Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

http://pages.stat.wisc.edu/~mchung/teaching/MIA/reading/fMRI.woolrich.NI.2001.pdf

Temporal autocorrelation in univariate linear modeling of FMRI data.

A number of supervised learning methods have been introduced in the last decade. Unfortunately, the last comprehensive empirical evaluation of supervised learning was the Statlog Project in the early 90's. We present a large-scale empirical comparison between ten supervised learning methods: SVMs, neural nets, logistic regression, naive bayes, memory-based learning, random forests, decision trees, bagged trees, boosted trees, and boosted stumps. We also examine the effect that calibrating the models via Platt Scaling and Isotonic Regression has on their performance. An important aspect of our study is the use of a variety of performance criteria to evaluate the learning methods.

/pdf/an-empirical-comparison-of-supervised-learning-algorithms-3sk6pzs7c3.pdf

An empirical comparison of supervised learning algorithms

Recently, a lot of attention has been paid to regularization based methods for sparse signal reconstruction (e.g., basis pursuit denoising and compressed sensing) and feature selection (e.g., the Lasso algorithm) in signal processing, statistics, and related fields. These problems can be cast as -regularized least-squares programs (LSPs), which can be reformulated as convex quadratic programs, and then solved by several standard methods such as interior-point methods, at least for small and medium size problems. In this paper, we describe a specialized interior-point method for solving large-scale -regularized LSPs that uses the preconditioned conjugate gradients algorithm to compute the search direction. The interior-point method can solve large sparse problems, with a million variables and observations, in a few tens of minutes on a PC. It can efficiently solve large dense problems, that arise in sparse signal recovery with orthogonal transforms, by exploiting fast algorithms for these transforms. The method is illustrated on a magnetic resonance imaging data set.

/pdf/an-interior-point-method-for-large-scale-ell-1-regularized-4i61kdl13o.pdf

An Interior-Point Method for Large-Scale $\ell_1$ -Regularized Least Squares

We examine the relationship between the predictions made by different learning algorithms and true posterior probabilities. We show that maximum margin methods such as boosted trees and boosted stumps push probability mass away from 0 and 1 yielding a characteristic sigmoid shaped distortion in the predicted probabilities. Models such as Naive Bayes, which make unrealistic independence assumptions, push probabilities toward 0 and 1. Other models such as neural nets and bagged trees do not have these biases and predict well calibrated probabilities. We experiment with two ways of correcting the biased probabilities predicted by some learning methods: Platt Scaling and Isotonic Regression. We qualitatively examine what kinds of distortions these calibration methods are suitable for and quantitatively examine how much data they need to be effective. The empirical results show that after calibration boosted trees, random forests, and SVMs predict the best probabilities.

/pdf/predicting-good-probabilities-with-supervised-learning-3wtlkxwd3v.pdf

Predicting good probabilities with supervised learning

This paper provides tests of hypotheses and confidence intervals for the scale and shape parameters singly of the Birnbaum-Saunders fatigue life model, each with the other an unknown nuisance parameter The procedure for each parameter is based on its maximum likelihood estimate, although in the case of the scale parameter it is necessary to consider conditional procedures

Inferences on the Parameters of the Birnbaum-Saunders Fatigue Life Distribution Based on Maximum Likelihood Estimation

A problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard. We consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control. A number of statistical procedures have been proposed for this situation, of which the best known is Dunnett's (1955) multiple comparison procedure. Dunnett's test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located “symmetrically” within the alternative region. We study a large class of such tests, which includes Dunnett's test as a particular case. One of these tests, which is based on the maximum contrast of the data vector with several orthogonal vectors, is very easy to implement and has an uncomplicated, good, and fairly uniform power function over t...

Comparison of Several Treatments with a Control Using Multiple Contrasts

Most algorithms in order restricted statistical inference express the estimates in terms of certain summary statistics computed from pooled samples. These algorithms may or may not yield optimal estimates depending on whether or not the Cauchy mean value property holds strictly for the summary statistics. In this paper a minimum lower sets algorithm, which holds generally, is described and used to prove the optimality of estimates described by a max-min formula.

/pdf/algorithms-in-order-restricted-statistical-inference-and-the-32c8dp1iyc.pdf

Algorithms in Order Restricted Statistical Inference and the Cauchy Mean Value Property

Suppose that for each real number $t$ in [0, 1] we have a distribution with distribution function $F_t(\bullet)$, mean $\mu(t)$ and median $m(t) (\mu(t)$ and $m(t)$ are referred to as regression functions). Consider the problems of estimating $\mu(\bullet)$ and $m(\bullet)$. In this paper we propose and discuss an estimator, $\hat{m}(\bullet)$, of $m(\bullet)$ which is monotone. This estimator is analogous to the estimator $\hat{\mu}(\bullet)$ of $\mu(\bullet)$ which was explored by Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press, 177-195). Rates for the convergence of $\hat{m}(\bullet)$ to $m(\bullet)$ are given and a simulation study, where $\hat{m}(\bullet), \hat{\mu}(\bullet)$ and the least squares linear estimator are compared, is discussed.

/pdf/monotone-median-regression-2whoo19maw.pdf

Monotone Median Regression

: Consider the hypothesis H(1):theta(1) or = theta(2) or = ... or = theta(k) regarding a collection, theta(1), theta(2),...theta(k), of unknown parameters. It is clear that this trend is reflected in certain possible parameter sets more than in others. A quantification of this notion of conformity to a trend, which was suggested by Barlow and Brunk (1972), is studied. Applications of the resulting theory to several order restricted hypothesis tests are presented. (Author)

/pdf/on-measuring-the-conformity-of-a-parameter-set-to-a-trend-1nv93b44jh.pdf

On Measuring the Conformity of a Parameter Set to a Trend, with Applications

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