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Background Individual and environmental health outcomes are frequently linked to changes in the diversity of associated microbial communities. Thus, deriving health indicators based on microbiome diversity measures is essential. While microbiome data generated using high-throughput 16S rRNA marker gene surveys are appealing for this purpose, 16S surveys also generate a plethora of spurious microbial taxa. Results When this artificial inflation in the observed number of taxa is ignored, we find that changes in the abundance of detected taxa confound current methods for inferring differences in richness. Experimental evidence, theory-guided exploratory data analyses, and existing literature support the conclusion that most sub-genus discoveries are spurious artifacts of clustering 16S sequencing reads. We proceed to model a 16S survey’s systematic patterns of sub-genus taxa generation as a function of genus abundance to derive a robust control for false taxa accumulation. These controls unlock classical regression approaches for highly flexible differential richness inference at various levels of the surveyed microbial assemblage: from sample groups to specific taxa collections. The proposed methodology for differential richness inference is available through an R package, Prokounter. Conclusions False species discoveries bias richness estimation and confound differential richness inference. In the case of 16S microbiome surveys, supporting evidence indicate that most sub-genus taxa are spurious. Based on this finding, a flexible method is proposed and is shown to overcome the confounding problem noted with current approaches for differential richness inference. Package availability: https://github.com/mskb01/prokounter

It is proved under broad regularity conditions that there is no unbiased estimating function depending only on the structural parameter (the coefficients) in the Cox proportional hazards model with nuisance survival hazard parameters and possibly right-censored survival times.

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.

The notion of respondent contact burden in sample surveys is defined, and a multi-stage process to develop policies for curtailing nonresponse follow-up is described with the goal of reducing this burden on prospective survey respondents. The method depends on contact history paradata containing information about contact attempts both for respondents and for sampled nonrespondents. By analysis of past data, policies to stop case follow-up based on control variables measured in paradata can be developed by calculating propensities to respond for paradata-defined subgroups of sampled cases. Competing policies can be assessed by comparing outcomes (lost interviews, numbers of contacts, patterns of reluctant participation, or refusal to participate) as if these stopping policies had been followed in past data. Finally, embedded survey experiments may be used to assess contact-burden reduction policies when these are implemented in the field. The multi-stage method described here abstracts the stages followed in a series of research studies aimed at reducing contact burden in the Computer Assisted Telephone Interview (CATI) and Computer Assisted Personal Interview (CAPI) modes of the American Community Survey (ACS), which culminated in implementation of policy changes in the ACS.

Bioequivalence (BE) studies are an essential part of the evaluation of generic drugs. The most common in vivo BE study design is the two-period two-treatment crossover design. AUC (area under the concentration-time curve) and Cmax (maximum concentration) are obtained from the observed concentration-time profiles for each subject from each treatment under each sequence. In the BE evaluation of pharmacokinetic crossover studies, the normality of the univariate response variable, e.g. log(AUC)1 or log(Cmax), is often assumed in the literature without much evidence. Therefore, we investigate the distributional assumption of the normality of response variables, log(AUC) and log(Cmax), by simulating concentration-time profiles from two-stage pharmacokinetic models (commonly used in pharmacokinetic research) for a wide range of pharmacokinetic parameters and measurement error structures. Our simulations show that, under reasonable distributional assumptions on the pharmacokinetic parameters, log(AUC) has heavy tails and log(Cmax) is skewed. Sensitivity analyses are conducted to investigate how the distribution of the standardized log(AUC) (or the standardized log(Cmax)) for a large number of simulated subjects deviates from normality if distributions of errors in the pharmacokinetic model for plasma concentrations deviate from normality and if the plasma concentration can be described by different compartmental models.

