Abstract.  We investigate non-parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non-decreasing baseline hazard function are proposed. We derive the non-parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non-increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non-increasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.

Abstract.  A common practice in obtaining an efficient semiparametric estimate is through iteratively maximizing the (penalized) full log-likelihood w.r.t. its Euclidean parameter and functional nuisance parameter. A rigorous theoretical study of this semiparametric iterative estimation approach is the main purpose of this study. We first show that the grid search algorithm produces an initial estimate with the proper convergence rate. Our second contribution is to provide a formula in calculating the minimal number of iterations k^* needed to produce an efficient estimate . We discover that (i) k^* depends on the convergence rates of the initial estimate and the nuisance functional estimate, and (ii) k^* iterations are also sufficient for recovering the estimation sparsity in high dimensional data. The last contribution is the novel construction of which does not require knowing the explicit expression of the efficient score function. The above general conclusions apply to semiparametric models estimated under various regularizations, for example, kernel or penalized estimation. As far as we are aware, this study provides a first general theoretical justification for the ‘one-/two-step iteration’ phenomena observed in the semiparametric literature.

Abstract.  To use control charts in practice, the in-control state usually has to be estimated. This estimation has a detrimental effect on the performance of control charts, which is often measured by the false alarm probability or the average run length. We suggest an adjustment of the monitoring schemes to overcome these problems. It guarantees, with a certain probability, a conditional performance given the estimated in-control state. The suggested method is based on bootstrapping the data used to estimate the in-control state. The method applies to different types of control charts, and also works with charts based on regression models. If a non-parametric bootstrap is used, the method is robust to model errors. We show large sample properties of the adjustment. The usefulness of our approach is demonstrated through simulation studies.

Abstract.  For several independent multivariate bioassays performed at different laboratories or locations, the problem of testing the homogeneity of the relative potencies is addressed, assuming the usual slope-ratio or parallel line assay model. When the homogeneity hypothesis holds, interval estimation of the common relative potency is also addressed. These problems have been investigated in the literature using likelihood-based methods, under the assumption of a common covariance matrix across the different studies. This assumption is relaxed in this investigation. Numerical results show that the usual likelihood-based procedures are inaccurate for both of the above problems, in terms of providing inflated type I error probabilities for the homogeneity test, and providing coverage probabilities below the nominal level for the interval estimation of the common relative potency, unless the sample sizes are large, as expected. Correction based on small sample asymptotics is investigated in this article, and this provides significantly more accurate results in the small sample scenario. The results are also illustrated with examples.

Abstract.  This study gives a generalization of Birch's log-linear model numerical invariance result. The generalization is given in the form of a sufficient condition for numerical invariance that is simple to verify in practice and is applicable for a much broader class of models than log-linear models. Unlike Birch's log-linear result, the generalization herein does not rely on any relationship between sufficient statistics and maximum likelihood estimates. Indeed the generalization does not rely on the existence of a reduced set of sufficient statistics. Instead, the concept of homogeneity takes centre stage. Several examples illustrate the utility of non-log-linear models, the invariance (and non-invariance) of fitted values, and the invariance (and non-invariance) of certain approximating distributions.

Abstract.  A substantive problem in neuroscience is the lack of valid statistical methods for non-Gaussian random fields. In the present study, we develop a flexible, yet tractable model for a random field based on kernel smoothing of a so-called Lévy basis. The resulting field may be Gaussian, but there are many other possibilities, including random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. It is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of test statistics commonly used in neuroscience. We give a concrete example of magnetic resonance imaging scans that are non-Gaussian. For these data, simulations under the fitted models show that traditional methods based on Gaussian random field theory may leave small, but significant changes in signal level undetected, while these changes are detectable under a non-Gaussian Lévy model.

Abstract.  A simple summary of a treatment effect is attractive, which is part of the explanation of the success of the Cox model when analysing time-to-event data since the relative risk measure is such a convenient summary measure. In practice, however, the Cox model may fail to give a reasonable fit, very often because of time-changing treatment effect. The Aalen additive hazards model may be a good alternative as time-changing effects are easily modelled within this model, but results are then evidently more complicated to communicate. In such situations, the odds of concordance measure (OC) is a convenient way of communicating results, and recently Martinussen & Pipper (2012) showed how a variant of the OC measure may be estimated based on the Aalen additive hazards model. In this study, we propose an estimator that should be preferred in observational studies as it always estimates the causal effect on the chosen scale, only assuming that there are no un-measured confounders. The resulting estimator is shown to be consistent and asymptotically normal, and an estimator of its limiting variance is provided. Two real applications are provided.

Abstract.  General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non-Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M-estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non-Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996), we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M-estimator. We also consider bootstrapping the M-estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M-estimation and bootstrap procedures. An empirical example of financial time series is also provided.

Abstract.  Similar to variable selection in the linear model, selecting significant components in the additive model is of great interest. However, such components are unknown, unobservable functions of independent variables. Some approximation is needed. We suggest a combination of penalized regression spline approximation and group variable selection, called the group-bridge-type spline method (GBSM), to handle this component selection problem with a diverging number of correlated variables in each group. The proposed method can select significant components and estimate non-parametric additive function components simultaneously. To make the GBSM stable in computation and adaptive to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.

