Anomalous effective action, Noether current, Virasoro algebra and Horizon entropy

Several investigations show that in a very small length scale there exists corrections to the entropy of black hole horizon. Due to fluctuations of the background metric and the external fields the action incorporates corrections. In the low energy regime, the one loop effective action in four dimensions leads to trace anomaly. We start from the Noether current corresponding to the Einstein-Hilbert plus the one loop effective action to calculate the charge for the diffeomorphisms which preserve the Killing horizon structure. Then a bracket among the charges is calculated. We show that the Fourier modes of the bracket is exactly similar to Virasoro algebra. Then using Cardy formula the entropy is evaluated. Finally, the explicit terms of the entropy expression is calculated for a classical background. It turns out that the usual expression for entropy; i.e. the Bekenstein-Hawking form, is not modified.Comment: Minor modifications, to appear in EPJ

A Study of Non-regularity in Dynamic Treatment Regimes and Some Design Considerations for Multicomponent Interventions.

This dissertation investigates two methodological problems. The first problem concerns developing and optimizing multicomponent interventions. The traditional approach to this problem is to conduct a two-group randomized trial of a "likely best" intervention vs. control, followed by observational analyses. In this approach, all inferences about individual components and their interactions are typically based on observational analyses, and hence are subject to confounding bias. An emerging approach called the Multiphase Optimization Strategy (MOST) addresses the above problem by including two evidentiary phases of randomized experiments to precede and inform a confirmatory two-group randomized trial. Full and fractional factorial designs are useful tools in this approach. However there exists a lot of criticism in the clinical and behavioral intervention trials literature regarding their use. In this dissertation, we address these criticisms in the context of the MOST framework. Furthermore, we provide an operationalization of the screening phase of MOST using fractional factorial designs. Also to strengthen the case for MOST as the "gold standard" for designing multicomponent intervention trials, we provide an illustrative simulation study comparing MOST with the traditional approach. The second problem investigated in this dissertation is that of non-regularity that arises in the estimation of the optimal dynamic treatment regimes (DTR). DTRs are multistage, individualized treatment rules that are useful for treating chronic disorders. In the estimation of the optimal DTRs, the treatment effect parameters at any stage prior to the last can be non-regular under certain distributions of the data. This results in biased estimates and invalid confidence intervals for the treatment effect parameters. To address the problem of non-regularity, we propose a shrinkage estimator called the soft-threshold estimator. We derive this as an empirical Bayes estimator under a hierarchical Bayesian model. We also provide an extensive simulation study to compare the soft-threshold estimator with other available estimators that attempt to address non-regularity. Analysis of data from a smoking cessation trial is provided as an illustration.Ph.D.StatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64656/1/bibhas_1.pd

Incorporating Participants' Welfare into Sequential Multiple Assignment Randomized Trials

Dynamic treatment regimes (DTRs) are sequences of decision rules that recommend treatments based on patients' time-varying clinical conditions. The sequential multiple assignment randomized trial (SMART) is an experimental design that can provide high-quality evidence for constructing optimal DTRs. In a conventional SMART, participants are randomized to available treatments at multiple stages with balanced randomization probabilities. Despite its relative simplicity of implementation and desirable performance in comparing embedded DTRs, the conventional SMART faces inevitable ethical issues including assigning many participants to the empirically inferior treatment or the treatment they dislike, which might slow down the recruitment procedure and lead to higher attrition rates, ultimately leading to poor internal and external validities of the trial results. In this context, we propose a SMART under the Experiment-as-Market framework (SMART-EXAM), a novel SMART design that holds the potential to improve participants' welfare by incorporating their preferences and predicted treatment effects into the randomization procedure. We describe the steps of conducting a SMART-EXAM and evaluate its performance compared to the conventional SMART. The results indicate that the SMART-EXAM can improve the welfare of the participants enrolled in the trial, while also achieving a desirable ability to construct an optimal DTR when the experimental parameters are suitably specified. We finally illustrate the practical potential of the SMART-EXAM design using data from a SMART for children with attention-deficit/hyperactivity disorder (ADHD)

Incorporating nonparametric methods for estimating causal excursion effects in mobile health with zero-inflated count outcomes

In the domain of mobile health, tailoring interventions for real-time delivery is of paramount importance. Micro-randomized trials have emerged as the "gold-standard" methodology for developing such interventions. Analyzing data from these trials provides insights into the efficacy of interventions and the potential moderation by specific covariates. The "causal excursion effect", a novel class of causal estimand, addresses these inquiries, backed by current semiparametric inference techniques. Yet, existing methods mainly focus on continuous or binary data, leaving count data largely unexplored. The current work is motivated by the Drink Less micro-randomized trial from the UK, which focuses on a zero-inflated proximal outcome, the number of screen views in the subsequent hour following the intervention decision point. In the current paper, we revisit the concept of causal excursion effects, specifically for zero-inflated count outcomes, and introduce novel estimation approaches that incorporate nonparametric techniques. Bidirectional asymptotics are derived for the proposed estimators. Through extensive simulation studies, we evaluate the performance of the proposed estimators. As an illustration, we also employ the proposed methods to the Drink Less trial data.Comment: 37pages,2 figure