Dengue fever is a viral disease transmitted by mosquitoes. In recent decades, dengue fever has spread throughout the world. In 2014 and 2015, southern Taiwan experienced its most serious dengue outbreak in recent years. Some statistical models have been established in the past, however, these models may not be suitable for predicting huge outbreaks in 2014 and 2015. The control of dengue fever has become the primary task of local health agencies. This study attempts to predict the occurrence of dengue fever in order to achieve the purpose of timely warning. We applied a newly developed autoregressive model (AR model) to assess the association between daily weather variability and daily dengue case number in 2014 and 2015 in Kaohsiung, the largest city in southern Taiwan. This model also contained additional lagged weather predictors, and developed 5-day-ahead and 15-day-ahead predictive models. Our results indicate that numbers of dengue cases in Kaohsiung are associated with humidity and the biting rate (BR). Our model is simple, intuitive and easy to use. The developed model can be embedded in a "real-time" schedule, and the data (at present) can be updated daily or weekly based on the needs of public health workers. In this study, a simple model using only meteorological factors performed well. The proposed real-time forecast model can help health agencies take public health actions to mitigate the influences of the epidemic., Author summary Meteorological conditions are the most frequently mentioned factors in the study of dengue fever. Some of the main factors other than the purely meteorological about which the public-health authorities might have data, such as numbers of cases or other current measurements of dengue outbreaks in neighboring cities, had been used in some of the past dengue studies. In this study, we developed models for predicting dengue case number based on past dengue case data and meteorological data. The goal of the models is to provide early warning of the occurrence of dengue fever to assist public health agencies in preparing an epidemic response plan.

Drawing on a long history in macroecology, correlation analysis of microbiome datasets is becoming a common practice for identifying relationships or shared ecological niches among bacterial taxa. However, many of the statistical issues that plague such analyses in macroscale communities remain unresolved for microbial communities. Here, we discuss problems in the analysis of microbial species correlations based on presence-absence data. We focus on presence-absence data because this information is more readily obtainable from sequencing studies, especially for whole-genome sequencing, where abundance estimation is still in its infancy. First, we show how Pearson's correlation coefficient (r) and Jaccard's index (J)-two of the most common metrics for correlation analysis of presence-absence data-can contradict each other when applied to a typical microbiome dataset. In our dataset, for example, 14% of species-pairs predicted to be significantly correlated by r were not predicted to be significantly correlated using J, while 37.4% of species-pairs predicted to be significantly correlated by J were not predicted to be significantly correlated using r. Mismatch was particularly common among species-pairs with at least one rare species (

The Survey of Income and Program Participation (SIPP) is a survey with a longitudinal structure and complex nonignorable design, for which correct estimation requires using the weights. The longitudinal setting also suggests conditional-independence relations between survey variables and early- versus late-wave employment classifications. We state original assumptions justifying an extension of the partially model-based approach of Pfeffermann, Skinner and Humphreys [J. Roy. Statist. Soc. Ser. A 161 (1998) 13–32], accounting for the design of SIPP and similar longitudinal surveys. Our assumptions support the use of log-linear models of longitudinal survey data. We highlight the potential they offer for simultaneous bias-control and reduction of sampling error relative to direct methods when applied to small subdomains and cells. Our assumptions allow us to innovate by showing how to rigorously use only a longitudinal survey to estimate a complex log-linear longitudinal association structure and embed it in cross-sectional totals to construct estimators that can be more efficient than direct estimators for small cells.

Background Count data derived from high-throughput deoxy-ribonucliec acid (DNA) sequencing is frequently used in quantitative molecular assays. Due to properties inherent to the sequencing process, unnormalized count data is compositional, measuring relative and not absolute abundances of the assayed features. This compositional bias confounds inference of absolute abundances. Commonly used count data normalization approaches like library size scaling/rarefaction/subsampling cannot correct for compositional or any other relevant technical bias that is uncorrelated with library size. Results We demonstrate that existing techniques for estimating compositional bias fail with sparse metagenomic 16S count data and propose an empirical Bayes normalization approach to overcome this problem. In addition, we clarify the assumptions underlying frequently used scaling normalization methods in light of compositional bias, including scaling methods that were not designed directly to address it. Conclusions Compositional bias, induced by the sequencing machine, confounds inferences of absolute abundances. We present a normalization technique for compositional bias correction in sparse sequencing count data, and demonstrate its improved performance in metagenomic 16s survey data. Based on the distribution of technical bias estimates arising from several publicly available large scale 16s count datasets, we argue that detailed experiments specifically addressing the influence of compositional bias in metagenomics are needed. Electronic supplementary material The online version of this article (10.1186/s12864-018-5160-5) contains supplementary material, which is available to authorized users.

