arXiv subjects

Soumendra N. Lahiri

Publications and source records attributed to Soumendra N. Lahiri.

18 recordsLinked to original sources

arXiv2026

Large Sample Properties of Higher Order Markov Models

We study large-sample properties of higher-order Markov chains on a finite alphabet $Σ$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $Σ^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals $\sum_{t}\! g_n(Y_t^{(n)})$ under natural ergodicity and sparsity conditions. The normalization involves the stationary return time to a suitably chosen state and accommodates triangular arrays with $m_n\!\to\!\infty$ and $m_n/n\!\to\!0$. We further illustrate the assumptions in a binary variable length Markov chain (VLMC), deriving explicit lower bounds on stationary masses that yield a concrete growth regime (e.g., $m_n\log m_n/n \to 0$) ensuring the CLT. These results provide asymptotic foundations for inference in sparse/partitioned higher-order models; including VLMCs and sparse Markov models (SMMs) where the effective dimensionality grows with the sample size.

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arXiv2026

On Optimal Data Splitting for Split Conformal Prediction

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

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arXiv2026

Multi-Resolution Analysis of Variable Selection for Road Safety in St. Louis and Its Neighboring Area

Generally, Lasso, Adaptive Lasso, and SCAD are standard approaches in variable selection in the presence of a large number of predictors. In recent years, during intensity function estimation for spatial point processes with a diverging number of predictors, many researchers have considered these penalized methods. But we have discussed a multi-resolution perspective for the variable selection method for spatial point process data. Its advantage is twofold: it not only efficiently selects the predictors but also provides the idea of which points are liable for selecting a predictor at a specific resolution. Actually, our research is motivated by the crime and accident occurrences in St. Louis and its neighborhoods. It is more relevant to select predictors at the local level, and thus we get the idea of which set of predictors is relevant for the occurrences of crime or accident in which parts of St. Louis. We describe the simulation results to justify the accuracy of local-level variable selection during intensity function estimation.

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arXiv2022

On optimal block resampling for Gaussian-subordinated long-range dependent processes

Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functionals are also considered. Unlike weak dependence, the properties of resampling estimators under strong dependence are shown to depend intricately on the nature of non-linearity in the time series (beyond Hermite ranks) in addition the long-memory coefficient and block size. Additionally, the intuition has often been that optimal block sizes should be larger under strong dependence (say $O(n^{1/2})$ for a sample size $n$) than the optimal order $O(n^{1/3})$ known under weak dependence. This intuition turns out to be largely incorrect, though a block order $O(n^{1/2})$ may be reasonable (and even optimal) in many cases, owing to non-linearity in a long-memory time series. While optimal block sizes are more complex under long-range dependence compared to short-range, we provide a consistent data-driven rule for block selection, and numerical studies illustrate that the guides for block selection perform well in other block-based problems with long-memory time series, such as distribution estimation and strategies for testing Hermite rank.

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arXiv2019

Simulating Markov random fields with a conclique-based Gibbs sampler

For spatial and network data, we consider models formed from a Markov random field (MRF) structure and the specification of a conditional distribution for each observation. Fast simulation from such MRF models is often an important consideration, particularly when repeated generation of large numbers of data sets is required. However, a standard Gibbs strategy for simulating from MRF models involves single-site updates, performed with the conditional univariate distribution of each observation in a sequential manner, whereby a complete Gibbs iteration may become computationally involved even for moderate samples. As an alternative, we describe a general way to simulate from MRF models using Gibbs sampling with "concliques" (i.e., groups of non-neighboring observations). Compared to standard Gibbs sampling, this simulation scheme can be much faster by reducing Gibbs steps and independently updating all observations per conclique at once. The speed improvement depends on the number of concliques relative to the sample size for simulation, and order-of-magnitude speed increases are possible with many MRF models (e.g., having appropriately bounded neighborhoods). We detail the simulation method, establish its validity, and assess its computational performance through numerical studies, where speed advantages are shown for several spatial and network examples.

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arXiv2019

On Statistical Properties of A Veracity Scoring Method for Spatial Data

Measuring veracity or reliability of noisy data is of utmost importance, especially in the scenarios where the information are gathered through automated systems. In a recent paper, Chakraborty et. al. (2019) have introduced a veracity scoring technique for geostatistical data. The authors have used a high-quality `reference' data to measure the veracity of the varying-quality observations and incorporated the veracity scores in their analysis of mobile-sensor generated noisy weather data to generate efficient predictions of the ambient temperature process. In this paper, we consider the scenario when no reference data is available and hence, the veracity scores (referred as VS) are defined based on `local' summaries of the observations. We develop a VS-based estimation method for parameters of a spatial regression model. Under a non-stationary noise structure and fairly general assumptions on the underlying spatial process, we show that the VS-based estimators of the regression parameters are consistent. Moreover, we establish the advantage of the VS-based estimators as compared to the ordinary least squares (OLS) estimator by analyzing their asymptotic mean squared errors. We illustrate the merits of the VS-based technique through simulations and apply the methodology to a real data set on mass percentages of ash in coal seams in Pennsylvania.

