An explanation of generalized profile likelihoods

Let X, T, Y be random vectors such that the distribution of Y conditional on covariates partitioned into the vectors X = x and T = t is given by f(y; x, ), where = (, (t)). Here is a parameter vector and (t) is a smooth, real–valued function of t. The joint distribution of X and T is assumed to be independent of and . This semiparametric model is called conditionally parametric because the conditional distribution f(y; x, ) of Y given X = x, T = t is parameterized by a finite dimensional parameter = (, (t)). Severini and Wong (1992. Annals of Statistics 20: 1768–1802) show how to estimate and (·) using generalized profile likelihoods, and they also provide a review of the literature on generalized profile likelihoods. Under specified regularity conditions, they derive an asymptotically efficient estimator of and a uniformly consistent estimator of (·). The purpose of this paper is to provide a short tutorial for this method of estimation under a likelihood–based model, reviewing results from Stein (1956. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, pp. 187–196), Severini (1987. Ph.D Thesis, The University of Chicago, Department of Statistics, Chicago, Illinois), and Severini and Wong (op. cit.).     

Semiparametric Efficient Estimation in the Generalized Odds-Rate Class of Regression Models for Right-Censored Time-to-Event Data

The generalized odds-rate class of regression models for time to event data is indexed by a non-negative constant and assumes thatg(S(t|Z)) = (t) + Zwhere g(s) = log(^-1(s-) for > 0, g₀(s) = log(- log s), S(t|Z) is the survival function of the time to event for an individual with qx1 covariate vector Z, is a qx1 vector of unknown regression parameters, and (t) is some arbitrary increasing function of t. When =0, this model is equivalent to the proportional hazards model and when =1, this model reduces to the proportional odds model. In the presence of right censoring, we construct estimators for and exp((t)) and show that they are consistent and asymptotically normal. In addition, we show that the estimator for is semiparametric efficient in the sense that it attains the semiparametric variance bound.     

Generating random numbers of prescribed distribution using physical sources

When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B _j are extracted and combined into the machine number . In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X _n (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U _n : = X _{2n – 1}/(X _{2n – 1} + X _2n ) is uniform in [0, 1]. In the practical application X _n can only be measured up to a given precision (in terms of the expectation of the X _n ); it is shown that the distribution function obtained by calculating U _n from these measurements differs from the uniform by less than /2.We compare this deviation with the error resulting from the use of biased bits B _j with P {B _j = 1{ = (where ] – [) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm Q ^TV _p = ( |Q()| ^p )^1/p (p 1) we have P _Y – P ⁰ _Y ^TV _p (c _n · )^1/p with c _n p for n . For the distribution function F _Y – F ⁰ _Y 2(1 – 2^–n )|| holds.     

The Statistics of Simulating Chaos

When simulating a dynamical system, the computation is actually of a spatially discretized system, because finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretized simulations often have collapsing effects, to a fixed point or to short cycles. Statistical properties of these phenomena can be modelled with random mappings with an absorbing centre. The model gives results which are very much in line with computational experiments. The effects are discussed with special reference to the family of mappings f (x)=1-|1-2x|,x [0,1],1,<,,<,. Computer experiments show close agreement with predictions of the model.     

Principal points of univariate continuous distributions

The K principal points of a p-variate random variable X are defined as those points ₁,..., _K which minimize the expected squared distance of X from the nearest of the _k . This paper reviews some of the theory of principal points and presents a method of determining principal points of univariate continuous distributions. The method is applied to the uniform distribution, to the normal distribution and to the exponential distribution.     

Calculating the density and distribution function for the singly and doubly noncentral F

Simple, closed form saddlepoint approximations for the distribution and density of the singly and doubly noncentral F distributions are presented. Their overwhelming accuracy is demonstrated numerically using a variety of parameter values. The approximations are shown to be uniform in the right tail and the associated limitating relative error is derived. Difficulties associated with some algorithms used for exact computation of the singly noncentral F are noted.     

The exponentiation formula of reliability and survival: Does it always hold?

In reliability and biometry, it is common practice to choose a failure model by first assessing the failure rate function subjectively, and then invoking the well known exponentiation formula. The derivation of this formula is based on the assumption that the underlying failure distribution be absolutely continuous. Thus, implicit in the above approach is the understanding that the selected failure distribution will be absolutely continuous. The purpose of this note is to point out that the absolute continuity may fail when the failure rate is assessed conditionally, and in particular when it is conditioned on certain types of covariates, called internal covariates. When such is the case, the exponentiation formula should not be used.     