Linear mixed models (LMMs) are widely used for regression analysis of data that are assumed to be clustered or correlated. Assessing model fit is important for valid inference but to date no confirmatory tests are available to assess the adequacy of the fixed effects part of LMMs against general alternatives. We therefore propose a class of goodness-of-fit tests for the mean structure of LMMs. Our test statistic is a quadratic form of the difference between observed values and the values expected under the estimated model in cells defined by a partition of the covariate space. We show that this test statistic has an asymptotic chi-squared distribution when model parameters are estimated by maximum likelihood or by least squares and method of moments, and study its power under local alternatives both analytically and in simulations. Data on repeated measurements of thyroglobulin from individuals exposed to the accident at the Chernobyl power plant in 1986 are used to illustrate the proposed test.

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.

This article reflects the views of the authors and should not be construed to be those of the US Food and Drug Administration.

Summary The problem of accurately estimating the mean-squared error of small area estimators within a Fay–Herriot normal error model is studied theoretically in the common setting where the model is fitted to a logarithmically transformed response variable. For bias-corrected empirical best linear unbiased predictor small area point estimators, mean-squared error formulae and estimators are provided, with biases of smaller order than the reciprocal of the number of small areas. The performance of these mean-squared error estimators is illustrated by a simulation study and a real data example relating to the county level estimation of child poverty rates in the US Census Bureau's on-going ‘Small area income and poverty estimation’ 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.

This paper studies the representation and large-sample consistency for non- parametric maximum likelihood estimators (NPMLEs) of an unknown baseline continuous cumu- lative-hazard-type function and parameter of group survival difference, based on right-censored two-sample survival data with marginal survival function assumed to follow a transformation model, a slight generalization of the class of frailty survival regression models. The paper's main theoretical results are existence and unique a.s. limit, characterized variationally, for large data samples of the NPMLE of baseline nuisance function in an appropriately defined neighbourhood of the true function when the group difference parameter is fixed, leading to consistency of the NPMLE when the difference parameter is fixed at a consistent estimator of its true value. The joint NPMLE is also shown to be consistent. An algorithm for computing it numerically, based directly on likelihood equations in place of the expectation-maximization (EM) algorithm, is illustrated with real data.

We describe an exchange market consisting of many agents with stochastic preferences for two goods. When individuals are indifferent between goods, statistical mechanics predicts that goods and wealth will have steady-state gamma distributions. Simulation studies show that gamma distributions arise for a broader class of preference distributions. We demonstrate this mathematically in the limit of large numbers of individual agents. These studies illustrate the potential power of a statistical mechanical approach to stochastic models in economics and suggest that gamma distributions will describe steady-state wealths for a class of stochastic models with periodic redistribution of conserved quantities. Journal of Economic Literature Classification Numbers: C15, C62, C73, D3, D5.

This paper uses principal components (PC) analysis to represent coronal tongue contours for the 11 vowels of English in two consonant contexts (/s/, /l/), based upon five replicated measurements in three sessions for each of 6 subjects. Curves from multiple sessions and speakers were overlaid before analysis onto a common (x, y) coordinate system by extensive preprocessing of the curves including: extension (padding) or truncation within session, translation, and truncation to a common x range. Four PCs plus a mean level allow accurate representation of coronal tongue curves, but PC shapes depend strongly on the degree of padding or truncation. The PCs successfully reduced the dimensionality of the curves and reflected vowel height, consonant context, and physiological features.

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.

This paper continues the study of the software reliability model of Fakhre- Zakeri & Slud (1995), an "exponential order statistic model" in the sense of Miller (1986) with general mixing distribution, imperfect debugging and large-sample asymptotics reflect- ing increase of the initial number of bugs with software size. The parameters of the model are 0 (proportional to the initial number of bugs in the software), G(., p) (the mixing df, with finite dimensional unknown parameter , for the rates A)i with which the bugs in the software cause observable system failures), and p (the probability with which a detected bug is instantaneously replaced with another bug instead of being removed). Maximum likelihood estimation theory for (0,p,p) is applied to construct a likelihood-based score test for large sample data of the hypothesis of "perfect debugging" (p = 0) vs "imperfect" (p > 0) within the models studied. There are important models (including the Jelinski-Moranda) under which the score statistics with 1/V/i normalization are asymptotically degenerate. These statistics, illustrated on a software reliability data of Musa (1980), can serve nevertheless as important diagnostics for inadequacy of simple models.

