GRID for model structure discovering in high dimensional regression

Given a nonparametric regression model, we assume that the number of covariates d → ∞ but only some of these covariates are relevant for the model. Our goal is to identify the relevant covariates and to obtain some information about the structure of the model. We propose a new nonparametric procedure, called GRID, having the following features: (a) it automatically identifies the relevant covariates of the regression model, also distinguishing the nonlinear from the linear ones (a covariate is defined linear/nonlinear depending on the marginal relation between the response variable and such a covariate); (b) the interactions between the covariates (mixed effect terms) are automatically identified, without the necessity of considering some kind of stepwise selection method. In particular, our procedure can identify the mixed terms of any order (two way, three way, ...) without increasing the computational complexity of the algorithm; (c) it is completely data-driven, so being easily implementable for the analysis of real datasets. In particular, it does not depend on the selection of crucial regularization parameters, nor it requires the estimation of the nuisance parameter 2 (self scaling). The acronym GRID has a twofold meaning: first, it derives from Gradient Relevant Identification Derivatives, meaning that the procedure is based on testing the significance of a partial derivative estimator; second, it refers to a graphical tool which can help in representing the identified structure of the regression model. The properties of the GRID procedure are investigated theoretically

On Inconsistency of the Jackknife-after-Bootstrap Bias Estimator for Dependent Data,

B. Efron introducedjackknife-after-bootstrapas a computationally efficient method for estimating standard errors of bootstrap estimators. In a recent paper consistency of the jackknife-after-bootstrap variance estimators has been established for different bootstrap quantities for independent and dependent data. In this paper, it is shown that in the dependent case, the standard jackknife-after-bootstrap estimator for the bias of block bootstrap quantities is inconsistent for almost any sensible choice of the blocking parameters. Some alternative bias estimators are proposed and shown to be consistent.jackknife block bootstrap consistency weak dependence

On Edgeworth Expansion and Moving Block Bootstrap for StudentizedM-Estimators in Multiple Linear Regression Models

This paper considers the multiple linear regression modelYi=xi'[beta]+[var epsilon]i,i=i, ..., n, wherexi's are knownp-1 vectors,[beta]is ap-1 vector of parameters, and[var epsilon]1,[var epsilon]2, ... are stationary, strongly mixing random variables. Let[beta]ndenote anM-estimator of[beta]corresponding to some score function[psi]. Under some conditions on[psi],xi's and[var epsilon]i's, a two-term Edgeworth expansion for Studentized multivariateM-estimator is proved. Furthermore, it is shown that the moving block bootstrap is second-order correct for some suitable bootstrap analog of Studentized[beta]n.Edgeworth expansion moving block bootstrap M-estimators multiple linear regression stationarity strong mixing Studentization (null)

Second order optimality of stationary bootstrap

This paper proves the second order correctness of the stationary bootstrap procedure for normalized, multivariate sample mean of weakly dependent observations. Similar results are shown to hold also for more general vector valued statistics based on sample means.Bootstrap Edgeworth expansion stationarity weak dependence normalized sample mean

Bootstrapping weighted empirical processes that do not converge weakly

We show that the bootstrap method provides valid approximations to the sampling distribution of a weighted empirical process on D[0,1] even in the cases where it fails to converge weakly. Furthermore, the result is applied to construct valid bootstrap confidence sets in such pathological cases.Weighted empirical process Bootstrap Weak convergence Confidence sets

In silico approach of receptor-ligand binding and interaction: Established phytoligands from Tagetes errecta Linn. against bacterial β-glucosidase receptor

The medicinal plant, Tagetes errecta Linn. is a common ornamental plant and leaves of this plant are containing phytochemicals (volatile oil) that inhibit the growth of bacteria, fungi and known natural antimicrobial agents. The objective of the present study was to detect receptor-ligand binding energy and interaction through molecular docking for phytoligands established in the leaves of T. errecta against β-glucosidase receptor (PDB ID: 3AHZ). Molecular docking was performed by using PyRx (Version 0.8) for the structure-based virtual screening and visualized the interaction in the molecular graphic laboratory (MGL) tool (Version 1.5.6). Among 25 phytochemicals and 2 synthetic compounds (Carbendazim and 2-Amino-2-hydroxymethyl-propane-1,3-diol), binding energy value was obtained highest in Bicyclogermacrene (-6.4 Kcal/mol) and lowest in Octanol (-4.4 Kcal/mol) and Carbendazim and 2-Amino-2-hydroxymethyl-propane-1,3-diol showed -6.7 Kcal/mol and -3.5 Kcal/mol all of these showed no hydrogen bonding. The binding interaction of target protein with this phytocompound found binding at the mouth of the active site may be treated as competitive inhibitor. In conclusion, phytocompound Bicyclogermacrene can be alternative of synthetic fungicide as per binding energy value and interaction. It is suggesting further pharmacological and toxicological assay with this phytocompound after isolation from ornamental plant (T. errecta)

GRID: A Variable selection and structure discovery method for high dimensional nonparametric regression

We consider nonparametric regression in high dimensions where only a relatively small subset of a large number of variables are relevant and may have nonlinear effects on the response. We develop methods for variable selection, structure discovery and estimation of the true low-dimensional regression function, allowing any degree of interactions among the relevant variables that need not be specified a-priori. The proposed method, called the GRID, combines empirical likelihood based marginal testing with the local linear estimation machinery in a novel way to select the relevant variables. Further, it provides a simple graphical tool for identifying the low dimensional nonlinear structure of the regression function. Theoretical results establish consistency of variable selection and structure discovery, and also Oracle risk property of the GRID estimator of the regression function, allowing the dimension d of the covariates to grow with the sample size n at the rate $d = O(n^a)$ for any $ain (0;1)$ and the number of relevant covariates r to grow at a rate $r = O(n^gamma)$ for some $gammain (0; 1)$ under some regularity conditions that, in particular, require finiteness of certain absolute moments of the error variables depending on $a$. Finite sample properties of the GRID are investigated in a moderately large simulation study