Search

A recent paper [Thibaudeau, Slud, and Gottschalck (2017). Modeling log-linear conditional probabilities for estimation in surveys. The Annals of Applied Statistics, 11, 680–697] proposed a ‘hybrid’...

In many settings, multiple data collections and analyses on the same topic are summarised separately through statistical estimators of parameters and variances, and yet there are scientific...

This article proposes a new random field model to describe the spatial variation of rainfall amounts accumulated over short periods of time. The model is intended to satisfy a set of desiderata motivated by the understanding of rainfall generating mechanisms and exploratory data analysis of datasets of this type. First and second order properties of the proposed model are derived, including the mean and covariance functions, as well as the families of marginal and bivariate distributions. Properties of the proposed model are shown by a mix of analytical derivations and numerical exploration that use Gauss–Hermite quadrature to approximate the required integrals. The proposed model also satisfies a stochastic dominance property, which is argued to be sensible and consistent with most rainfall data of this type. A study of identifiability is carried out, which strongly suggests all model parameters are identifiable. The generalized method of moments is proposed to estimate the parameters, and the properties of these estimators are explored based on simulated data.

After a brief historical survey of parametric survival models, from actuarial, biomedical, demographical and engineering sources, this paper discusses the persistent reasons why parametric models still play an important role in exploratory statistical research. The phase-type models are advanced as a flexible family of latent-class models with interpretable components. These models are now supported by computational statistical methods that make numerical calculation of likelihoods and statistical estimation of parameters feasible in theory for quite complicated settings. However, consideration of Fisher Information and likelihood-ratio type tests to discriminate between model families indicates that only the simplest phase-type model topologies can be stably estimated in practice, even on rather large datasets. An example of a parametric model with features of mixtures, multiple stages or 'hits', and a trapping-state is given to illustrate simple computational tools in R, both on simulated data and on a large SEER 1992-2002 breast-cancer dataset.

This paper develops methodology for survey estimation and small-area prediction using Fay–Herriot (1979) models in which the responses are left-censored. Parameter and small-area estimators are derived both by censored-data likelihoods and by an estimating-equation approach which adjusts a Fay–Herriot analysis restricted to the uncensored observations. Formulas for variances of estimators and mean-squared errors of small-area predictions are provided and supported by a simulation study. The methodology is applied to provide diagnostics for the left-censored Fay–Herriot model which are illustrated in the context of the Census Bureau's ongoing Small-Area Income and Poverty Estimation (SAIPE) project.

A new strategy is developed for obtaining large-sample efficient estimators of finite-dimensional parameters β within semiparametric statistical models. The key idea is to maximize over β a nonparametric log-likelihood with the infinite-dimensional nuisance parameter λ replaced by a consistent preliminary estimator λ ˜ β of the Kullback–Leibler minimizing value λ β for fixed β . It is shown that the parametric submodel with Kullback–Leibler minimizer substituted for λ is generally a least-favorable model. Results extending those of Severini and Wong (Ann. Statist. 20 (1992) 1768) then establish efficiency of the estimator of β maximizing log-likelihood with λ replaced for fixed β by λ ˜ β . These theoretical results are specialized to censored linear regression and to a class of semiparametric survival analysis regression models including the proportional hazards models with unobserved random effect or `frailty', the latter through results of Slud and Vonta (Scand. J. Statist. 31 (2004) 21) characterizing the restricted Kullback–Leibler information minimizers.

The number of subjects in a pharmacokinetic two-period two-treatment crossover bioequivalence study is typically small, most often less than 60. The most common approach to testing for bioequivalence is the two one-sided tests procedure. No explicit mathematical formula for the power function in the context of the two one-sided tests procedure exists in the statistical literature, although the exact power based on Owen's special case of bivariate noncentral t-distribution has been tabulated and graphed. Several approximations have previously been published for the probability of rejection in the two one-sided tests procedure for crossover bioequivalence studies. These approximations and associated sample size formulas are reviewed in this article and compared for various parameter combinations with exact power formulas derived here, which are computed analytically as univariate integrals and which have been validated by Monte Carlo simulations. The exact formulas for power and sample size are shown to improve markedly in realistic parameter settings over the previous approximations.

In many clinical trials involving survival endpoints, one has additional data on some binary indicator of'response', such as initial tumour shrinkage in cancer trials. This paper studies the case of randomised clinical trials where the response indicator is available shortly after randomisation, and where one can assume that, within each stratum defined by the response indicator, a two treatment-group proportional-hazards model holds. The same model may also describe some incompletely randomised or observational studies. Asymptotic relative efficiencies for Kaplan-Meier-based estimators versus maximum partial likelihood estimators are examined under this model for estimating either the difference in survival probabilities at a specified time or the parameter estimated by the logrank numerator. It is shown that the efficiency gains using the model are more promising when estimating the difference in survival probabilities. An example is given comparing the long-term survival experience of two groups of patients with advanced Hodgkin's disease.

Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

In the context of parametric inference for a scalar parameter β in the presence of a finite-dimensional nuisance parameter λ based on a large random sample X 1 , …, X n , this paper calculates an exact one-sided inaccuracy rate for maximum-likelihood and M-estimators, as well as the Hodges-Lehmann (1956) large deviation rate for type-II error probabilities under fixed alternatives. The method is to couple the large-deviation theorems of Groeneboom et al. (1979) for empirical measures with a characterization via the Implicit Function Theorem of ‘least favorable measures’ extremizing the Kullback-Leibler information functional over statistically interesting sets of measures.

The Maximin Efficiency-Robust Test idea of Gastwirth (1966) was to maximize the minimum asymptotic power (for fixed size) versus special local families of alternatives over some specially chosen families of score statistics. This approach is reviewed from a general decision-theoretical perspective, including some Bayesian variants. For two-sample censored-data rank tests and stochastically ordered but not proportional-hazard alternatives, the MERT approach leads to customized weighted-logrank tests for which the weights depend on estimated random-censoring distributions. Examples include statistics which perform well against both Lehmann and logistic alternatives or against families of alternatives which include increasing, decreasing, and ‘bathtub-shaped’ hazards.

Let (Ω, F , P ) be a probability space equipped with two filtrations { F t } and { G t } satisfying the usual conditions. Assume that X is a semimartingale and that h is locally bounded and predictable for each of the two filtrations { F t } and { G t }. New examples of such processes are given. Utilizing and extending partial results of Zheng (1982), this paper extends the available results on the relationship between the stochastic integral processes ∫ t h s d X s taken respectively in the sense of { F t } and of { G t }. In particular, it is shown that these stochastic integrals differ at most by a continuous process with quadratic variation defined and equal to 0. If both stochastic integrals are { F t ∩ F t } semimartingales, then it is proved that the stochastic integral ∫ t h s d X s taken in { F t } sense is indistinguishable from that taken in { G t } sense.

Consistency issues related to autocorrelation estimation for Gaussian processes with mixed spectra are clarified. The sample autocorrelation is known not to be consistent when the spectrum contains spectral atoms. This fact is verified by computing explicitly its mean-square limit in terms of the random measure assigned to the atoms of the process. The alternative estimator constructed from the zero-crossing rate is likewise not consistent in general. However, it is consistent for a spectrum supported at a single frequency, or a spectrum for which the ‘signal’ and ‘noise’ have the exact same first-order autocorrelation. This is proved, using recent results from multiple Wiener-Ito integral expansions for level-crossing counts, by direct computation of the spectrum of the zero-crossing indicator process when the underlying process has a mixed spectrum. In general, regardless of the spectral type, the asymptotic zero-crossing rate admits values between the lowest and highest positive frequencies with probability one. For band-limited processes, this fact provides an easy way to assess the precision of functions of the zero-crossing rate.

The prevalence of survival analyses based on a fixed duration of time-on-test, together with the need for generally powerful two-sample censored-data rank tests against stochastically ordered but not proportional-hazard alternatives, are used to motivate an extension of ‘precedence tests’ (Nelson (1963), Lin and Sukhatme (1989)) to right-censored survival data. The idea is to compare the r -th Kaplan-Meier quantile from the first sample with the s -th Kaplan-Meier quantile from the second sample, where r and s are nearby values chosen to give size α and best power against local proportional-hazards alternatives.

This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.

SUMMARY For large-sample clinical trials with independent individuals randomly allocated to two treatment groups, in which survival times follow a log linear multiplicative intensity model with treatment group as one covariate, this paper calculates the asymptotic relative efficiency of the log rank test for treatment effect as compared with the optimal score test. The method is to exhibit the failure hazard intensity, not of proportional hazards form, obtained by ignoring all covariates other than treatment group. The efficiency formulae are illustrated in two examples, and estimation from data of the loss of efficiency is illustrated for two clinical trial datasets.

Several new methodological issues that arise within two-way factorial designs for survival experiments are discussed within the framework of asymptotic theory for the proportional hazards model with two binary treatment covariates. These issues include: the proper formulation of null hypotheses and alternatives, the choice among log-rank and adjusted or stratified log-rank statistics, the asymptotic correlation between test statistics for the separate main effects, the asymptotic power (under the various possible methods of analysis) of tests to detect main effects and interactions, the comparison of power to detect main effects within a 2 x 2 factorial design with power in a three-group trial where no patients are randomized simultaneously to both treatments, and the problems of analysis arising when accrual or exposure to one of the treatments is terminated early for ethical reasons.

