On a Theorem of Grzegorek and Labuda
This paper presents a generalized version of a theorem of Grzegorek and Labuda in category bases and also endeavours to establish a variant formulation of the same in Marczewski structures.
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arXiv subjects
Sanjib Basu
Publications and source records attributed to Sanjib Basu.
Addressing Positivity Violations in Extending Inference to a Target Population
Enhancing the external validity of trial results is essential for their applicability to real-world populations. However, violations of the positivity assumption can limit both the generalizability and transportability of findings. To address positivity violations in estimating the average treatment effect for a target population, we propose a framework that integrates characterizing the underrepresented group and performing sensitivity analysis for inference in the original target population. Our approach helps identify limitations in trial sampling and improves the robustness of trial findings for real-world populations. We apply this approach to extend findings from phase IV trials of treatments for opioid use disorder to a real-world population based on the 2021 Treatment Episode Data Set.
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On non-Baire rare sets in category bases
In this paper, we deal with non-Baire rare sets in category bases which forms $\aleph_0$-independent family, where a rare set is a common generalization of both Luzin and Sierpinski set.
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A Kuratowski theorem revisited
If $f : X\mapsto Y$ is a function having Baire property from a metric space $X$ into a separable metric space $Y$ , then $f$ is continuous except on a set of first category. Kuratowski asked whether the condition of separability could be removed. Several attempts were done in the past to solve this problem. In fact, the first impressive attempt was initiated by Kunugi. This paper is aimed towards solving the problem of Kuratowski in category bases which generalizes the result of Kunugi.
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Sum of two rare sets in a category base can be absolutely non-baire
In this paper, we give generalized version in category bases of a result of Kharazishvili dealing with absolute nonmeasurability of the Minkowski sum of certain universal measure zero sets which were based on an earlier result of Erdos, Kunen and Mauldin in the real line.
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Restricted Mean Survival Time Estimation Using Bayesian Nonparametric Dependent Mixture Models
Restricted mean survival time (RMST) is an intuitive summary statistic for time-to-event random variables, and can be used for measuring treatment effects. Compared to hazard ratio, its estimation procedure is robust against the non-proportional hazards assumption. We propose nonparametric Bayeisan (BNP) estimators for RMST using a dependent stick-breaking process prior mixture model that adjusts for mixed-type covariates. The proposed Bayesian estimators can yield both group-level causal estimate and subject-level predictions. Besides, we propose a novel dependent stick-breaking process prior that on average results in narrower credible intervals while maintaining similar coverage probability compared to a dependent probit stick-breaking process prior. We conduct simulation studies to investigate the performance of the proposed BNP RMST estimators compared to existing frequentist approaches and under different Bayesian modeling choices. The proposed framework is applied to estimate the treatment effect of an immuno therapy among KRAS wild-type colorectal cancer patients.
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On topological sigma-ideals
The concept of $\mathcal S$-topological $σ$-ideal in measurable space $(X, \mathcal S)$ was introduced by Hejduk and using a theorem of Wagner on convergence of measurable functions characterized $\mathcal S$-topological $σ$-ideals. In this paper, we give a general construction of $\mathcal S$-topological $σ$-ideals from structures induced by $σ$-algebras and weakly upper semicontinuous $ω$-small systems. We also show that instead of weak upper semicontinuity, if we use upper semicontinuity, we get $\mathcal S$-uniformizable $σ$-ideals. This generalizes the approach of Wagner and Wilczynski metrizing Boolean Lattice of measurable functions
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Bayesian Arc Length Survival Analysis Model (BALSAM): Theory and Application to an HIV/AIDS Clinical Trial
Stochastic volatility often implies increasing risks that are difficult to capture given the dynamic nature of real-world applications. We propose using arc length, a mathematical concept, to quantify cumulative variations (the total variability over time) to more fully characterize stochastic volatility. The hazard rate, as defined by the Cox proportional hazards model in survival analysis, is assumed to be impacted by the instantaneous value of a longitudinal variable. However, when cumulative variations pose a significant impact on the hazard, this assumption is questionable. Our proposed Bayesian Arc Length Survival Analysis Model (BALSAM) infuses arc length into a united statistical framework by synthesizing three parallel components (joint models, distributed lag models, and arc length). We illustrate the use of BALSAM in simulation studies and also apply it to an HIV/AIDS clinical trial to assess the impact of cumulative variations of CD4 count (a critical longitudinal biomarker) on mortality while accounting for measurement errors and relevant variables.
