arXiv subjects
Publications and source records attributed to Michael Woodroofe.
We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are o(n), there is a CLT with a convergence towards N(0,1) when dividing by standard deviation of the partial sums, and the CLT is not quenched. The weak invariance principle does not hold.
We consider the problem of estimating the region on which a non-parametric regression function is at its baseline level in two dimensions. The baseline level typically corresponds to the minimum/maximum of the function and estimating such regions or their complements is pertinent to several problems arising in edge estimation, environmental statistics, fMRI and related fields. We assume the baseline region to be convex and estimate it via fitting a `stump' function to approximate $p$-values obtained from tests for deviation of the regression function from its baseline level. The estimates, obtained using an algorithm originally developed for constructing convex contours of a density, are studied in two different sampling settings, one where several responses can be obtained at a number of different covariate-levels (dose-response) and the other involving limited number of response values per covariate (standard regression). The shape of the baseline region and the smoothness of the regression function at its boundary play a critical role in determining the rate of convergence of our estimate: for a regression function which is `p-regular' at the boundary of the convex baseline region, our estimate converges at a rate $N^{2/(4p+3)}$ in the dose-response setting, $N$ being the total budget, and its analogue in the standard regression setting converges at a rate of $N^{1/(2p+2)}$. Extensions to non-convex baseline regions are explored as well.
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).
We establish a central limit theorem and an invariance principle for stationary random fields, with projective-type conditions. Our result is obtained via an m-dependent approximation method. As applications, we establish invariance principles for orthomartingales and functionals of linear random fields.
We establish sufficient conditions for the asymptotic normality of kernel density estimators, applied to causal linear random fields. Our conditions on the coefficients of linear random fields are weaker than known results, although our assumption on the bandwidth is not minimal. The proof is based on the $m$-approximation method. As a key step, we prove a central limit theorem for triangular arrays of stationary $m$-dependent random fields with unbounded $m$. We also apply a moment inequality recently established for stationary random fields.
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 $\tilde{F}_n$.
A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to those given by Ibragimov for linear processes. When specializing this result to the case when sums are being taken over rectangles, a complete analogue of Ibragimov result is obtained with a lot of uniformity.
It is shown that the existence of an L^1 co boundary does not imply the quenched version of the central limit theorem. In another result it is shown that Hannan's condition does imply quenched convergence for an appropriately centered version of the sum.
Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to κ\in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.
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.
Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values.
Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $π$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $π$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\lim_{n\to \infty}E(S_n^2)/n<\infty$. Assuming the first of these, let $\Vert g\Vert^2_+=\limsup_{n\to \infty}E(S_n^2)/n$; then $\Vert\cdot\Vert_+$ defines a pseudo norm on the subspace of $L^2(π)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\Vert\cdot\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(π)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(π)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis.
(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).
There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $R_n$ is a remainder term for which $E(R_n^2)=o(n)$. Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting $\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition for the partial sums of a stationary process to have the form $S_n=M_n+R_n$ is that $n^{-3/2}\Vert E(S_n|X_0,X_{-1},...)\Vert$ be summable. A sufficient condition for the LIL is only slightly stronger, requiring $n^{-3/2}\log^{3/2}(n)\Vert E(S_n|X_0,X_{-1},...)\Vert$ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.
We extend the isotonic analysis for Wicksell's problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer--Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate $n^{-1}\log n$ faster than $n^{-2/3}\log n$. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and smooth estimators.
We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF_2 itself.
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 .
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.