator is known to inflate the magnitude of its coefficients if the sample size n is small, the dimension p is (moderately) large or the signal-to-noise ratio \(1/\sigma \) is large (probabilities of observing a label are close to 0 or 1). With this in mind, we study the logistic regression estimator with \(p\ll n/\log n\) , assuming Gaussian covariates and labels generated by the Gaussian link function, with a mild optimization constraint on the estimator’s length to ensure existence. We provide finite sample guarantees for its direction, which serves as a classifier, and its Euclidean norm, which is an estimator for the signal-to-noise ratio. We distinguish between two regimes. In the low-noise/small-sample regime ( \(\sigma \lesssim (p\log n)/n\) ), we show that the estimator’s direction (and consequentially the classification error) achieve the rate \((p\log n)/n\) - up to the log term as if the problem was noiseless. In this case, the norm of the estimator is at least of order \(n/(p\log n)\) . If instead \((p\log n)/n\lesssim \sigma \lesssim 1\) , the estimator’s direction achieves the rate \(\sqrt{\sigma p\log n/n}\) , whereas its norm converges to the true norm at the rate \(\sqrt{p\log n/(n\sigma ^3)}\) . As a corollary, the data are not linearly separable with high probability in this regime. In either regime, logistic regression provides a competitive classifier.
PubDate: 2024-05-21
DOI: 10.1007/s13171-024-00358-6
- Learning Statistics From Counterexamples
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Abstract: Abstract The title of this article is (essentially) the same as the famous paper Basu (2011b). Basu often opined that counterexamples were the best way to learn limitations of theories or methods and I have followed his directive in my own teaching. A number of counterexamples I use extensively in teaching are collected here.
PubDate: 2024-05-11
DOI: 10.1007/s13171-024-00356-8
- On the Feasibility of Parsimonious Variable Selection for
Hotelling’s $$T^2$$ -test-
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Abstract: Abstract Hotelling’s \(T^2\) -test for the mean of a multivariate normal distribution is one of the triumphs of classical multivariate analysis. It is uniformly most powerful among invariant tests, and admissible, proper Bayes, and locally and asymptotically minimax among all tests. Nonetheless, investigators often prefer non-invariant tests, especially those obtained by selecting only a small subset of variables from which the \(T^2\) -statistic is to be calculated, because such reduced statistics are more easily interpretable for their specific application. Thus it is relevant to ask the extent to which power is lost when variable selection is limited to very small subsets of variables, e.g. of size one (yielding univariate Student- \(t^2\) tests) or size two (yielding bivariate \(T^2\) -tests). This study presents preliminary evidence suggesting that in some cases, no power may be lost, in fact may be gained, over a wide range of alternatives.
PubDate: 2024-05-11
DOI: 10.1007/s13171-024-00357-7
- Kernel-based Measures of Association Between Inputs and Outputs Using
ANOVA-
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Abstract: Abstract ANOVA decomposition of a function with random input variables provides ANOVA functionals (AFs), which contain information about the contributions of the input variables on the output variable(s). By embedding AFs into an appropriate reproducing kernel Hilbert space regarding their distributions, we propose an efficient statistical test of independence between the input variables and output variable(s). The resulting test statistic leads to new dependence measures of association between inputs and outputs that allow for i) dealing with any distribution of AFs, including the Cauchy distribution, ii) accounting for the necessary or desirable moments of AFs and the interactions among the input variables. In uncertainty quantification for mathematical models, a number of existing measures are special cases of this framework. We then provide unified and general global sensitivity indices and their consistent estimators, including asymptotic distributions. For Gaussian-distributed AFs, we obtain Sobol’ indices and dependent generalized sensitivity indices using quadratic kernels.
PubDate: 2024-05-07
DOI: 10.1007/s13171-024-00354-w
- Dirac-type Theorems for Inhomogenous Random Graphs
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Abstract: Abstract In this paper, we study Dirac-type theorems for an inhomogenous random graph \(G\) whose edge probabilities are not necessarily all the same. We obtain sufficient conditions for the existence of Hamiltonian paths and perfect matchings, in terms of the sum of edge probabilities. For edge probability assignments with two-sided bounds, we use Pósa rotation and single vertex exclusion techniques to show that \(G\) is Hamiltonian with high probability. For weaker one-sided bounds, we use bootstrapping techniques to obtain a perfect matching in \(G,\) with high probability. We also highlight an application of our results in the context of channel assignment problem in wireless networks.
