G. Jogesh Babu

Application of Bernstein Polynomials for smooth estimation of a distribution and density function

The empirical distribution function is known to have optimum properties as an estimator of the underlying distribution function. However, it may not be appropriate for estimating continuous distributions because of its jump discontinuities. In this paper, we consider the application of Bernstein polynomials for approximating a bounded and continuous function and show that it can be naturally adapted for smooth estimation of a distribution function concentrated on the interval [0,1] by a continuous approximation of the empirical distribution function. The smoothness of the approximating polynomial is further used in deriving a smooth estimator of the corresponding density. The asymptotic properties of the resulting estimators are investigated. Specifically, we obtain strong consistency and asymptotic normality under appropriate choice of the degree of the polynomial. The case of distributions with other compact and non-compact support can be dealt through transformations. Thus, this paper gives a general method for non-parametric density estimation as an alternative to the current estimators. A small numerical investigation shows that the estimator proposed here may be preferable to the popular kernel-density estimator.

Joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population

The joint asymptotic distributions of the marginal quantiles and quantile functions in samples from a p-variate population are derived. Of particular interest is the joint asymptotic distribution of the marginal sample medians, on the basis of which tests of significance for population medians are developed. Methods of estimating unknown nuisance parameters are discussed. The approach is completely nonparametric.

The Astrophysical Multimessenger Observatory Network (AMON)

We summarize the science opportunity, design elements, current and projected partner observatories, and anticipated science returns of the Astrophysical Multimessenger Observatory Network (AMON). AMON will link multiple current and future high-energy, multimessenger, and follow-up observatories together into a single network, enabling near real-time coincidence searches for multimessenger astrophysical transients and their electromagnetic counterparts. Candidate and high-confidence multimessenger transient events will be identified, characterized, and distributed as AMON alerts within the network and to interested external observers, leading to follow-up observations across the electromagnetic spectrum. In this way, AMON aims to evoke the discovery of multimessenger transients from within observatory subthreshold data streams and facilitate the exploitation of these transients for purposes of astronomy and fundamental physics. As a central hub of global multimessenger science, AMON will also enable cross-collaboration analyses of archival datasets in search of rare or exotic astrophysical phenomena.

Horizontal Branch Morphology of Globular Clusters: A Multivariate Statistical Analysis

The proper interpretation of horizontal branch (HB) morphology is crucial to the understanding of the formation history of stellar populations. In the present study a multivariate analysis is used (principal component analysis) for the selection of appropriate HB morphology parameter, which, in our case, is the logarithm of effective temperature extent of the HB (log Teff HB). Then this parameter is expressed in terms of the most significant observed independent parameters of Galactic globular clusters (GGCs) separately for coherent groups, obtained in a previous work, through a stepwise multiple regression technique. It is found that, metallicity ([Fe/H]), central surface brightness (μv), and core radius (rc) are the significant parameters to explain most of the variations in HB morphology (multiple R 2 ∼ 0.86) for GGC elonging to the bulge/disk while metallicity ([Fe/H]) and absolute magnitude (Mv) are responsible for GGC belonging to the inner halo (multiple R 2 ∼ 0.52). The robustness is tested by taking 1000 bootstrap samples. A cluster analysis is performed for the red giant branch (RGB) stars of the GGC belonging to Galactic inner halo (Cluster 2). A multi-episodic star formation is preferred for RGB stars of GGC belonging to this group. It supports the asymptotic giant branch (AGB) model in three episodes instead of two as suggested by Carretta et al. for halo GGC while AGB model is suggested to be revisited for bulge/disk GGC.

Data skeletons: simultaneous estimation of multiple quantiles for massive streaming datasets with applications to density estimation

Abstract We consider the problem of density estimation when the data is in the form of a continuous stream with no fixed length. In this setting, implementations of the usual methods of density estimation such as kernel density estimation are problematic. We propose a method of density estimation for massive datasets that is based upon taking the derivative of a smooth curve that has been fit through a set of quantile estimates. To achieve this, a low-storage, single-pass, sequential method is proposed for simultaneous estimation of multiple quantiles for massive datasets that form the basis of this method of density estimation. For comparison, we also consider a sequential kernel density estimator. The proposed methods are shown through simulation study to perform well and to have several distinct advantages over existing methods.

Resampling Methods for Model Fitting and Model Selection

Resampling procedures for fitting models and model selection are considered in this article. Nonparametric goodness-of-fit statistics are generally based on the empirical distribution function. The distribution-free property of these statistics does not hold in the multivariate case or when some of the parameters are estimated. Bootstrap methods to estimate the underlying distributions are discussed in such cases. The results hold not only in the case of one-dimensional parameter space, but also for the vector parameters. Bootstrap methods for inference, when the data is from an unknown distribution that may or may not belong to a specified family of distributions, are also considered. Most of the information criteria-based model selection procedures such as the Akaike information criterion, Bayesian information criterion, and minimum description length use estimation of bias. The bias, which is inevitable in model selection problems, arises mainly from estimating the distance between the “true” model and an estimated model. A jackknife type procedure for model selection is discussed, which instead of bias estimation is based on bias reduction.

Confidence limits to the distance of the true distribution from a misspecified family by bootstrap

In statistical practice, an estimated distribution function (d.f.) from a specified family is used for taking decisions. When the true d.f. from which samples are drawn does not belong to the specified family, it is of interest to know how close the true d.f. is to the specified family. In this paper, we use non-parametric bootstrap to obtain confidence limits to the difference between the true d.f. and a member of the specified family closest to it in the sense of Kullback-Leibler measure.

A NEW STATISTICAL METHOD FOR FILTERING AND ENTROPY ESTIMATION OF A CHAOTIC MAP FROM NOISY DATA

We consider a discrete time deterministic chaotic dynamical system, xn+1=τ(xn), where τ is a nonlinear map of the unit interval into itself. We assume that τ is piecewise expanding and piecewise C2. The effects of noise contamination are modeled by xn+1=τ(xn)+ξn, where ξn is an independent random variable with small noise amplitude. A new statistical method is presented for filtering τ and estimating the metric entropy of τ from observed noisy data.

Nonparametric Estimation of Survival Functions under Dependent Competing Risks

In a certain target population, the individuals will die due to either Cause 1 or Cause 2 with probabilities π and (1-π), respectively. Let F1 and F2 be the life time distributions of individuals who die off due to Causes 1 and 2, respectively. In any random sample of individuals from the population, subjects can leave the study at random times. In this paper, we derive nonparametric estimates of π, F1 and F2 using such censored data and study some of their properties. The model that suggests itself encapsulating the essentials of the problem is more general than the usual competing risks model.

VOStat: A Statistical Web Service for Astronomers

VOStat is a Web service providing interactive statistical analysis of astronomical tabular datasets. It is integrated into the suite of analysis and visualization tools associated with the international Virtual Observatory (VO) through the SAMP communication system. A user supplies VOStat with a dataset extracted from the VO, or otherwise acquired, and chooses from among ∼60 statistical functions. These include data transformations, plots and summaries, density estimation, one- and two-sample hypothesis tests, global and local regressions, multivariate analysis and clustering, spatial analysis, directional statistics, survival analysis (for censored data like upper limits), and time series analysis. The statistical operations are performed using the public domain R statistical software environment, including a small fraction of its >4000 CRAN add-on packages. The purpose of VOStat is to facilitate a wider range of statistical analyses than are commonly used in astronomy, and to promote use of more advanced methodology in R and CRAN. Online material: color figures