arXiv subjects

Rahul Mukerjee

Publications and source records attributed to Rahul Mukerjee.

At least 19 recordsLinked to original sources

arXiv2026

Efficient partially replicated block designs with each replication number one or two

We investigate block designs, under the A- and MV-criteria, when each treatment can have only one or two replications due to resource constraints, as can happen, for example, in early generation varietal trials. While these are commonly known as partially replicated designs, a key new feature of the present work is that no restriction about a constant block size is imposed on the subdesign consisting of the twice replicated treatments. This makes the derivation more challenging but allows us to entertain a wider class of competing designs and hence increases the flexibility of the results. Considering all treatments as equally important, design-independent, sharp lower bounds on the A- and MV-criteria are derived, so as to find highly efficient designs over this wider class. The roles of (a) linked block designs, (b) designs in an online catalog designtheory.org, and (c) partially balanced incomplete block designs, or duals thereof, as adapted to our setup, are explored at length. Illustrative examples are presented.

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arXiv2023

Efficient augmented block designs for unreplicated test treatments

Augmented block designs for unreplicated test treatments are investigated under the A- and MV-criteria with respect to control versus control, test versus test and control versus test comparisons. We derive design-independent lower bounds on these criteria over a wide class of competing designs. These bounds are useful benchmarks and the resulting expressions for efficiencies enable objective assessment of any given design under the A- and MV-criteria. It is seen that the use of BIB or PBIB designs or duals thereof often leads to very high efficiencies, which compare extremely well with algorithmic constructions. Our findings also cover the case of partial replication of some test treatments. Illustrative examples, including large-scale ones, are presented.

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arXiv2022

A Statistical Approach to Broken Stick Problems

Let a stick be broken at random at n-1 points to form n pieces. We consider three problems on forming k-gons with k out of these n pieces, and show how a statistical approach, through a linear transformation of variables, yields simple solutions that also allow fast computation.

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arXiv2022

Variance Analysis of Multiple Importance Sampling Schemes

Multiple importance sampling (MIS) is an increasingly used methodology where several proposal densities are used to approximate integrals, generally involving target probability density functions. The use of several proposals allows for a large variety of sampling and weighting schemes. Then, the practitioner must choose a given scheme, i.e., sampling mechanism and weighting function. A variance analysis has been proposed in Elvira et al (2019, Statistical Science 34, 129-155), showing the superiority of the balanced heuristic estimator with respect to other competing schemes in some scenarios. However, some of their results are valid only for two proposals. In this paper, we extend and generalize these results, providing novel proofs that allow to determine the variance relations among MIS schemes.

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arXiv2021

Improving upon the effective sample size based on Godambe information for block likelihood inference

We consider the effective sample size, based on Godambe information, for block likelihood inference which is an attractive and computationally feasible alternative to full likelihood inference for large correlated datasets. With reference to a Gaussian random field having a constant mean, we explore how the choice of blocks impacts this effective sample size. It is seen that spreading out the spatial points within each block, instead of keeping them close together, can lead to considerable gains while retaining computational simplicity. Analytical results in this direction are obtained under the AR(1) model. The insights so found facilitate the study of other models, including correlation models on a plane, where closed form expressions are intractable.

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arXiv2019

Causal Inference from Possibly Unbalanced Split-Plot Designs: A Randomization-based Perspective

Split-plot designs find wide applicability in multifactor experiments with randomization restrictions. Practical considerations often warrant the use of unbalanced designs. This paper investigates randomization based causal inference in split-plot designs that are possibly unbalanced. Extension of ideas from the recently studied balanced case yields an expression for the sampling variance of a treatment contrast estimator as well as a conservative estimator of the sampling variance. However, the bias of this variance estimator does not vanish even when the treatment effects are strictly additive. A careful and involved matrix analysis is employed to overcome this difficulty, resulting in a new variance estimator, which becomes unbiased under milder conditions. A construction procedure that generates such an estimator with minimax bias is proposed.

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arXiv2018

Causal Inference from Strip-Plot Designs in a Potential Outcomes Framework

Strip-plot designs are very useful when the treatments have a factorial structure and the factors levels are hard-to-change. We develop a randomization-based theory of causal inference from such designs in a potential outcomes framework. For any treatment contrast, an unbiased estimator is proposed, an expression for its sampling variance is worked out, and a conservative estimator of the sampling variance is obtained. This conservative estimator has a nonnegative bias, and becomes unbiased under between-block additivity, a condition milder than Neymannian strict additivity. A minimaxity property of this variance estimator is also established. Simulation results on the coverage of resulting confidence intervals lend support to theoretical considerations.

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arXiv2018

Design of Order-of-Addition Experiments

In an order-of-addition experiment, each treatment is a permutation of m components. It is often unaffordable to test all the m! treatments, and the design problem arises. We consider a model that incorporates the order of each pair of components and can also account for the distance between the two components in every such pair. Under this model, the optimality of the uniform design measure is established, via the approximate theory, for a broad range of criteria. Coupled with an eigen-analysis, this result serves as a benchmark that paves the way for assessing the efficiency and robustness of any exact design. The closed-form construction of a class of robust optimal fractional designs is then explored and illustrated.