Abstract.  In the context of multivariate mean regression, we propose a new method to measure and estimate the inadequacy of a given parametric model. The measure is basically the missed fraction of variation after adjusting the best possible parametric model from a given family. The proposed approach is based on the minimum L²-distance between the true but unknown regression curve and a given model. The estimation method is based on local polynomial averaging of residuals with a polynomial degree that increases with the dimension d of the covariate. For any d  1 and under some weak assumptions we give a Bahadur-type representation of the estimator from which -consistency and asymptotic normality are derived for strongly mixing variables. We report the outcomes of a simulation study that aims at checking the finite sample properties of these techniques. We present the analysis of a dataset on ultrasonic calibration for illustration.

Abstract. We give a rigorous study of weak convergence of the wild bootstrap for non-parametric estimation of the cumulative event probability of a competing risk. The data may be subject to independent left-truncation and right-censoring. Inclusion of left-truncation is motivated by a study on pregnancy outcomes. The wild bootstrap includes as one case a popular resampling technique, where the limit distribution is approximated by repeatedly generating standard normal variates, while the data are kept fixed. Simulation results and a data example are also presented.

ABSTRACT. In cancer diagnosis studies, high-throughput gene profiling has been extensively conducted, searching for genes whose expressions may serve as markers. Data generated from such studies have the ‘large d, small n’ feature, with the number of genes profiled much larger than the sample size. Penalization has been extensively adopted for simultaneous estimation and marker selection. Because of small sample sizes, markers identified from the analysis of single data sets can be unsatisfactory. A cost-effective remedy is to conduct integrative analysis of multiple heterogeneous data sets. In this article, we investigate composite penalization methods for estimation and marker selection in integrative analysis. The proposed methods use the minimax concave penalty (MCP) as the outer penalty. Under the homogeneity model, the ridge penalty is adopted as the inner penalty. Under the heterogeneity model, the Lasso penalty and MCP are adopted as the inner penalty. Effective computational algorithms based on coordinate descent are developed. Numerical studies, including simulation and analysis of practical cancer data sets, show satisfactory performance of the proposed methods.

Abstract.  In this paper, we consider two kinds of collapsibility, that is, the model-collapsibility and the estimate-collapsibility, of conditional graphical models for multidimensional contingency tables. We show that these two definitions are equivalent, and propose a sufficient and necessary condition for them in terms of the interaction graph, which allows the collapsibility to be characterized and judged intuitively and conveniently.

Abstract. Motivated by applications of Poisson processes for modelling periodic time-varying phenomena, we study a semi-parametric estimator of the period of cyclic intensity function of a non-homogeneous Poisson process. There are no parametric assumptions on the intensity function which is treated as an infinite dimensional nuisance parameter. We propose a new family of estimators for the period of the intensity function, address the identifiability and consistency issues and present simulations which demonstrate good performance of the proposed estimation procedure in practice. We compare our method to competing methods on synthetic data and apply it to a real data set from a call center.

Abstract.  Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.

Abstract.  Many statistical models arising in applications contain non- and weakly-identified parameters. Due to identifiability concerns, tests concerning the parameters of interest may not be able to use conventional theories and it may not be clear how to assess statistical significance. This paper extends the literature by developing a testing procedure that can be used to evaluate hypotheses under non- and weakly-identifiable semiparametric models. The test statistic is constructed from a general estimating function of a finite dimensional parameter model representing the population characteristics of interest, but other characteristics which may be described by infinite dimensional parameters, and viewed as nuisance, are left completely unspecified. We derive the limiting distribution of this statistic and propose theoretically justified resampling approaches to approximate its asymptotic distribution. The methodology's practical utility is illustrated in simulations and an analysis of quality-of-life outcomes from a longitudinal study on breast cancer.

Abstract.  Parameter estimation in diffusion processes from discrete observations up to a first-passage time is clearly of practical relevance, but does not seem to have been studied so far. In neuroscience, many models for the membrane potential evolution involve the presence of an upper threshold. Data are modelled as discretely observed diffusions which are killed when the threshold is reached. Statistical inference is often based on a misspecified likelihood ignoring the presence of the threshold causing severe bias, e.g. the bias incurred in the drift parameters of the Ornstein–Uhlenbeck model for biological relevant parameters can be up to 25–100 per cent. We compute or approximate the likelihood function of the killed process. When estimating from a single trajectory, considerable bias may still be present, and the distribution of the estimates can be heavily skewed and with a huge variance. Parametric bootstrap is effective in correcting the bias. Standard asymptotic results do not apply, but consistency and asymptotic normality may be recovered when multiple trajectories are observed, if the mean first-passage time through the threshold is finite. Numerical examples illustrate the results and an experimental data set of intracellular recordings of the membrane potential of a motoneuron is analysed.

ABSTRACT.  We develop exact Markov chain Monte Carlo methods for discretely sampled, directly and indirectly observed diffusions. The qualification ‘exact’ refers to the fact that the invariant and limiting distribution of the Markov chains is the posterior distribution of the parameters free of any discretization error. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. The article introduces various methods to boost the performance of the basic scheme, including reparametrizations and auxiliary Poisson sampling. We contrast both theoretically and empirically how this new approach compares to irreducible high frequency imputation, which is the state-of-the-art alternative for the class of processes we consider, and we uncover intriguing connections. All methods discussed in the article are tested on typical examples.

Abstract.  We consider N independent stochastic processes (X_i (t), t ∈  [0,T_i]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ_i. The distribution of the random effect φ_i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ_i and φ_i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T_i=T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.

Abstract.  Let M be an isotonic real-valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection-type estimators are r_n-equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability-weighted moment, quantile) and biometric (mean remaining lifetime) functions.