Data depths satisfying the projection property

Summary: Data depth is a concept that measures the centrality of a point in a given data cloud x ₁, x ₂,...,x _n or in a multivariate distribution P ^X on ^{d d} . Every depth defines a family of so–called trimmed regions. The –trimmed region is given by the set of points that have a depth of at least . Data depth has been used to define multivariate measures of location and dispersion as well as multivariate dispersion orders.If the depth of a point can be represented as the minimum of the depths with respect to all unidimensional projections, we say that the depth satisfies the (weak) projection property. Many depths which have been proposed in the literature can be shown to satisfy the weak projection property. A depth is said to satisfy the strong projection property if for every the unidimensional projection of the –trimmed region equals the –trimmed region of the projected distribution.After a short introduction into the general concept of data depth we formally define the weak and the strong projection property and give necessary and sufficient criteria for the projection property to hold. We further show that the projection property facilitates the construction of depths from univariate trimmed regions. We discuss some of the depths proposed in the literature which possess the projection property and define a general class of projection depths, which are constructed from univariate trimmed regions by using the above method.Finally, algorithmic aspects of projection depths are discussed. We describe an algorithm which enables the approximate computation of depths that satisfy the projection property.     

Sampling based approach for one-hit and multi-hit models in quantal bioassay

Bayesian methods for estimating the dose response curves with the one-hit model, the gamma multi-hit model, and their modified versions with Abbott's correction are studied. The Gibbs sampling approach with data augmentation and with the Metropolis algorithm is employed to compute the Bayes estimates of the potency curves. In addition, estimation of the relative additional risk and the virtually safe dose is studied. Model selection based on conditional predictive ordinates from cross-validated data is developed.     

Pitfalls in linear models for style analysis

We discuss the statistical properties of return-based OLS style analysis introduced by Sharpe (1992). The aim of style analysis is to infer a fund managers investment decisions using only publicly available data on the fund performance and on the time evolution of market indexes. We show that the model proposed by Sharpe suffers of relevant drawbacks, most notably that it fails to yield correct results even in the simple case of a buy-and-hold strategy that only invests in the market indexes. Under this hypothesis we show that a model linear in index levels, as opposed to index returns, estimated via a Kalman filter avoids Sharpes model drawbacks. We further extend our analysis to strategies where the fund manager policy changes with time and the asset classes in which the fund manager invests are not known exactly. In this last case we show that a style analysis is possible only conditional to either an orthogonality hypothesis on the active investment strategy, or by the introduction of suitable instrumental variables.The authors are grateful to the editor and an anonymous referee for many comments which greatly helped in improving the paper. The authors are, obviously, fully responsible for any remaining error.     

Computation of exact confidence intervals from discrete data using studentized test statistics

A new area of research interest is the computation of exact confidence limits or intervals for a scalar parameter of interest from discrete data by inverting a hypothesis test based on a studentized test statistic. See, for example, Chan and Zhang (1999), Agresti and Min (2001) and Agresti (2003) who deal with a difference of binomial probabilities and Agresti and Min (2002) who deal with an odds ratio. However, neither (1) a detailed analysis of the computational issues involved nor (2) a reliable method of computation that deals effectively with these issues is currently available. In this paper we solve these two problems for a very broad class of discrete data models. We suppose that the distribution of the data is determined by (,) where is a nuisance parameter vector. We also consider six different studentized test statistics. Our contributions to (1) are as follows. We show that the P-value resulting from the hypothesis test, considered as a function of the null-hypothesized value of , has both jump and drop discontinuities. Numerical examples are used to demonstrate that these discontinuities lead to the failure of simple-minded approaches to the computation of the confidence limit or interval. We also provide a new method for efficiently computing the set of all possible locations of these discontinuities. Our contribution to (2) is to provide a new and reliable method of computing the confidence limit or interval, based on the knowledge of this set.     

Historical controls and modern survival analysis

Comparison of observed mortality with known, background, or standard rates has taken place for several hundred years. With the developments of regression models for survival data, an increasing interest has arisen in individualizing the standardisation using covariates of each individual. Also, account sometimes needs to be taken of random variation in the standard group.Emphasizing uses of the Cox regression model, this paper surveys a number of critical choices and pitfalls in this area. The methods are illustrated by comparing survival of liver patients after transplantation with survival after conservative treatment.     

Applications of a general propagation algorithm for probabilistic expert systems

A probabilistic expert system provides a graphical representation of a joint probability distribution which can be used to simplify and localize calculations. Jensenet al. (1990) introduced a flow-propagation algorithm for calculating marginal and conditional distributions in such a system. This paper analyses that algorithm in detail, and shows how it can be modified to perform other tasks, including maximization of the joint density and simultaneous fast retraction of evidence entered on several variables.     

Jerome H. Friedman and Nicholas I. Fisher

Many data analytic questions can be formulated as (noisy) optimization problems. They explicitly or implicitly involve finding simultaneous combinations of values for a set of (input) variables that imply unusually large (or small) values of another designated (output) variable. Specifically, one seeks a set of subregions of the input variable space within which the value of the output variable is considerably larger (or smaller) than its average value over the entire input domain. In addition it is usually desired that these regions be describable in an interpretable form involving simple statements (rules) concerning the input values. This paper presents a procedure directed towards this goal based on the notion of patient rule induction. This patient strategy is contrasted with the greedy ones used by most rule induction methods, and semi-greedy ones used by some partitioning tree techniques such as CART. Applications involving scientific and commercial data bases are presented.     

