Richard A. Lockhart

A significance test for the lasso

In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an Exp(1) asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix X. On the other hand, our proof for a general step in the lasso path places further technical assumptions on X and the generative model, but still allows for the important high-dimensional case p > n, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables. Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a [Formula: see text] distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than [Formula: see text] under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter λ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the [Formula: see text] penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties-adaptivity and shrinkage-and its null distribution is tractable and asymptotically Exp(1).

Cramer-von Mises statistics for discrete distributions

Cramer-von Mises statistics are developed for use in testing for discrete distributions, and tables are given for tests for the discrete uniform distribution. The Cram6r-von Mises family of goodness-of-fit statistics is a well-known group of statistics used to test fit to a continuous distribution. In this article we extend the family to provide tests for discrete distributions. The statistics examined are the analogues of those called Cramer-von Mises, Watson, and Anderson-Darling, namely W2, U2 and A2 respectively, and their components. We provide formulae for the test statistics, and asymptotic percentage points for the test for a uniform distribution with k cells. The tests are based on the empirical distribution function (EDF) of the sample. They are closely related to Pearsons X2 test, and to Neyman-Barton smooth tests; in particular, all the tests can be broken down into components, as has been observed by many authors. It is suggested that A2 be used to test the overall null hypothesis in general, and U2 for the particular case where observations are counts around a circle. Their components can be used to test for particular types of departure from the null. In Section 2, we define the test statistics and give the general distribution theory. In Section 3 the solution of the uniform case is given, together with two examples; in Section 4 modified versions of the statistics are discussed. In Section 5 power studies are given which show that A2 is a good omnibus test statistic. Finally, in Section 6 we discuss the use of components as individual test statistics and demonstrate the use of a graphical procedure called the Z-plot to determine, when a statistic is found to be significant, the type of departure from the null.

Finding the location of a signal: a bayesian analysis

Abstract We study the problem of determining the location of an emergency transmitter in a downed aircraft. The observations are bearings read at fixed stations. A Bayesian approach, yielding a posterior map of probable locations, seems reasonable in this situation. We therefore develop conjugate prior distributions for the von Mises distribution, which we use to compute a posterior distribution of the location. An approximation to the posterior distribution yields accurate, rapidly computable answers. A common problem with this kind of data is the possibility that signals will reflect off orographic terrain features, resulting in wild bearings. Such bearings can affect the posterior distribution severely. We develop a sensitivity analysis, based on the idea of predictive distribution, to reject wild bearings. The method, which is based on an asymptotic argument, nonetheless performs well in a small simulation study. When the preceding approximation is used, the sensitivity analysis is practical in terms ...

Regression and error analysis applied to the dose-response curves in thermoluminescence dating

Abstract In the dating of Quaternary geological materials by thermoluminescence (TL), one frequently encounters sublinear TL vs applied dose curves for which accurate extrapolation is required. For dating the last exposure to sunlight of unheated Quaternary sediments, the intersection of two sublinear dose-response curves is usually sought. Adequate methods for estimating the uncertainty in this intersection have not been previously described. Here we outline an iteratively reweighted least-squares procedure for calculating the coefficients of second-order polynomials as well as saturating exponentials, and a variance-covariance method of error analysis appropriate for the intersection of two dose-response curves. Applications to several data sets are presented. The procedures described here are equally applicable, with trivial changes, to extrapolations of single sublinear dose-response curves (the additive dose technique).

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramér-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.

Bent-Cable Regression Theory and Applications

We use the so-called “bent-cable” model to describe natural phenomena that exhibit a potentially sharp change in slope. The model comprises two linear segments, joined smoothly by a quadratic bend. The class of bent cables includes, as a limiting case, the popular piecewise-linear model (with a sharp kink), otherwise known as the broken stick. Associated with bent-cable regression is the estimation of the bend-width parameter, through which the abruptness of the underlying transition may be assessed. We present worked examples and simulations to demonstrate the regularity and irregularity of bent-cable regression encountered in finite-sample settings. We also extend existing bent-cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1. Practical conditions on the design are given to ensure regularity of the full bent-cable estimation problem if the underlying bend segment has nonzero width. Under such conditions, the least-squares estimators are shown to be consistent and to asymptotically follow a multivariate normal distribution. Furthermore, the deviance statistic (or the likelihood ratio statistic, if the random errors are normally distributed) is shown to have an asymptotic chi-squared distribution.

Box-Cox transformations in linear models: large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box-Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non-normal errors, they give adjustments to the log-likelihood and to asymptotic standard errors.

Tests of Independence in Time Series

Tests of the hypothesis that a seriesX1...,Xn is a sequence of independent and identically distributed observations are investigated. The tests are based on a smoothing of the plotXi+1 againstXi. Asymptotic distribution theory on the null hypothesis and under contiguous alternatives is derived. The resulting limit distributions are available in published tables. Omnibus quadratic statistics of the Anderson–Darling type are compared with some optimal statistics and are shown to have good asymptotic properties. A Monte Carlo study is provided to show that the tests have good small sample properties

ON THE NON-EXISTENCE OF CONSISTENT ESTIMATES IN GALTON-WATSON PROCESSES

Estimation of the offspring distribution from a single realization of a supercritical Galton-Watson process is studied. It is shown that, based on population totals, a parameter of the offspring distribution cannot be estimated unless it is determined by the mean, variance, lattice size and lattice offset of the offspring distribution. CONSISTENT ESTIMATION; SUPERCRITICAL GALTON-WATSON PROCESSES

Tests for the response distribution in a Poisson regression model

A test of fit of a Poisson regression model to a set of data is composite, with two components: the test that the regression model is correct, and the test that the response has a Poisson distribution. Here, we give tests which are effective for the second assumption. The tests are based on the residuals after the model has been fitted, and the statistics are Cramer–von Mises statistics. Formulas are given to calculate the statistics, and a combination of asymptotic theory and Monte Carlo methods is used to find the estimated p-value of the test statistic. An example is given involving cancer data for workers at an aluminum plant. Power comparisons are made with other tests for the Poisson regression model. The results show that the statistics are good overall against alternatives to the Poisson assumption.