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arXiv2016

Two Stage Non-penalized Corrected Least Squares for High Dimensional Linear Models with Measurement error or Missing Covariates

This paper provides an alternative to penalized estimators for estimation and vari- able selection in high dimensional linear regression models with measurement error or missing covariates. We propose estimation via bias corrected least squares after model selection. We show that by separating model selection and estimation, it is possible to achieve an improved rate of convergence of the L2 estimation error compared to the rate sqrt{s log p/n} achieved by simultaneous estimation and variable selection methods such as L1 penalized corrected least squares. If the correct model is selected with high probability then the L2 rate of convergence for the proposed method is indeed the oracle rate of sqrt{s/n}. Here s, p are the number of non zero parameters and the model dimension, respectively, and n is the sample size. Under very general model selection criteria, the proposed method is computationally simpler and statistically at least as efficient as the L1 penalized corrected least squares method, performs model selection without the availability of the bias correction matrix, and is able to provide estimates with only a small sub-block of the bias correction covariance matrix of order s x s in comparison to the p x p correction matrix required for computation of the L1 penalized version. Furthermore we show that the model selection requirements are met by a correlation screening type method and the L1 penalized corrected least squares method. Also, the proposed methodology when applied to the estimation of precision matrices with missing observations, is seen to perform at least as well as existing L1 penalty based methods. All results are supported empirically by a simulation study.

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arXiv2015

A frequency domain empirical likelihood method for irregularly spaced spatial data

This paper develops empirical likelihood methodology for irregularly spaced spatial data in the frequency domain. Unlike the frequency domain empirical likelihood (FDEL) methodology for time series (on a regular grid), the formulation of the spatial FDEL needs special care due to lack of the usual orthogonality properties of the discrete Fourier transform for irregularly spaced data and due to presence of nontrivial bias in the periodogram under different spatial asymptotic structures. A spatial FDEL is formulated in the paper taking into account the effects of these factors. The main results of the paper show that Wilks' phenomenon holds for a scaled version of the logarithm of the proposed empirical likelihood ratio statistic in the sense that it is asymptotically distribution-free and has a chi-squared limit. As a result, the proposed spatial FDEL method can be used to build nonparametric, asymptotically correct confidence regions and tests for covariance parameters that are defined through spectral estimating equations, for irregularly spaced spatial data. In comparison to the more common studentization approach, a major advantage of our method is that it does not require explicit estimation of the standard error of an estimator, which is itself a very difficult problem as the asymptotic variances of many common estimators depend on intricate interactions among several population quantities, including the spectral density of the spatial process, the spatial sampling density and the spatial asymptotic structure. Results from a numerical study are also reported to illustrate the methodology and its finite sample properties.

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arXiv2014

Convergence rates of empirical block length selectors for block bootstrap

We investigate the accuracy of two general non-parametric methods for estimating optimal block lengths for block bootstraps with time series - the first proposed in the seminal paper of Hall, Horowitz and Jing (Biometrika 82 (1995) 561-574) and the second from Lahiri et al. (Stat. Methodol. 4 (2007) 292-321). The relative performances of these general methods have been unknown and, to provide a comparison, we focus on rates of convergence for these block length selectors for the moving block bootstrap (MBB) with variance estimation problems under the smooth function model. It is shown that, with suitable choice of tuning parameters, the optimal convergence rate of the first method is $O_p(n^{-1/6})$ where $n$ denotes the sample size. The optimal convergence rate of the second method, with the same number of tuning parameters, is shown to be $O_p(n^{-2/7})$, suggesting that the second method may generally have better large-sample properties for block selection in block bootstrap applications beyond variance estimation. We also compare the two general methods with other plug-in methods specifically designed for block selection in variance estimation, where the best possible convergence rate is shown to be $O_p(n^{-1/3})$ and achieved by a method from Politis and White (Econometric Rev. 23 (2004) 53-70).

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arXiv2014

A nonstandard empirical likelihood for time series

Standard blockwise empirical likelihood (BEL) for stationary, weakly dependent time series requires specifying a fixed block length as a tuning parameter for setting confidence regions. This aspect can be difficult and impacts coverage accuracy. As an alternative, this paper proposes a new version of BEL based on a simple, though nonstandard, data-blocking rule which uses a data block of every possible length. Consequently, the method does not involve the usual block selection issues and is also anticipated to exhibit better coverage performance. Its nonstandard blocking scheme, however, induces nonstandard asymptotics and requires a significantly different development compared to standard BEL. We establish the large-sample distribution of log-ratio statistics from the new BEL method for calibrating confidence regions for mean or smooth function parameters of time series. This limit law is not the usual chi-square one, but is distribution-free and can be reproduced through straightforward simulations. Numerical studies indicate that the proposed method generally exhibits better coverage accuracy than standard BEL.

