arXiv subjects

Bodhisattva Sen

Publications and source records attributed to Bodhisattva Sen.

At least 73 records · Page 4Linked to original sources

arXiv2013

Model Based Bootstrap Methods for Interval Censored Data

We investigate the performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data. We show that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for both the current status and case 2 interval censoring models. A model based smoothed bootstrap procedure is proposed and shown to be consistent. In addition, simulation studies are conducted to illustrate the (in)-consistency of the bootstrap methods. Our conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.

stat.ME↗

arXiv2013

Bootstrapping a Change-Point Cox Model for Survival Data

This paper investigates the (in)-consistency of various bootstrap methods for making inference on a change-point in time in the Cox model with right censored survival data. A criterion is established for the consistency of any bootstrap method. It is shown that the usual nonparametric bootstrap is inconsistent for the maximum partial likelihood estimation of the change-point. A new model-based bootstrap approach is proposed and its consistency established. Simulation studies are carried out to assess the performance of various bootstrap schemes.

stat.ME↗

arXiv2013

A Mass-Dependent Yield Origin of Neutron-Capture Element Abundance Distributions in Ultra-Faint Dwarfs

One way to constrain the nature of the high-redshift progenitors of the Milky Way is to look at the low-metallicity stellar populations of the different Galactic components today. For example, high-resolution spectroscopy of very metal poor (VMP) stars demonstrates remarkable agreement between the distribution of [Ti/Fe] in the stellar populations of the Milky Way halo (MW) and ultra-faint dwarf (UFD) galaxies. In contrast, for the neutron capture (nc) abundance ratio distributions [(Sr,Ba)/Fe], the peak of the small UFD sample (6 stars) exhibits a signicant under-abundance relative to the VMP stars in the larger MW halo sample (~ 300 stars). We present a simple scenario that can simultaneously explain these similarities and differences by assuming: (i) that the MW VMP stars were predominately enriched by a prior generation of stars which possessed a higher total mass than the prior generation of stars that enriched the UFD VMP stars; and (ii) a much stronger mass-dependent yield (MDY) for nc-elements than for the (known) MDY for Ti. Simple statistical tests demonstrate that conditions (i) and (ii) are consistent with the observed abundance distributions, albeit without strong constraints on model parameters. A comparison of the broad constraints for these nc-MDY with those derived in the literature seems to rule out Ba production from low-mass SNs and affirms models that primarily generate yields from high-mass SN. Our scenario can be confirmed by a relatively modest (factor of ~ 3-4) increase in the number of high-resolution spectra of VMP stars in UFDs.

astro-ph.GA↗

arXiv2012

Bootstrap confidence intervals for isotonic estimators in a stereological problem

Let $\mathbf{X}=(X_1,X_2,X_3)$ be a spherically symmetric random vector of which only $(X_1,X_2)$ can be observed. We focus attention on estimating F, the distribution function of the squared radius $Z:=X_1^2+X_2^2+X_3^2$, from a random sample of $(X_1,X_2)$. Such a problem arises in astronomy where $(X_1,X_2,X_3)$ denotes the three dimensional position of a star in a galaxy but we can only observe the projected stellar positions $(X_1,X_2)$. We consider isotonic estimators of F and derive their limit distributions. The results are nonstandard with a rate of convergence $\sqrt{n/{\log n}}$. The isotonized estimators of F have exactly half the limiting variance when compared to naive estimators, which do not incorporate the shape constraint. We consider the problem of constructing point-wise confidence intervals for F, state sufficient conditions for the consistency of a bootstrap procedure, and show that the conditions are met by the conventional bootstrap method (generating samples from the empirical distribution function).

math.ST↗

arXiv2012

Covering Numbers for Convex Functions

In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $ε$-covering number of $\C([a, b]^d, B)$, in the $L_p$-metric, $1 \le p < \infty$, in terms of the relevant constants, where $d \geq 1$, $a < b \in \mathbb{R}$, $B>0$, and $\C([a,b]^d, B)$ denotes the set of all convex functions on $[a, b]^d$ that are uniformly bounded by $B$. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.

cs.IT↗

arXiv2012

Nonparametric Least Squares Estimation of a Multivariate Convex Regression Function

This paper deals with the consistency of the least squares estimator of a convex regression function when the predictor is multidimensional. We characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs. Mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided. We also consider a regression function which is known to be convex and componentwise nonincreasing and discuss the characterization, computation and consistency of its least squares estimator.