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.

A broad array of data series have embedded discrete event?occurrences, cumulative counts of which can be viewed as superpositions $$N(t) = \sum {_{i = 1}^n } N_i (t)$$ of independent identically distributed (iid) counting processes with intensities of the form $$\lambda _i (t) = h(Z_i ,V(t),\vartheta )$$ , where ? is unknown, n may be unknown, Z i are unobservable iid vectors, and V(t) is an observable "environmental" vector process, but the function h is known or hypothesized. It is shown how such data?structures and models arise naturally within the application areas of environmental epidemiology and software reliability, and lead to statistical analyses using novel forms of Poisson regression models with random effects. Such models are potentially very useful in analysis and characterization of teletraffic patterns.

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.

A class of software-reliability mixture-type models is introduced in which individual bugs come with i.i.d. random failure-causation rates /spl lambda/, and have conditional hazard function /spl phi/(t|/spl lambda/) for software failure times. The models allow the possibility of imperfect debugging, in that at each failure a new bug (possibly with another rate-parameter /spl lambda/) is introduced, statistically independently of the past, with probability p. For /spl phi/(t|/spl lambda/)=/spl lambda/, it is shown that the unknown parameters p, n/sub 0/ (the initial number of bugs), and G (the Cdf for /spl lambda/) are uniquely determined from the probability law of the failure-count function (N(t), 0/spl les/t/spl les/spl delta/), for arbitrary /spl delta/>0. The parameters (n/sub 0/,G) are also uniquely determined by the mean failure-count function E{N(t)} when p is known (e.g., is assumed to be 0), but not when p is unknown. For special parametric classes of G, the parameters (n/sub 0/,p,G) are uniquely determined by (E{N(t)}, 0/spl les/t/spl les/spl delta/). >

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.

Randomized two-group clinical survival experiments now commonly allow at least one interim look, enabling possible early stopping in order to meet ethical concerns. Various authors have also studied the possibility of interim design modifications to adapt to unexpected accrual or control-group mortality rates. This chapter formulates trial design as a decision-theoretic problem with a finite number of interim looks and a large class of loss functions, in the setting of a statistic with the asymptotic behavior of Brownian motion with drift, as in Leifer and Slud (2002). A more general action space can specify adaptive designs allowing the option of continued follow-up without new accrual past an interim look, as was introduced in Koutsoukos et al. (1998). An optimal two-look design is displayed in the first formulation, and a seven-look design in the second, and both types of adaptation are given a unified decision-theoretic motivation.

In many areas of application, one searches within finite populations for items of interest, where the probability of sampling an item is proportional to a random size attribute from an i.i.d. superpopulation of attributes which may or may not be observable upon discovery. Here we treat the problem of asymptotically optimal stopping rules for size-dependent searches of this type, as the size of the underlying population grows, where the loss function includes an asymptotically smooth time-dependent cost, a constant cost per item sampled and a cost per undiscovered item which may depend on the size attribute of the undiscovered item. Under some regularity and convexity conditions related to the asymptotic expected loss, we characterize asymptotically optimal rules even when the initial population size and the distribution of size attributes are unknown. We direct especial attention to applications in software reliability, where the items of interest are software faults ("bugs"). In this setting, the size attributes will not be observable when faults are found, and, in addition, our search model allows new bugs to be introduced into the software when faults are detected "imperfect debugging"). Our results extend those of Dalal and Mallows and Kramer and Starr, and are illustrated in the perfect-debugging case on a previously analyzed dataset of Musa.

This paper discusses two mechanisms with time-dependent covariates for dependence between competing-risk latent failure times under which the marginal survival functions are identifiable. The first is the finite-state nonhomogeneous Markov chain with two absorbing failure states, A and B, where “marginal survival function for A” is the probability that failure due to A does not occur before t when transitions to B are suppressed. Even in highly stratified models, the nonparametric survival estimator due to Aalen & Johansen (1978) is readily computable when transitions between each pair (i,j) states can occur only in one direction.

