Charles J. Stone

A study of logspline density estimation

A method of estimating an unknown density function @? based on sample data is studied. Our approach is to use maximum likelihood etimation to estimate log(@?) by a function s from a space of cubic splines that have a finite number of prespecified knots and are linear in the tails. The knots are placed at selected order statistics of the sample data. The number of knots can be determined either by a simple rule or by minimizing a variant of AIC. Examples using both simulated and real data show that the method works well both in obtaining smooth estimates and in picking up small details. The method is fully automatic and can easily be extended to yield estimates and confidence bounds for quantiles.

Logspline Density Estimation for Censored Data

Abstract Logspline density estimation is developed for data that may be right censored, left censored, or interval censored. A fully automatic method, which involves the maximum likelihood method and may involve stepwise knot deletion and either the Akaike information criterion (AIC) or Bayesian information criterion (BIC), is used to determine the estimate. In solving the maximum likelihood equations, the Newton–Raphson method is augmented by occasional searches in the direction of steepest ascent. Also, a user interface based on S is described for obtaining estimates of the density function, distribution function, and quantile function and for generating a random sample from the fitted distribution.

OPTIMAL UNIFORM RATE OF CONVERGENCE FOR NONPARAMETRIC ESTIMATORS OF A DENSITY FUNCTION OR ITS DERIVATIVES

Publisher Summary This chapter discusses optimal uniform rate of convergence for the nonparametric estimators of a density function or its derivatives. It describes the optimal uniform rate of convergence of an arbitrary estimator of T(f) for various choices of F . It is called the optimal rate of convergence, if it is both a lower and an achievable rate of convergence.

Robust confidence bounds for extreme upper quantiles

Four related methods are discussed for obtaining robust confidence bounds for extreme upper quantiles of the unknown distribution of a positive random variable. These methods are designed to work when the upper tail of the distribution is neither too heavy nor too light in comparison to the exponential distribution. An extensive simulated study is described, which compares the performance of nominal 90% upper confidence bounds corresponding to the four methods over a wide variety of distributions having light to heavy upper tails, ranging from a half-normal distribution to a heavy-tailed lognormal distribution.

Uniform Error Bounds Involving Logspline Models

Publisher Summary This chapter discusses uniform error bounds involving logspline models. Splines are of increasing importance in statistical theory and methodology. The chapter discusses exponential families of densities in which the logarithm of the density is a spline and the corresponding exponential response models. The chapter in each context presents the use of an extension of a key result of de Boor to obtain a bound on the L∞ norm of the approximation error associated with maximizing the associated expected log-likelihood.

The L2 Rate of Convergence for Event History Regression with Time‐dependent Covariates

Consider repeated events of multiple kinds that occur according to a right-continuous semi-Markov process whose transition rates are influenced by one or more time-dependent covariates. The logarithms of the intensities of the transitions from one state to another are modelled as members of a linear function space, which may be finite- or infinite-dimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finite-dimensional approximating spaces. It is shown that the L2 rates of convergence of the maximum likelihood estimates are determined by the approximation power and dimension of the approximating spaces. The theory is applied to a functional ANOVA model, where the logarithms of the intensities are approximated by functions having the form of a specified sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). It is shown that the curse of dimensionality can be ameliorated if only main effects and low-order interactions are considered in functional ANOVA models.

Local asymptotic admissibility of a generalization of Akaike′s model selection rule

SummaryA model selection rule of the form minimize [−2 log (maximized likelihood)+complexity] is considered, which is equivalent to Akaikes minimum AIC rule if the complexity of a model is defined to be twice the number of independently adjusted parameters of the model. Under reasonable assumptions, when applied to a locally asymptotically normal sequence of experiments, the model selection rule is shown to be locally asymptotically admissible with respect to a loss function of the form [inaccuracy+complexity], where the inaccuracy is defined as twice the Kullback-Leibler measure of the discrepancy between the true model and the fitted version of the selected model.

Free knot splines in concave extended linear modeling

Many problems of practical interest can be formulated as the estimation of a certain function such as a regression function, logistic or other generalized regression function, density function, conditional density function, hazard function, or conditional hazard function. Extended linear modeling provides a convenient framework for using polynomial splines and their tensor products in such function estimation problems. Huang (Statist. Sinica 11 (2001) 173) has given a general treatment of the rates of convergence of maximum likelihood estimation in the context of concave extended linear modeling. Here these results are generalized to let the approximation space used in the fitting procedure depend on a vector of parameters. More detailed treatments are given for density estimation and generalized regression (including ordinary regression) on the one hand and for approximation spaces whose components are suitably regular free knot splines and their tensor products on the other hand.

Statistical modeling of diffusion processes with free knot splines

We consider the nonparametric estimation of the drift coefficient in a diffusion type process in which the diffusion coefficient is known and the drift coefficient depends in an unknown manner on a vector of time-dependent covariates. Based on many continuous realizations of the process, the estimator is constructed using the method of maximum likelihood, where the maximization is taken over a finite dimensional estimation space whose dimension grows with the sample size n. We focus on estimation spaces of polynomial splines. We obtain rates of convergence of the spline estimates when the knot positions are prespecified but the number of knots increases with the sample size. We also give the rates of convergence for free knot spline estimates, in which the knot positions of splines are treated as free parameters that are determined by data.

Comparison of Parametric and Bootstrap Approaches to Obtaining Confidence Intervals for Logspline Density Estimation

In earlier articles, we developed an automated methodology for using cubic splines with tail linear constraints to model the logarithm of a univariate density function. This methodology was subsequently modified so that the knots were determined by stepwise addition-deletion and the remaining coefficients were determined by maximum likelihood estimation. An alternative approach, referred to as the free knot spline procedure, is to use the maximum likelihood method to estimate the knot locations as well as the remaining coefficients. This article compares various approaches to constructing confidence intervals for logspline density estimates, for both the stepwise procedure and the free knot procedure. It is concluded that a variation of the bootstrap, in which only a limited number of bootstrap simulations are used to estimate standard errors that are combined with standard normal quantiles, seems to perform the best, especially when coverages and computing time are both taken into account.