math.ST↗

arXiv2011

Uncloaking globular clusters in the inner Galaxy

Extensive photometric studies of the globular clusters located towards the center of the Milky Way have been historically neglected. The presence of patchy differential reddening in front of these clusters has proven to be a significant obstacle to their detailed study. We present here a well-defined and reasonably homogeneous photometric database for 25 of the brightest Galactic globular clusters located in the direction of the inner Galaxy. These data were obtained in the B, V, and I bands using the Magellan 6.5m telescope and the Hubble Space Telescope. A new technique is extensively used in this paper to map the differential reddening in the individual cluster fields, and to produce cleaner, dereddened color-magnitude diagrams for all the clusters in the database. Subsequent papers will detail the astrophysical analysis of the cluster populations, and the properties of the obscuring material along the clusters' lines of sight.

astro-ph.GA↗

arXiv2011

Threshold estimation based on a p-value framework in dose-response and regression settings

We use p-values to identify the threshold level at which a regression function takes off from its baseline value, a problem motivated by applications in toxicological and pharmacological dose-response studies and environmental statistics. We study the problem in two sampling settings: one where multiple responses can be obtained at a number of different covariate-levels and the other the standard regression setting involving limited number of response values at each covariate. Our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value and then computing the potentially approximate p-value of the test. An estimate of the threshold is obtained by fitting a piecewise constant function with a single jump discontinuity, otherwise known as a stump, to these observed p-values, as they behave in markedly different ways on the two sides of the threshold. The estimate is shown to be consistent and its finite sample properties are studied through simulations. Our approach is computationally simple and extends to the estimation of the baseline value of the regression function, heteroscedastic errors and to time-series. It is illustrated on some real data applications.

stat.ME↗

arXiv2011

A continuous mapping theorem for the smallest argmax functional

This paper introduces a version of the argmax continuous mapping theorem that applies to M-estimation problems in which the objective functions converge to a limiting process with multiple maximizers. The concept of the smallest maximizer of a function in the d-dimensional Skorohod space is introduced and its main properties are studied. The resulting continuous mapping theorem is applied to three problems arising in change-point regression analysis. Some of the results proved in connection to the d-dimensional Skorohod space are also of independent interest.

math.ST↗

arXiv2011

Mapping differential reddening in the inner Galactic globular cluster system

A serious limitation in the study of many globular clusters -- especially those located near the Galactic Center -- has been the existence of large and differential extinction by foreground dust. In a series of papers we intend to map the differential extinction and remove its effects, using a new dereddening technique, in a sample of clusters in the direction of the inner Galaxy, observed using the Magellan 6.5m telescope and the Hubble Space Telescope. These observations and their analysis will let us produce high quality color-magnitude diagrams of these poorly studied clusters that will allow us to determine these clusters' relative ages, distances and chemistry and to address important questions about the formation and the evolution of the inner Galaxy. We also intend to use the maps of the differential extinction to sample and characterize the interstellar medium along the numerous low latitude lines of sight where the clusters in our sample lie. In this first paper we describe in detail our dereddening method along with the powerful statistics tools that allow us to apply it, and we show the kind of results that we can expect, applying the method to M62, one of the clusters in our sample. The width of the main sequence and lower red giant branch narrows by a factor of 2 after applying our dereddening technique, which will significantly help to constrain the age, distance, and metallicity of the cluster.

astro-ph.GA↗

arXiv2011

Change-point in stochastic design regression and the bootstrap

In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its remarkable finite sample performance in our simulation study, and prove the consistency of the procedure. The $m$ out of $n$ bootstrap procedure is also considered and shown to be consistent. We also provide sufficient conditions for any bootstrap procedure to be consistent in this scenario.

math.ST↗

arXiv2010

Fractals with point impact in functional linear regression

This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a sensitive time point, is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.

math.ST↗

arXiv2010

Inconsistency of bootstrap: The Grenander estimator

In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function $f$ on $[0,\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0\in(0,\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function $\mathbb{F}_n$ or its least concave majorant $\tilde{F}_n$, does not have any weak limit in probability. We provide a set of sufficient conditions for the consistency of any bootstrap method in this example and show that bootstrapping from a smoothed version of $\tilde{F}_n$ leads to strongly consistent estimators. The $m$ out of $n$ bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_n$ and $\tilde{F}_n$.

math.ST↗

arXiv2010

Threshold estimation based on a P-value framework

We use p-values as a discrepancy criterion for identifying the threshold value at which a regression function takes off from its baseline value -- a problem that is motivated by applications in omics experiments, systems engineering, pharmacological dose-response studies and astronomy. In this paper, we study the problem in a controlled sampling setting, where multiple responses, discrete or continuous, can be obtained at a number of different covariate-levels. Our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value using the sampled responses at that value and then computing the p-value of the test. An estimate of the threshold is provided by fitting a stump, i.e., a piecewise constant function with a single jump discontinuity, to the observed p-values, since the corresponding p-values behave in markedly different ways on different sides of the threshold. The estimate is shown to be consistent, as both the number of covariate values and the number of responses sampled at each value become large, and its finite sample properties are studied through an extensive simulation study. Our approach is computationally simple and can also be used to estimate the baseline value of the regression function. The procedure is illustrated on two motivating real data applications. Extensions to multiple thresholds are also briefly investigated.