Let $\mathbf{X} = (X_t, t \geq 0)$ be a stationary Gaussian process with zero mean, continuous spectral distribution and twice-differentiable correlation function. An explicit representation is given for the number $N_\psi(T)$ of crossings of a $C^1$ curve $\psi$ by $\mathbf{X}$ on the bounded interval $\lbrack 0, T\rbrack$, in a multiple Wiener-Ito integral expansion. This continues work of the author in which the result was given for $\psi \equiv 0$. The representation is applied to prove new central and noncentral limit theorems for numbers of crossings of constant levels, and some consequences for asymptotic variances are given in mixed-spectrum settings.

Let $Y$ be a random variable defined by a polynomial $p(W)$ of degree $n$ in finitely many normally distributed variables. This paper studies which such variables $Y$ are "determinate," i.e., have probability laws uniquely determined by their moments. Extending results of Berg, which applied to powers of a single normal variable, we prove that (a) $Y$ is determinate if $n = 1, 2$ or if $n = 4$, with the essential support of the law of $Y$ strictly smaller than the real line, and (b) $Y$ is not determinate either if $n$ is odd $\geq 3$ or if $n$ is even $\geq 6$ such that $p(\mathbf{w})$ attains a finite minimum value. Some other polynomials $Y = p(\mathbf{W})$ with even degree $n \geq 4$ are proved not to be determinate.

It is well known that survival data randomly censored from the right by deaths from a competing risk do not allow nonparametric identifiability of marginal survival distributions when survival times and competing-risk censoring times are dependent (Tsiatis 1975). Parametric models for the joint distribution of survival and competing-risk censoring times cannot solve the problem since the goodness of fit of such models cannot be tested with observable data. Nevertheless, there are many such settings where marginal “latent” survival distributions are a desirable object of inference, expressing most clearly the underlying biological failure mechanism disentangled from physiologically distinct effects. One hope to overcome the obstacle of nonidentifiability is to make use of observable covariate data which are prognostic only for the latent survival times and not for the competing risk. In this paper, it is shown how the marginal distribution of the latent survival time T can be nonparametrically identifiable when only the data min(T, C), I[T≤C], and V are observed, where C is a latent competing-risk censoring time and V is an observed covariate such that C and V are conditionally independent given T.

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.

The asymptotic distribution theory of sequentially computed modified-Wilcoxon scores is developed for two-sample survival data with random staggered entry and random loss to follow-up. The asymptotic covariance indicates generally dependent modified-Wilcoxon increments, contradicting (the authors' reading of) Jones and Whitehead (1979). A repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory. The early stopping properties of this procedure are illustrated by a prostate cancer example.

Asymptotic properties of the partitions of the unit interval induced by some special random sequences of points are studied from two points of view. The partition-entropy and largest spacings for independent identically distributed (i.i.d.) uniform sequences in [0,1] are first compared with the corresponding quantities for a random sequence generated by a scheme of S. Kakutani. A general combinatorial technique exploiting the tree structure of interval splittings is then introduced to prove almost-sure (a.s.) equidistribution of the points of Kakutani's construction, along with a rate of convergence of empirical measures to Lebesgue measure and such other results as a.s. equidistribution of k-tuples in [0,1]k

Let X=(X t ,t∈ℝ) be a stationary Gaussian process on (Ω, ℱ,P), letH(X) be the Hilbert space of variables inL 2 (Ω,P) which are measurable with respect toX, and let (U s ,s∈ℝ) be the associated family of time-shift operators. We sayY∈H(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively, ], where $$Y_T (t) \equiv {{\int\limits_0^{Tt} {U_s \circ Yds} } \mathord{\left/ {\vphantom {{\int\limits_0^{Tt} {U_s \circ Yds} } {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right. \kern-\nulldelimiterspace} {\left\{ {Var\left( {\int\limits_0^T {U_s \circ Yds} } \right)} \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}$$ andW(•) is a standard Wiener process on [0,1]. This paper provides some general sufficient conditions onX andY ensuring thatY satisfies the CLT or FCLT. Examples ofY are given which satisfy the CLT but not the FCLT. This work extends CLT's of Maruyama (1976) and Breuer and Major (1983).