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arXiv2013

A penalized empirical likelihood method in high dimensions

This paper formulates a penalized empirical likelihood (PEL) method for inference on the population mean when the dimension of the observations may grow faster than the sample size. Asymptotic distributions of the PEL ratio statistic is derived under different component-wise dependence structures of the observations, namely, (i) non-Ergodic, (ii) long-range dependence and (iii) short-range dependence. It follows that the limit distribution of the proposed PEL ratio statistic can vary widely depending on the correlation structure, and it is typically different from the usual chi-squared limit of the empirical likelihood ratio statistic in the fixed and finite dimensional case. A unified subsampling based calibration is proposed, and its validity is established in all three cases, (i)-(iii). Finite sample properties of the method are investigated through a simulation study.

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arXiv2012

Goodness of fit tests for a class of Markov random field models

This paper develops goodness of fit statistics that can be used to formally assess Markov random field models for spatial data, when the model distributions are discrete or continuous and potentially parametric. Test statistics are formed from generalized spatial residuals which are collected over groups of nonneighboring spatial observations, called concliques. Under a hypothesized Markov model structure, spatial residuals within each conclique are shown to be independent and identically distributed as uniform variables. The information from a series of concliques can be then pooled into goodness of fit statistics. Under some conditions, large sample distributions of these statistics are explicitly derived for testing both simple and composite hypotheses, where the latter involves additional parametric estimation steps. The distributional results are verified through simulation, and a data example illustrates the method for model assessment.

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arXiv2008

Estimation of distributions, moments and quantiles in deconvolution problems

When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, $W$, say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of $W$ can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators.

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arXiv2007

A frequency domain empirical likelihood for short- and long-range dependence

This paper introduces a version of empirical likelihood based on the periodogram and spectral estimating equations. This formulation handles dependent data through a data transformation (i.e., a Fourier transform) and is developed in terms of the spectral distribution rather than a time domain probability distribution. The asymptotic properties of frequency domain empirical likelihood are studied for linear time processes exhibiting both short- and long-range dependence. The method results in likelihood ratios which can be used to build nonparametric, asymptotically correct confidence regions for a class of normalized (or ratio) spectral parameters, including autocorrelations. Maximum empirical likelihood estimators are possible, as well as tests of spectral moment conditions. The methodology can be applied to several inference problems such as Whittle estimation and goodness-of-fit testing.

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arXiv2006

Resampling Based Empirical Prediction: An Application to Small Area Estimation

Best linear unbiased prediction is well known for its wide range of applications including small area estimation. While the theory is well established for mixed linear models and under normality of the error and mixing distributions, the literature is sparse for nonlinear mixed models under nonnormality of the error or of the mixing distributions. This article develops a resampling based unified approach for predicting mixed effects under a generalized mixed model set up. Second order accurate nonnegative estimators of mean squared prediction errors are also developed. Given the parametric model, the proposed methodology automatically produces estimates of the small area parameters and their MSPEs, without requiring explicit analytical expressions for the MSPE.

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arXiv2006

Nonnegative mean squared prediction error estimation in small area estimation

Small area estimation has received enormous attention in recent years due to its wide range of application, particularly in policy making decisions. The variance based on direct sample size of small area estimator is unduly large and there is a need of constructing model based estimator with low mean squared prediction error (MSPE). Estimation of MSPE and in particular the bias correction of MSPE plays the central piece of small area estimation research. In this article, a new technique of bias correction for the estimated MSPE is proposed. It is shown that that the new MSPE estimator attains the same level of bias correction as the existing estimators based on straight Taylor expansion and jackknife methods. However, unlike the existing methods, the proposed estimate of MSPE is always nonnegative. Furthermore, the proposed method can be used for general two-level small area models where the variables at each level can be discrete or continuous and, in particular, be nonnormal.

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arXiv2006

A Sub-Gaussian Berry-Esseen Theorem for the Hypergeometric Distribution

In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and the Normal distribution functions are shown to decay at a sub-Gaussian rate.

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arXiv2005

On optimal spatial subsample size for variance estimation

We consider the problem of determining the optimal block (or subsample) size for a spatial subsampling method for spatial processes observed on regular grids. We derive expansions for the mean square error of the subsampling variance estimator, which yields an expression for the theoretically optimal block size. The optimal block size is shown to depend in an intricate way on the geometry of the spatial sampling region as well as characteristics of the underlying random field. Final expressions for the optimal block size make use of some nontrivial estimates of lattice point counts in shifts of convex sets. Optimal block sizes are computed for sampling regions of a number of commonly encountered shapes. Numerical studies are performed to compare subsampling methods as well as procedures for estimating the theoretically best block size.

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