Nonparametric Estimation Procedure Research Articles

On the optimal reconstruction of partially observed functional data

We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed at $m_{i}$, $i=1,\dots ,n$, discretization points. We derive rates of consistency for our nonparametric estimation procedures using a double asymptotic. For data situations, as in our real data application where $m_{i}$ is considerably smaller than $n$, we show that our functional principal components based estimator can provide better rates of convergence than conventional nonparametric smoothing methods.

Adaptive nonparametric regression on finite support

In this paper, we propose an adaptive nonparametric regression estimation procedure when the covariate is supported over a finite interval. Unlike the classical symmetric kernel regression, the proposed kernel estimator is based on the Beta density function. Its large sample properties, including the asymptotic normality and the uniform convergence, are thoroughly investigated. Meanwhile, general data-driven guidelines for the bandwidth selection are suggested. The finite sample performance of the proposed estimator is evaluated via a simulation study and a real data application.

Bootstrap bandwidth selection in time-varying coefficient models with jumps

The time-varying coefficient models (TVCMs) are very important tools to explore the hidden structure between the time series response variable and its predictors. The assumption that the coefficient functions are smooth can be restrictive in practice. The TVCMs with jumps are needed to be considered. Selection of smoothing parameters plays a critical role in assessing the performance of estimations of coefficient functions. In this article, we first develop a nonparametric two-step estimation procedure for estimating a finite number of jumps of the coefficient functions. Then, based on the estimated jumps, we propose a bootstrapping bandwidth selection procedure in the TVCMs with jumps. Monte Carlo simulations are conducted to evaluate the finite sample performance of the proposed data-driven estimation and bootstrap bandwidth selection procedures. As an illustration, the proposed methodologies are further applied to a real data example.

PAFit: An R Package for the Non-Parametric Estimation of Preferential Attachment and Node Fitness in Temporal Complex Networks

Many real-world systems are profitably described as complex networks that grow over time. Preferential attachment and node fitness are two simple growth mechanisms that not only explain certain structural properties commonly observed in real-world systems, but are also tied to a number of applications in modeling and inference. While there are statistical packages for estimating various parametric forms of the preferential attachment function, there is no such package implementing non-parametric estimation procedures. The non-parametric approach to the estimation of the preferential attachment function allows for comparatively finer-grained investigations of the `rich-get-richer' phenomenon that could lead to novel insights in the search to explain certain nonstandard structural properties observed in real-world networks. This paper introduces the R package PAFit, which implements non-parametric procedures for estimating the preferential attachment function and node fitnesses in a growing network, as well as a number of functions for generating complex networks from these two mechanisms. The main computational part of the package is implemented in C++ with OpenMP to ensure scalability to large-scale networks. We first introduce the main functionalities of PAFit through simulated examples, and then use the package to analyze a collaboration network between scientists in the field of complex networks. The results indicate the joint presence of `rich-get-richer' and `fit-get-richer' phenomena in the collaboration network. The estimated attachment function is observed to be near-linear, which we interpret as meaning that the chance an author gets a new collaborator is proportional to their current number of collaborators. Furthermore, the estimated author fitnesses reveal a host of familiar faces from the complex networks community among the field's topmost fittest network scientists.

Model-free approach to quantifying the proportion of treatment effect explained by a surrogate marker.

In randomized clinical trials, the primary outcome, Y, often requires long-term follow-up and/or is costly to measure. For such settings, it is desirable to use a surrogate marker, S, to infer the treatment effect on Y, Δ. Identifying such an S and quantifying the proportion of treatment effect on Y explained by the effect on S are thus of great importance. Most existing methods for quantifying the proportion of treatment effect are model based and may yield biased estimates under model misspecification. Recently proposed nonparametric methods require strong assumptions to ensure that the proportion of treatment effect is in the range [0, 1]. Additionally, optimal use of S to approximate Δ is especially important when S relates to Y nonlinearly. In this paper we identify an optimal transformation of S, g opt(·), such that the proportion of treatment effect explained can be inferred based on g opt(S). In addition, we provide two novel model-free definitions of proportion of treatment effect explained and simple conditions for ensuring that it lies within [0, 1]. We provide nonparametric estimation procedures and establish asymptotic properties of the proposed estimators. Simulation studies demonstrate that the proposed methods perform well in finite samples. We illustrate the proposed procedures using a randomized study of HIV patients.