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A note on sets avoiding rational distances in category bases
Michalski gave a short and elegant proof of a theorem of A. Kumar which states that for each set A in R, there exists a subset B of A which is full in A and such that no distance between points in B is a rational number. He also proved a similar theorem for sets in the Euclidean plane. In this paper, we generalize these results in some special types of category bases.
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Generalization of two theorems of Steinhaus in category bases
Here we unify two results of Steinhaus and their corresponding category analogues by extending them in the settings of category bases. We further show that in any perfect translation base, every abundant Baire set contains a full subset for which our second theorem fails.
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Bayesian Functional Data Analysis over Dependent Regions and Its Application for Identification of Differentially Methylated Regions
We consider a Bayesian functional data analysis for observations measured as extremely long sequences. Splitting the sequence into a number of small windows with manageable length, the windows may not be independent especially when they are neighboring to each other. We propose to utilize Bayesian smoothing splines to estimate individual functional patterns within each window and to establish transition models for parameters involved in each window to address the dependent structure between windows. The functional difference of groups of individuals at each window can be evaluated by Bayes Factor based on Markov Chain Monte Carlo samples in the analysis. In this paper, we examine the proposed method through simulation studies and apply it to identify differentially methylated genetic regions in TCGA lung adenocarcinoma data.
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Category bases that are equivalent to topologies
In view of the fact that many of the most familiar examples of category bases are equivalent to some topology, it is natural to ask whether category bases are always topological in nature. The answer is in the negative. In this paper, we show that under certain circumstances, a category base can be equivalent to a topology. So this work may be considered a continuation of similar type of works done earlier in this area.
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A few remarks on non-Baire sets in category bases
In this paper, we first establish some equivalent formulations of non-Baire sets in category bases. We then introduce the notion of an uniform non-Baire family of sets and show that there is an uniform non-Baire family inducing a decomposition of the whole space. This phenomenon is then interpreted in the context of the famous Banah-Mazur game.
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On completely non-Baire union in category bases
In this paper, we intend to show that under not too restrictive conditions, results much stronger than the one obtained earlier by Hejduk could be established in category bases.
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On non-Baire sets in category bases
In this paper, we prove a result on non-Baire sets in category bases which when applied together with a result of Grzegorek yeilds a comparatively stronger version of a decomposition theorem due to Ulam.
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A Model-free Approach for Testing Association
The question of association between outcome and feature is generally framed in the context of a model on functional and distributional forms. Our motivating application is that of identifying serum biomarkers of angiogenesis, energy metabolism, apoptosis, and inflammation, predictive of recurrence after lung resection in node-negative non-small cell lung cancer patients with tumor stage T2a or less. We propose an omnibus approach for testing association that is free of assumptions on functional forms and distributions and can be used as a black box method. This proposed maximal permutation test is based on the idea of thresholding, is readily implementable and is computationally efficient. We illustrate that the proposed omnibus tests maintain their levels and have strong power as black box tests for detecting linear, nonlinear and quantile-based associations, even with outlier-prone and heavy-tailed error distributions and under nonparametric setting. We additionally illustrate the use of this approach in model-free feature screening and further examine the level and power of these tests for binary outcome. We compare the performance of the proposed omnibus tests with comparator methods in our motivating application to identify preoperative serum biomarkers associated with non-small cell lung cancer recurrence in early stage patients.
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On a theorem of Pelc and Prikry on the nonexistence of invariant extensions of Borel measures
There are certain countably generated sigma-algebras of sets in the real line which do not admit any non-zero, sigma-finite, diffused (or, continuous) measure. Such countably generated sigma-algebras can be obtained by the use of some special types of infinite matrix known as the Banach-Kuratowski matrix and the same may be used in deriving a generalized version of Pelc and Prikry's theorem as shown by Kharazishvili. In this paper, using some methods of combinatorial set theory and some modified version of the notion of small sets originally introduced by Riecan, Riecan and Neubrunn, we give an abstract and generalized formulation of Pelc and Prikry's theorem in spaces with transformation groups.
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A Nonseparable Invariant Extension of Lebesgue Measure -- A Generalized and Abstract Approach
Here using some methods of combinatorial set theory, particularly the ones related to the construction of independent families of sets and some modified version of the notion of small sets originally introduced by Riecan, Riecan and Neubrunn, we give abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby relating to nonseparable extension of Lebesgue measure in spaces with transformation groups
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