PubDate: 2024-04-22
DOI: 10.1007/s13171-024-00353-x
- Efficiency Bound Under Identifiability Constraints in Semiparametric
Models-
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Abstract: Abstract The purpose of this work is to define an adequate efficiency bound in some models presenting some identification problems. We show how it is possible to define such bounds in some regular semi-parametric models (in the sense of Le Cam) when an identifying constraint is available, despite the degeneracy of the information matrix. We establish a new convolution theorem in this context. We illustrate the computation of the information bound for some standard identifiability constraints, in some interesting models, including probit, single-index, and ANOVA models. We also show how a two-step procedure still based on a preliminary estimator satisfying approximately the constraint, allows us to obtain an efficient estimator of the parameters.
PubDate: 2024-04-22
DOI: 10.1007/s13171-024-00352-y
- Finite Sequences Representing Expected Order Statistics
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Abstract: Abstract Characterizations of finite sequences \(\beta _{1}<\cdots <\beta _{n}\) representing expected values of order statistics from a random sample of size n are given. As a by-product, a characterization of binomial mixtures, when the mixing random variable is supported in the open interval (0, 1), is presented; this enables the exact description of the convex hull of the open binomial curve, as well as the open moment curve.
PubDate: 2024-04-20
DOI: 10.1007/s13171-024-00343-z
- Statistical Analysis of Improved Type-II Adaptive Progressive Hybrid
Censored NH Data-
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Abstract: Abstract Recently, improved Type-II adaptive progressive censoring has been introduced to ensure that the experimental duration does not exceed a certain time and that the test concludes once a predetermined number of failures are recorded. This paper addresses the problem of estimating the unknown parameters as well as the reliability and hazard rate functions of the proposed lifetime Nadarajah-Haghighi distribution when the collected data are obtained from the proposed censoring plan. For each unknown parameter of life, using maximum likelihood and Bayes inference methods, both point and interval estimators are derived. The approximate confidence intervals are acquired based on the asymptotic normality of the maximum likelihood estimators. Under the assumption of independent gamma priors, the Bayes estimators cannot be obtained in closed form, therefore, the Markov-Chain Monte-Carlo approximation technique via the Metropolis–Hastings algorithm is utilized to evaluate the Bayes point estimates and to create their credible interval estimates. To compare the efficiency of the different proposed estimators, in terms of root mean squared-error, mean relative absolute bias, and average interval length values, extensive Monte Carlo simulations are implemented. Ultimately, to show how the acquired estimators can be applied in a real-life engineering scenario, a real data set consisting of eighteen failure times for electronic devices is analyzed.
PubDate: 2024-04-17
DOI: 10.1007/s13171-024-00345-x
- Distributional Approximation for General Curie–Weiss Models with
Size-dependent Inverse Temperatures-
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Abstract: Abstract The Curie–Weiss model is a statistical physics model that describes the behavior of a system of particles with mutual interactions. In this paper, we apply Stein’s method to establish Berry–Esseen bounds for both normal and non-normal approximations of a broad types of Curie–Weiss model, incorporating a size-dependent inverse temperature. Our result encompasses the Blumer-Emery-Griffiths model as a particular instance, while surpassing the convergence rate of earlier findings by Eichelsbacher and Martschink (2014). By using Stein’s method, we provide a comprehensive analysis of the Curie–Weiss model, offering improved bounds on the rate of convergence.
PubDate: 2024-04-11
DOI: 10.1007/s13171-024-00351-z
- On the Convolution of Scaled Sibuya Distributions
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Abstract: Abstract We introduce a new heavy tailed distribution on \(\mathbb {Z}_+\) that arises as the infinite convolution of scaled Sibuya distributions. We provide closed form expressions for its probability mass function, its cumulative distribution function, and its probability generating function. We interpret our main results in terms of the weak convergence of partial sums of a binomially thinned sequence of i.i.d. random variables with a common scaled Sibuya distribution. Properties of infinite divisibility and discrete self-decomposability of the new distribution are also discussed. As an application, we briefly describe an integer-valued autoregressive process of order one with a scaled Sibuya innovation sequence. Finally, we discuss some partial extensions of our results to the case of the generalized Sibuya distribution introduced by Kozubowski and Podgórski., Ann. of the Inst. Statist. Math., 70(4), 855-887., 2018.
PubDate: 2024-04-11
DOI: 10.1007/s13171-024-00346-w
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