Analysis and data-driven reconstruction of bivariate jump-diffusion processes

We introduce the bivariate jump-diffusion process, consisting of two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. We present a data-driven, nonparametric estimation procedure of higher-order (up to 8) Kramers-Moyal coefficients that allows one to reconstruct relevant aspects of the underlying jump-diffusion processes and to recover the underlying parameters. The procedure is validated with numerically integrated data using synthetic bivariate time series from continuous and discontinuous processes. We further evaluate the possibility of estimating the parameters of the jump-diffusion model via data-driven analyses of the higher-order Kramers-Moyal coefficients, and the limitations arising from the scarcity of points in the data or disproportionate parameters in the system.

Regression analysis of misclassified current status data

Misclassified current status data occur when each subject under study is observed only once and the failure status at the observation time is determined by a diagnostic test with imperfect sensitivity and specificity. In this article, we provide a methodology for the analysis of such data under a wide class of flexible semiparametric transformation models. For inference, a nonparametric maximum likelihood estimation procedure is proposed along with the development of an EM algorithm. Furthermore, we show that the resulting estimators of regression parameters are consistent, asymptotically normal and semiparametrically efficient. A simulation study and a real data application demonstrate that the proposed approach performs well in practice and has substantial superiority over the naive method that ignores the misclassification.

Quantifying treatment effects using the personalized chance of longer survival.

The hazard ratio is widely used to measure or to summarize the magnitude of treatment effects, but it is justifiably difficult to interpret in a meaningful way to patients and perhaps for clinicians as well. In addition, it is most meaningful when the hazard functions are approximately proportional over time. We propose a new measure, termed personalized chance of longer survival. The measure, which quantifies the probability of living longer with one treatment over the another, accounts for individualized characteristics to directly address personalized treatment effects. Hence, the measure is patient focused, which can be used to evaluate subgroups easily. We believe it is intuitive to understand and clinically interpretable in the presence of nonproportionality. Furthermore, because it estimates the probability of living longer by some fixed amount of time, it encodes the probabilistic part of treatment effect estimation. We provide nonparametric estimation and inference procedures that can accommodate censored survival outcomes. We conduct extensive simulation studies, which characterize performance of the proposed method, and data from a large randomized Phase III clinical trial (SWOG S0819) are analyzed using the proposed method.

Dimension reduction for integrating data series in Bayesian inversion of geostatistical models

This study explores methods with which multidimensional data, e.g. time series, can be effectively incorporated into a Bayesian framework for inferring geostatistical parameters. Such series are difficult to use directly in the likelihood estimation procedure due to their high dimensionality; thus, a dimension reduction approach is taken to utilize these measurements in the inference. Two synthetic scenarios from hydrology are explored in which pumping drawdown and concentration breakthrough curves are used to infer the global mean of a log-normally distributed hydraulic conductivity field. Both cases pursue the use of a parametric model to represent the shape of the observed time series with physically-interpretable parameters (e.g. the time and magnitude of a concentration peak), which is compared to subsets of the observations with similar dimensionality. The results from both scenarios highlight the effectiveness for the shape-matching models to reduce dimensionality from 100+ dimensions down to less than five. The models outperform the alternative subset method, especially when the observations are noisy. This approach to incorporating time series observations in the Bayesian framework for inferring geostatistical parameters allows for high-dimensional observations to be faithfully represented in lower-dimensional space for the non-parametric likelihood estimation procedure, which increases the applicability of the framework to more observation types. Although the scenarios are both from hydrogeology, the methodology is general in that no assumptions are made about the subject domain. Any application that requires the inference of geostatistical parameters using series in either time of space can use the approach described in this paper.

Using a surrogate marker for early testing of a treatment effect.

