arXiv subjects

Ryan Martin

Publications and source records attributed to Ryan Martin.

115 records · Page 7Linked to original sources

arXiv2011

Semiparametric inference in mixture models with predictive recursion marginal likelihood

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered.

stat.ME↗

arXiv2011

On the edit distance from $K_{2,t}$-free graphs II: Cases $t\geq 5$

The edit distance between two graphs on the same vertex set is defined to be size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, $\mathcal{H}$, is a function of $p$ and measures, asymptotically, the furthest graph with edge density $p$ from $\mathcal{H}$ under this metric. The edit distance function has proven to be difficult to compute for many hereditary properties. Some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz, have been uncovered in attempts to compute various edit distance functions. In this paper, we address the hereditary property $\forb(K_{2,t})$ when $t\geq5$, the property of having no induced copy of the complete bipartite graph with 2 vertices in one class and $t$ in the other. This work continues from a prior paper by the authors. Employing an assortment of techniques and colored regularity graph constructions, we are able to extend the interval over which the edit distance function for this hereditary property is generally known and determine its maximum value for all odd $t$. We also explore several constructions to improve upon known upper bounds for the function.

math.CO↗

arXiv2011

Stochastic Approximation and Newton's Estimate of a Mixing Distribution

Many statistical problems involve mixture models and the need for computationally efficient methods to estimate the mixing distribution has increased dramatically in recent years. Newton [Sankhya Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating the mixing distribution, which we study as a special case of stochastic approximation (SA). We begin with a review of SA, some recent statistical applications, and the theory necessary for analysis of a SA algorithm, which includes Lyapunov functions and ODE stability theory. Then standard SA results are used to prove consistency of Newton's estimate in the case of a finite mixture. We also propose a modification of Newton's algorithm that allows for estimation of an additional unknown parameter in the model, and prove its consistency.

stat.ME↗

arXiv2010

Dempster--Shafer Theory and Statistical Inference with Weak Beliefs

The Dempster--Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combining evidence. DS theory has been widely used in computer science and engineering applications, but has yet to reach the statistical mainstream, perhaps because the DS belief functions do not satisfy long-run frequency properties. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, that can incorporate desirable frequency properties into the DS framework by systematically enlarging the focal elements. The present paper reviews and extends this WB approach. We present a general description of WB in the context of inferential models, its interplay with the DS calculus, and the maximal belief solution. New applications of the WB method in two high-dimensional hypothesis testing problems are given. Simulations show that the WB procedures, suitably calibrated, perform well compared to popular classical methods. Most importantly, the WB approach combines the probabilistic reasoning of DS with the desirable frequency properties of classical statistics.

stat.ME↗

arXiv2010

Lower bounds for identifying codes in some infinite grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

math.CO↗

arXiv2009

Consistency of a recursive estimate of mixing distributions

Mixture models have received considerable attention recently and Newton [Sankhyā Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao--Blackwell argument is used to prove consistency in probability of this alternative estimate. Several simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate.

math.ST↗

arXiv2009

Results from the Neutral Current Detector phase of the Sudbury Neutrino Observatory

The Sudbury Neutrino Observatory (SNO) was a heavy water Cerenkov detector designed to solve the long-standing ``solar neutrino problem''; a discrepancy between the measured and predicted flux of electron-flavour solar neutrinos. SNO measured the rate of charged-current and neutral-current reactions of neutrinos in heavy water and was able to demonstrate that neutrinos from the Sun, produced in the electron flavour eigenstate, undergo flavour change on their way to the Earth, thus resolving the solar neutrino problem. The experiment was conducted in three phases, differing by the method for measuring the neutral current rate. This short paper summarizes results from the third phase of the experiment, which used an array of 36 strings of proportional counters filled with $^3$He to detect neutrons from the neutral-current reaction. When the data from the three phases is combined with solar and the KamLAND neutrino oscillation experiments, the resulting limits on the solar neutrino mixing angle and mass-squared difference are $θ_{12} = 34.4^{+1.3}_{-1.2}$ degrees and $Δm_{12}^2 =7.59^{+0.19}_{-0.21}\times 10^{-5}$eV$^2$, respectively.

hep-ex↗