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arXiv2016

Causal Inference in Rebuilding and Extending the Recondite Bridge between Finite Population Sampling and Experimental Design

This article considers causal inference for treatment contrasts from a randomized experiment using potential outcomes in a finite population setting. Adopting a Neymanian repeated sampling approach that integrates such causal inference with finite population survey sampling, an inferential framework is developed for general mechanisms of assigning experimental units to multiple treatments. This framework extends classical methods by allowing the possibility of randomization restrictions and unequal replications. Novel conditions that are "milder" than strict additivity of treatment effects, yet permit unbiased estimation of the finite population sampling variance of any treatment contrast estimator, are derived. The consequences of departures from such conditions are also studied under the criterion of minimax bias, and a new justification for using the Neymanian conservative sampling variance estimator in experiments is provided. The proposed approach can readily be extended to the case of treatments with a general factorial structure.

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arXiv2014

Efficient Designs in Small Blocks for Comparing Consecutive Pairs of Treatments

Optimal block designs in small blocks are explored when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first develop an approximate theory which leads to a convenient multiplicative algorithm for obtaining optimal design measures. This, in turn, yields highly efficient exact designs even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available possibly in several stages. Illustrative examples are given. Tables of optimal design measures are also provided.

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arXiv2014

Approximate theory-aided robust efficient factorial fractions under baseline parametrization

With reference to a baseline parametrization, we explore highly efficient fractional factorial designs for inference on the main effects and, perhaps, some interactions. Our tools include approximate theory together with certain carefully devised discretization procedures. The robustness of these designs to possible model misspecification is investigated using a minimaxity approach. Examples are given to demonstrate that our technique works well even when the run size is quite small.

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arXiv2014

Optimal Design Measures under Asymmetric Errors, with Application to Binary Design Points

We study the optimal design problem under second-order least squares estimation which is known to outperform ordinary least squares estimation when the error distribution is asymmetric. First, a general approximate theory is developed, taking due cognizance of the nonlinearity of the underlying information matrix in the design measure. This yields necessary and sufficient conditions that a D- or A-optimal design measure must satisfy. The results are then applied to find optimal design measures when the design points are binary. The issue of reducing the support size of the optimal design measure is also addressed.

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arXiv2013

A complementary set theory for quaternary code designs

Quaternary code (QC) designs form an attractive class of nonregular factorial fractions. We develop a complementary set theory for characterizing optimal QC designs that are highly fractionated in the sense of accommodating a large number of factors. This is in contrast to existing theoretical results which work only for a relatively small number of factors. While the use of imaginary numbers to represent the Gray map associated with QC designs facilitates the derivation, establishing a link with foldovers of regular fractions helps in presenting our results in a neat form.

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arXiv2013

Variance and Covariance Inequalities for Truncated Joint Normal Distribution via Monotone Likelihood Ratio and Log-concavity

Let X Nv(0, Λ) be a normal vector in v dimensions, where Λ is diagonal. With reference to the truncated distribution of X on the interior of a v-dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by F. Palombi and S. Toti [J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure the convergence of an algorithm for the reconstruction of Λ only on the basis of the covariance matrix of X truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove and utilize the fact that the cumulative distribution function of any positive linear combination of independent chi-square variates is log-concave, even though the same may not be true for the corresponding density function.

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arXiv2012

Highly Efficient Factorial Designs for cDNA Microarray Experiments: Use of Approximate Theory Together with a Step-up Step-down Procedure

A general method for obtaining highly efficient factorial designs of relatively small sizes is developed for cDNA microarray experiments. The method allows the main effects and interactions of successive orders to be of possibly unequal importance. First, the approximate theory is em-ployed to get an optimal design measure which is then discretized. It is, however, observed that a naïve discretization may fail to yield an exact design of the stipulated size and, even when it yields such an exact design, there is often scope for improvement in efficiency. To address these issues, we propose a step-up/down procedure which is seen to work very well. The resulting highly efficient designs are found to remain almost free from possible dye-color effects under a suitable dye-color assignment. They are also seen to be quite robust to heteroscedasticity as may be caused by biological variability. We focus on the baseline and all-to-next parametrizations but our method works equally well also for hybrids of the two and other parametrizations.

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arXiv2011

Key Predistribution Schemes for Distributed Sensor Networks

Key predistribution schemes for distributed sensor networks have received significant attention in the recent literature. In this paper we propose a new construction method for these schemes based on combinations of duals of standard block designs. Our method is a broad spectrum one which works for any intersection threshold. By varying the initial designs, we can generate various schemes and this makes the method quite flexible. We also obtain explicit algebraic expressions for the metrics for local connectivity and resiliency. These schemes are quite efficient with regard to connectivity and resiliency and at the same time they allow a straightforward shared-key discovery.

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arXiv2011

Improving Anonymity in Shared Key Primitives Based on Perfect Hash Families

We propose a new scheme for sharing symmetric key operations among a set of participants according to a (t,n) threshold access structure. We focus on anonymity properties of this scheme and show that this scheme provides improved values of anonymity measures than the existing ones. In particular, the scheme can provide optimal and equitable participant anonymity when it is based on balanced perfect hash families.

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arXiv2011

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.

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