The development of methods to identify, validate, and use surrogate markers to test for a treatment effect has been an area of intense research interest given the potential for valid surrogate markers to reduce the required costs and follow-up times of future studies. Several quantities and procedures have been proposed to assess the utility of a surrogate marker. However, few methods have been proposed to address how one might use the surrogate marker information to test for a treatment effect at an earlier time point, especially in settings where the primary outcome and the surrogate marker are subject to censoring. In this paper, we propose a novel test statistic to test for a treatment effect using surrogate marker information measured prior to the end of the study in a time-to-event outcome setting. We propose a robust nonparametric estimation procedure and propose inference procedures. In addition, we evaluate the power for the design of a future study based on surrogate marker information. We illustrate the proposed procedure and relative power of the proposed test compared to a test performed at the end of the study using simulation studies and an application to data from the Diabetes Prevention Program.

Time‐change models for asymmetric processes

AbstractMany records in environmental sciences exhibit asymmetric trajectories. The physical mechanisms behind these records may lead for example to sample paths with different characteristics at high and low levels (up–down asymmetries) or in the ascending and descending phases leading to time irreversibility (front–back asymmetries). Such features are important for many applications, and there is a need for simple and tractable models that can reproduce them. In this paper, we explore original time‐change models where the clock is a stochastic process that depends on the observed trajectory. The ergodicity of the proposed model is established under general conditions, and this result is used to develop nonparametric estimation procedures based on the joint distribution of the process and its derivative. The methodology is illustrated on meteorological and oceanographic data sets. We show that, combined with a marginal transformation, the proposed methodology is able to reproduce important characteristics of the data set such as marginal distributions, up‐crossing intensity, and up–down and front–back asymmetries.

A comparative study on nonparametric estimation procedures for survival quantiles

In survival or reliability data analysis, it is often useful to estimate the quantiles of the lifetime distribution, such as the median time to failure. Different nonparametric methods can construct confidence intervals for the quantiles of the lifetime distributions, some of which are implemented in commonly used statistical software packages. We here investigate the performance of different interval estimation procedures under a variety of settings with different censoring schemes. Our main objectives in this paper are to (i) evaluate the performance of confidence intervals based on the transformation approach commonly used in statistical software, (ii) introduce a new density-estimation-based approach to obtain confidence intervals for survival quantiles, and (iii) compare it with the transformation approach. We provide a comprehensive comparative study and offer some useful practical recommendations based on our results. Some numerical examples are presented to illustrate the methodologies developed.

Auctions with ex post uncertainty

AbstractUncertainty about ex post realized values is an inherent component in many auction environments. In this article, we develop a structural framework to analyze auction data subject to ex post uncertainty as a pure risk. We consider a low‐price sealed‐bid auction model with heterogeneous bidders' preferences and ex post uncertainty. The uncertainty can be common to all bidders or idiosyncratic. We derive the model restrictions and study nonparametric and semiparametric identification of the model primitives under exogenous and endogenous participation. We then develop multistep nonparametric and semiparametric estimation procedures in both cases.

Time‐Dependent Dual‐Frequency Coherence in Multivariate Non‐Stationary Time Series

Coherence is one common metric for cross‐dependence in multichannel signals. However, standard coherence does not sufficiently model many biological signals with complex dependence structures such as cross‐oscillatory interactions between a low‐frequency component in one signal and a high‐frequency component in another. The notion of cross‐dependence between low‐ and high‐frequency components, as defined in classical harmonizable processes, is still inadequate because it assumes time invariance and thus cannot capture cross‐frequency interactions that evolve over time. We construct a novel framework for modeling and estimating these dependencies under the replicated time series setting. Under this framework, we establish the novel concept of evolutionary dual‐frequency coherence and develop time‐localized estimators based on dual‐frequency local periodograms. The proposed nonparametric estimation procedure does not suffer from model misspecification. It uses the localized fast Fourier transform and hence is able to handle massive data. When applied to electroencephalogram data recorded in a motor intention experiment, the proposed method uncovers new and interesting cross‐oscillatory interactions that have been overlooked by the standard approaches.