A Diagnostic for Association in Bivariate Survival Models

We propose exploratory, easily implemented methods for diagnosing the appropriateness of an underlying copula model for bivariate failure time data, allowing censoring in either or both failure times. It is found that the proposed approach effectively distinguishes gamma from positive stable copula models when the sample is moderately large or the association is strong. Data from the Womens Health and Aging Study (WHAS, Guralnik et al., The Womenss Health and Aging Study: Health and Social Characterisitics of Older Women with Disability. National Institute on Aging: Bethesda, Mayland, 1995) are analyzed to demonstrate the proposed diagnostic methodology. The positive stable model gives a better overall fit to these data than the gamma frailty model, but it tends to underestimate association at the later time points. The finding is consistent with recent theory differentiating catastrophic from progressive disability onset in older adults. The proposed methods supply an interpretable quantity for copula diagnosis. We hope that they will usefully inform practitioners as to the reasonableness of their modeling choices.     

An algorithm for computing principal points with respect to a loss function in the unidimensional case

In Flury (1990) the k principal points of a random vector X are defned as the points p(1),..., p(k) minimizing EX–p(i)²; i=1,..., k. We extend this concept to that of k principal points with respect to a loss function L, and present an algorithm for their computation in the univariate case.     

Data depth and correlation

Summary: We describe depth–based graphical displays that show the interdependence of multivariate distributions. The plots involve one–dimensional curves or bivariate scatterplots, so they are easier to interpret than correlation matrices. The correlation curve, modelled on the scale curve of Liu et al. (1999), compares the volume of the observed central regions with the volume under independence. The correlation DD–plot is the scatterplot of depth values under a reference distribution against depth values under independence. The area of the plot gives a measure of distance from independence. Correlation curve and DD-plot require an independence model as a baseline: Besides classical parametric specifications, a nonparametric estimator, derived from the randomization principle, is used. Combining data depth and the notion of quadrant dependence, quadrant correlation trajectories are obtained which allow simultaneous representation of subsets of variables. The properties of the plots for the multivariate normal distribution are investigated. Some real data examples are illustrated. *This work was completed with the support of Ca Foscari University.     

Efficient computation of the discrete autocorrelation wavelet inner product matrix

Discrete autocorrelation (a.c.) wavelets have recently been applied in the statistical analysis of locally stationary time series for local spectral modelling and estimation. This article proposes a fast recursive construction of the inner product matrix of discrete a.c. wavelets which is required by the statistical analysis. The recursion connects neighbouring elements on diagonals of the inner product matrix using a two-scale property of the a.c. wavelets. The recursive method is an (log (N)³) operation which compares favourably with the (N log N) operations required by the brute force approach. We conclude by describing an efficient construction of the inner product matrix in the (separable) two-dimensional case.     

Threshold-range scaling of excitable cellular automata

Each cell of a two-dimensional lattice is painted one of colors, arranged in a color wheel. The colors advance (k tok+1 mod ) either automatically or by contact with at least a threshold number of successor colors in a prescribed local neighborhood. Discrete-time parallel systems of this sort in which color 0 updates by contact and the rest update automatically are called Greenberg-Hastings (GH) rules. A system in which all colors update by contact is called a cyclic cellular automation (CCA). Started from appropriate initial conditions, these models generate periodic traveling waves. Started from random configurations the same rules exhibit complex self-organization, typically characterized by nucleation of locally periodic ram's horns or spirals. Corresponding random processes give rise to a variety of forest fire equilibria that display large-scale stochastic wave fronts. This paper describes a framework, theoretically based, but relying on extensive interactive computer graphics experimentation, for investigation of the complex dynamics shared by excitable media in a broad spectrum of scientific contexts. By focusing on simple mathematical prototypes we hope to obtain a better understanding of the basic organizational principles underlying spatially distributed oscillating systems.     

Interpolation models with multiple hyperparameters

A traditional interpolation model is characterized by the choice of regularizer applied to the interpolant, and the choice of noise model. Typically, the regularizer has a single regularization constant , and the noise model has a single parameter . The ratio / alone is responsible for determining globally all these attributes of the interpolant: its complexity, flexibility, smoothness, characteristic scale length, and characteristic amplitude. We suggest that interpolation models should be able to capture more than just one flavour of simplicity and complexity. We describe Bayesian models in which the interpolant has a smoothness that varies spatially. We emphasize the importance, in practical implementation, of the concept of conditional convexity when designing models with many hyperparameters. We apply the new models to the interpolation of neuronal spike data and demonstrate a substantial improvement in generalization error.