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio 'age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy agedistribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Cs6rg6 et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals. RISK-RESERVE PROCESS; COMPOUND DELAYED-RENEWAL PROCESS; SUPEPPOSITION; STRONG APPROXIMATION

It is known that for any sequence X1, X2…, of identically distributed independent random matrices with a common distribution μ. the limitexists with probability 1. We study some conditions under which Λ(μk)→Λ(μ) provided μk → μ in the weak sense.

A new class of reliability point-process models for dependent components is introduced. The dependence is expressed through a regression, following a form suggested by Cox (1972) for survival data analysis involving the current life-length of the components. After formulating the current-life process as a Markov process with stationary transitions and stating some general results on asymptotic behavior, we describe the stationary distributions in some bivariate examples. Finally, we discuss statistical inference for the new models, exhibiting and justifying full- and partial-likelihood methods for their analysis.

SUMMARY In many contexts where there is interest in inferring the marginal distribution of a survival time T subject to censoring embodied in a latent waiting time C, the times T and C may not be independent. This paper presents a new class of nonparametric assumptions on the conditional distribution of T given C and shows how they lead to consistent generalizations of the Kaplan & Meier (1958) survival curve estimator. The new survival curve estimators are used under weak assumptions to construct bounds on the marginal survival which can be much narrower than those of Peterson (1976). In stratified populations where T and C are independent only within strata examples indicate that the Kaplan-Meier estimator is often approximately consistent.

The well-known submartingale maximal inquality of Birnbaum and Marshall (1961) is generalized to provide upper tail inequalities for suprema of processes which are products of a submartingale by a nonincreasing nonnegative predictable process. The new inequalities are proved by applying an inequality of Lenglart (1977), and are then used to provide best-possible universal growth-rates for a general submartingale in terms of the predictable compensator of its positive part. Applications of these growth rates include strong asymptotic upper bounds on solutions to certain stochastic differential equations, and strong asymptotic lower bounds on Brownian-motion occupation-times.

and let H(X) = L’(f), u(X), P) denote the space of square-integrable functionals of X. Say that Y t H(X) with EY =O satisfies the Central Limit Theorem (CLT) if A family of martingales (Z,(t), t 2 0) is exhibited for which 2, (“c) = Z,, and martingale tech- niques and results are used to provide suficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y. AMS Subject Ch\jfication.s: hOF05, 60G10, 60644 multiple Wiener-It8 integral 3 stochastic integral with respect to square-integrable martingale * predictable variance process * martingale Functional Central Limit Theorem * Band- weighted Central Limit Theorem

It is shown that the level-crossings process of zeroes and ones corresponding to a stationary but not independent Gaussian sequence can never be exactly ( m -step) Markov, although its correlation-sequence can agree exactly with that of a Markov sequence.

A new methodology is proposed for discrimination among stationary time-series. The time series are transformed into binary arrays by clipping (retaining only the signs of) the $j$th difference series, $j = 0, 1, 2, \cdots$. The degeneracy of clipped $j$th differences is studied as $j$ becomes large. A new goodness of fit statistic is defined as a quadratic form in the counts of axis-crossings by each of the first $k$ differences of the series. Simulations and the degeneracy of high-order differences justify fixing $k$ no larger than 10 for many processes. Empirical simulated distributions (with $k = 9$) of the goodness of fit statistic suggest a gamma approximation for its tail probabilities. Illustrations are given of discrimination between simple models with the new statistic.

It is shown, using the results of Slud and Rubinstein (1983, Biometrika 70, 643-649) in a specially constructed theoretical example, that competing latent failure times Ti and Ci and a two-level covariate Vi, if analyzed as though Ti and Ci are independent for each Vi level, can lead to exactly the wrong conclusion about the ordering of Pr(Ti greater than or equal to t[Vi = 1) and Pr(Ti greater than or equal to t[Vi = 0) for every t. This phenomenon can never be excluded on purely statistical grounds using such data and should be considered when interpreting data analyses involving competing risks.

The small-sample performance of some recently proposed nonparametric methods of constructing confidence intervals for the median survival time, based on randomly right-censored data, is compared with that of two new methods. Most of these methods are equivalent for large samples. All proposed intervals are either 'test-based' or 'reflected' intervals, in the sense defined in the paper. Coverage probabilities for the interval estimates were obtained by exact calculation for uncensored data, and by stimulation for three life distributions and four censoring patterns. In the range of situations studied, 'test-based' methods often have less than nominal coverage, while the coverage of the new 'reflected' confidence intervals is closer to nominal (although somewhat conservative), and these intervals are easy to compute.