CALIBRATION OF LÉVY PROCESSES USING OPTIMAL CONTROL OF KOLMOGOROV EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS

We present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4]. We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC) [12]. The first order necessary optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution.

Causal mediation analysis with multiple mediators in the presence of treatment noncompliance.

Randomized experiments are often complicated because of treatment noncompliance. This challenge prevents researchers from identifying the mediated portion of the intention-to-treated (ITT) effect, which is the effect of the assigned treatment that is attributed to a mediator. One solution suggests identifying the mediated ITT effect on the basis of the average causal mediation effect among compliers when there is a single mediator. However, considering the complex nature of the mediating mechanisms, it is natural to assume that there are multiple variables that mediate through the causal path. Motivated by an empirical analysis of a data set collected in a randomized interventional study, we develop a method to estimate the mediated portion of the ITT effect when both multiple dependent mediators and treatment noncompliance exist. This enables researchers to make an informed decision on how to strengthen the intervention effect by identifying relevant mediators despite treatment noncompliance. We propose a nonparametric estimation procedure and provide a sensitivity analysis for key assumptions. We conduct a Monte Carlo simulation study to assess the finite sample performance of the proposed approach. The proposed method is illustrated by an empirical analysis of JOBS II data, in which a job training intervention was used to prevent mental health deterioration among unemployed individuals.

Semiparametric regression analysis of clustered survival data with semi-competing risks

Analysis of semi-competing risks data is becoming increasingly important in medical research in which a subject may experience both nonterminal and terminal events, and the time to the intermediate nonterminal event (e.g. onset of a disease) is subject to dependent censoring by the terminal event (e.g. death) but not vice versa. Typically, both two types of events are dependent. In many applications, subjects may also be nested within clusters, such as patients in a multi-center study, leading to possible association among event times due to unobserved shared factors across subjects. To incorporate dependency within clusters and association between two types of event times, we propose a new flexible semiparametric modeling framework where a copula model is employed for the joint distribution of the nonterminal and terminal events, and their marginal distributions are modeled by Cox proportional hazards models with random effects. A nonparametric maximum likelihood estimation procedure is developed and implemented through a Monte Carlo EM algorithm. The proposed estimator is also shown to enjoy desirable asymptotic properties. Results from extensive simulation studies indicate that the proposed method performs very well in finite samples and is especially robust against misspecification of the random effects distribution. We further illustrate the practical utility of the method by analyzing data from a multi-institutional study of breast cancer.

Beyond precision: accelerated life testing for fuzzy life time data

Reliability analysis comprises statistical analysis techniques that make inferences based on life time data. Swift progress has been observed in life time data analyses during the last few decades. Accelerated life testing models are regarded as the most popular techniques for engineering life time data analysis. Their main aim is to model life times under different stress levels that are more severe than the usual stress level. The existing techniques consider life times as precise measurements and do not contemplate the imprecision of observations. In fact, life time measurements are not precise quantities but more or less fuzzy. Therefore, in addition to standard statistical tools, fuzzy model approaches are also essential. The current study generalizes some parametric and nonparametric classical estimation procedures for accelerated life testing in order to accommodate both fuzziness and random variation. The proposed estimators cover both uncertainties, which make them more applicable and practicable for life time analysis. The results of fuzzy life times are considered under various stress conditions, and comparisons with precise life time analysis are further presented in examples.

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

ABSTRACTIn earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.

Nonparametric estimation of multivariate quantiles

In many applications of hydrology, quantiles provide important insights in the statistical problems considered. In this paper, we focus on the estimation of multivariate quantiles based on copulas. We provide a nonparametric estimation procedure for a notion of multivariate quantiles, which has been used in a series of papers. These quantiles are based on particular level sets of copulas and admit the usual probabilistic interpretation that a p‐quantile comprises a probability mass p. We also explore the usefulness of a smoothed bootstrap in the estimation process. Our simulation results show that the nonparametric estimation procedure yields excellent results and that the smoothed bootstrap can be beneficially applied. The main purpose of our paper is to provide an easily applicable method for practitioners and applied researchers in domains such as hydrology and coastal engineering.