stat.ME↗

arXiv2009

Streaming motion in Leo I

Whether a dwarf spheroidal galaxy is in equilibrium or being tidally disrupted by the Milky Way is an important question for the study of its dark matter content and distribution. This question is investigated using 328 recent observations from the dwarf spheroidal Leo I. For Leo I, tidal disruption is detected, at least for stars sufficiently far from the center, but the effect appears to be quite modest. Statistical tools include isotonic and split point estimators, asymptotic theory, and resampling methods.

stat.AP↗

arXiv2008

Clean Kinematic Samples in Dwarf Spheroidals: An Algorithm for Evaluating Membership and Estimating Distribution Parameters When Contamination is Present

(abridged) We develop an algorithm for estimating parameters of a distribution sampled with contamination, employing a statistical technique known as ``expectation maximization'' (EM). Given models for both member and contaminant populations, the EM algorithm iteratively evaluates the membership probability of each discrete data point, then uses those probabilities to update parameter estimates for member and contaminant distributions. The EM approach has wide applicability to the analysis of astronomical data. Here we tailor an EM algorithm to operate on spectroscopic samples obtained with the Michigan-MIKE Fiber System (MMFS) as part of our Magellan survey of stellar radial velocities in nearby dwarf spheroidal (dSph) galaxies. These samples are presented in a companion paper and contain discrete measurements of line-of-sight velocity, projected position, and Mg index for ~1000 - 2500 stars per dSph, including some fraction of contamination by foreground Milky Way stars. The EM algorithm quantifies both dSph and contaminant distributions, returning maximum-likelihood estimates of the means and variances, as well as the probability that each star is a dSph member. Applied to our MMFS data, the EM algorithm identifies more than 5000 probable dSph members. We test the performance of the EM algorithm on simulated data sets that represent a range of sample size, level of contamination, and amount of overlap between dSph and contaminant velocity distributions. The simulations establish that for samples ranging from large (N ~3000) to small (N~30), the EM algorithm distinguishes members from contaminants and returns accurate parameter estimates much more reliably than conventional methods of contaminant removal (e.g., sigma clipping).

astro-ph↗

arXiv2007

Velocity Dispersion Profiles of Seven Dwarf Spheroidal Galaxies

We present stellar velocity dispersion profiles for seven Milky Way dwarf spheroidal (dSph) satellite galaxies. We have measured 8394 line-of-sight velocities (+/- 2.5 km/s) for 6804 stars from high-resolution spectra obtained at the Magellan and MMT telescopes. We combine these new data with previously published velocities to obtain the largest available kinematic samples, which include more than 5500 dSph members. All the measured dSphs have stellar velocity dispersion of order 10 km/s that remains approximately constant with distance from the dSph center, out to and in some cases beyond the radius at which the mean surface brightness falls to the background level. Assuming dSphs reside within dark matter halos characterized by the NFW density profile, we obtain reasonable fits to the empirical velocity dispersion profiles. These fits imply that, among the seven dSphs, M_vir ~ 10^[8-9] M_sun. The mass enclosed at a radius of 600 pc, the region common to all data sets, ranges from (2-7) x 10^7 M_sun .

astro-ph↗

arXiv2007

The Michigan/MIKE Fiber System Survey of Stellar Radial Velocities in Dwarf Spheroidal Galaxies: Acquisition and Reduction of Data

We introduce a stellar velocity survey of dwarf spheroidal galaxies, undertaken using the Michigan/MIKE Fiber System (MMFS) at the Magellan/Clay 6.5 m telescope at Las Campanas Observatory. As of 2006 November we have used MMFS to collect 6415 high-resolution (R= 20000-25000) spectra from 5180 stars in four dwarf spheroidal galaxies: Carina, Fornax, Sculptor and Sextans. Spectra sample the range 5140-5180 Angstroms, which includes the prominent magnesium triplet absorption feature. We measure radial velocity (RV) to a median precision of 2.0 km/s for stars as faint as V ~ 20.5. From the spectra we also are able to measure the strength of iron and magnesium absorption features using spectral indices that correlate with effective temperature, surface gravity and chemical abundance. Measurement of line strength allows us to identify interloping foreground stars independently of velocity, and to examine the metallicity distribution among dSph members. Here we present detailed descriptions of MMFS, our target selection and spectroscopic observations, the data reduction procedure, and error analysis. We compare our RV results to previously published measurements for individual stars. In some cases we find evidence for a mild, velocity-dependent offset between the RVs we measure using the magnesium triplet and previously published RV measurements derived from the infrared calcium triplet. In companion papers we will present the complete data sets and kinematic analyses of these new observations.

astro-ph↗