arXiv subjects

Mayukh Choudhury

Publications and source records attributed to Mayukh Choudhury.

4 recordsLinked to original sources

arXiv2026

Asymptotic Theory of Tail Dependence and Bootstrap for Checkerboard Copulas

A comprehensive asymptotic and bootstrap theory is established for checkerboard-based estimation of the copula and its lower and upper tail copula counterparts under unknown marginal distributions. The proposed estimator of the tail copula extends a local bilinear interpolation of the empirical copula to the tail region, providing a flexible nonparametric approach for modeling extremal dependence. Almost sure uniform consistency is established under mild conditions on the checkerboard grid. Weak convergence of the checkerboard copula process is derived, showing that smoothing preserves the first-order asymptotic limit of the empirical copula process, including the effect of marginal estimation. These results are further extended to lower and upper tail copula processes, yielding asymptotic normality for tail dependence measures. Since the limiting processes depend on unknown characteristics of the underlying true copula, a multiplier bootstrap procedure adapted to the checkerboard structure is proposed and shown to be asymptotically valid. Simulation studies and statistical applications validate our theoretical findings under a range of dependence structures. Although the limiting processes match with that observed for the empirical copula, the finite sample performance shows a noticeable improvement under checkerboard smoothing.

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arXiv2026

Asymptotic Theory of $K$-fold Cross-validation in Lasso and the validity of Bootstrap

Least absolute shrinkage and selection operator or Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). However, inferential properties, such as the variable selection consistency and $n^{1/2}-$consistency, of the $K-$fold CV based Lasso estimator and validity of the Bootstrap approximation are still unknown. In this paper, we consider the heteroscedastic linear regression model and show only under some moment type conditions that the Lasso estimator with $K$-fold CV based penalty is $n^{1/2}-$consistent, but not variable selection consistent. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. Therefore, our results theoretically justify the use of $K-$fold CV based Lasso estimator to perform statistical inference in linear regression. We validate our Bootstrap method for the $K-$fold CV based Lasso estimator in finite samples based on simulations. We also implement our Bootstrap based inference on a real data set.

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arXiv2026

High Dimensional Gaussian and Bootstrap Approximations in Generalized Linear Models

Generalized Linear Model (or GLM) extends the ordinary linear regression by linking the mean of the response variable to covariates through appropriate link functions. GLM is widely used in the analysis of datasets arising from diverse fields including medical sciences, clinical trials, population surveys and risk analysis. In this paper, we investigate the Gaussian and Bootstrap approximations of GLM under two separate high dimensional regimes: (I) when the dimension $d$ grows slower than $n$ and (II) when $d$ grows exponentially with $n$. Under regime (I), we essentially show that the Gaussian approximation holds over the collection of Borel convex sets when $d = o\big(n^{2/5}\big)$ and over the collection of Euclidean balls when $d = o\big(n^{1/2}\big)$. We further devise two high dimensional Bootstrap methods which are valid over the collections of Borel convex sets and Euclidean balls under the same dimension growth rates. Then we move to regime (II) where we invoke sparsity to GLM through Lasso. We show that the high dimensional Gaussian approximation fails under regime (II). However, the Bootstrap approximations over convex sets and Euclidean balls are valid for the relevant part of the GLM estimator provided $\log d = o\big(n^{2τ/3}\big)$ and the number of non-zero regression parameters is $o\big(n^{1/3- 4τ/3}\big)$, when the Lasso penalty $λ_n \sim n^{1/2 + τ}$, for some $τ\in (0, 1/4)$. Simulation studies confirm the strong finite-sample performance of our proposed Bootstrap methods under both regime (I) and (II). We also implement our methods on real datasets.

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arXiv2026

Bootstrapping Lasso in Generalized Linear Models

Generalized linear model or GLM constitutes a large class of models and essentially extends the ordinary linear regression by connecting the mean of the response variable with the covariate through appropriate link functions. On the other hand, Lasso is a popular and easy-to-implement penalization method in regression when not all covariates are relevant. However, the asymptotic distributional properties the Lasso estimator in GLM is still unknown. In this paper, we show that the Lasso estimator in GLM does not have a tractable form and subsequently, we develop two Bootstrap methods, namely the Perturbation Bootstrap and Pearson's Residual Bootstrap methods, for approximating the distribution of the Lasso estimator in GLM. As a result, our Bootstrap methods can be used to draw valid statistical inferences for any sub-model of GLM. We support our theoretical findings by showing good finite-sample properties of the proposed Bootstrap methods through a moderately large simulation study. We also implement one of our Bootstrap methods on a real data set.

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