For a one dimensional diffusion process $X=\{X(t) ; 0\leq t \leq T \}$, we suppose that $X(t)$ is hidden if it is below some fixed and known threshold $τ$, but otherwise it is visible. This means a partially hidden diffusion process. The problem treated in this paper is to estimate finite dimensional parameter in both drift and diffusion coefficients under a partially hidden diffusion process obtained by a discrete sampling scheme. It is assumed that the sampling occurs at regularly spaced time intervals of length $h_n$ such that $n h_n=T$. The asymptotic is when $h_n\to0$, $T\to\infty$ and $n h_n^2\to 0$ as $n\to\infty$. Consistency and asymptotic normality for estimators of parameters in both drift and diffusion coefficients are proved.
This paper introduces a family of recursively defined estimators of the parameters of a diffusion process. We use ideas of stochastic algorithms for the construction of the estimators. Asymptotic consistency of these estimators and asymptotic normality of an appropriate normalization are proved. The results are applied to two examples from the financial literature; viz., Cox-Ingersoll-Ross' model and the constant elasticity of variance (CEV) process illustrate the use of the technique proposed herein.
In this paper, we derive minimax rates for estimating both parametric and nonparametric components in partially linear additive models with high dimensional sparse vectors and smooth functional components. The minimax lower bound for Euclidean components is the typical sparse estimation rate that is independent of nonparametric smoothness indices. However, the minimax lower bound for each component function exhibits an interplay between the dimensionality and sparsity of the parametric component and the smoothness of the relevant nonparametric component. Indeed, the minimax risk for smooth non…
We consider parametric estimation and tests for multi-dimensional diffusion processes with a small dispersion parameter $\varepsilon$ from discrete observations. For parametric estimation of diffusion processes, the main target is to estimate the drift parameter and the diffusion parameter. In this paper, we propose two types of adaptive estimators for both parameters and show their asymptotic properties under $\varepsilon\to0$, $n\to\infty$ and the balance condition that $(\varepsilon n^ρ)^{-1} =O(1)$ for some $ρ>0$. Using these adaptive estimators, we also introduce consistent adaptive testi…
It is widely admitted that structured nonparametric modeling that circumvents the curse of dimensionality is important in nonparametric estimation. In this paper we show that the same holds for semi-parametric estimation. We argue that estimation of the parametric component of a semi-parametric model can be improved essentially when more structure is put into the nonparametric part of the model. We illustrate this for the partially linear model, and investigate efficiency gains when the nonparametric part of the model has an additive structure. We present the semi-parametric Fisher information…
We study semiparametric varying-coefficient partially linear models when some linear covariates are not observed, but ancillary variables are available. Semiparametric profile least-square based estimation procedures are developed for parametric and nonparametric components after we calibrate the error-prone covariates. Asymptotic properties of the proposed estimators are established. We also propose the profile least-square based ratio test and Wald test to identify significant parametric and nonparametric components. To improve accuracy of the proposed tests for small or moderate sample size…
We propose a spectral mean for closed curves described by sample points on its boundary subject to mis-alignment and noise. First, we ignore mis-alignment and derive maximum likelihood estimators of the model and noise parameters in the Fourier domain. We estimate the unknown curve by back-transformation and derive the distribution of the integrated squared error. Then, we model mis-alignment by means of a shifted parametric diffeomorphism and minimise a suitable objective function simultaneously over the unknown curve and the mis-alignment parameters. Finally, the method is illustrated on sim…
A partially linear probit model for spatially dependent data is considered. A triangular array setting is used to cover various patterns of spatial data. Conditional spatial heteroscedasticity and non-identically distributed observations and a linear process for disturbances are assumed, allowing various spatial dependencies. The estimation procedure is a combination of a weighted likelihood and a generalized method of moments. The procedure first fixes the parametric components of the model and then estimates the non-parametric part using weighted likelihood; the obtained estimate is then use…
The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed …
In a cross-sectional study, adolescent and young adult females were asked to recall the time of menarche, if experienced. Some respondents recalled the date exactly, some recalled only the month or the year of the event, and some were unable to recall anything. We consider estimation of the menarcheal age distribution from this interval censored data. A~complicated interplay between age-at-event and calendar time, together with the evident fact of memory fading with time, makes the censoring informative. We propose a model where the probabilities of various types of recall would depend on the …
We propose an empirical likelihood test that is able to test the goodness of fit of a class of parametric and semi-parametric multiresponse regression models. The class includes as special cases fully parametric models; semi-parametric models, like the multiindex and the partially linear models; and models with shape constraints. Another feature of the test is that it allows both the response variable and the covariate be multivariate, which means that multiple regression curves can be tested simultaneously. The test also allows the presence of infinite-dimensional nuisance functions in the mo…
In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-…
We consider parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) from high frequency data which are observed in time and space. By using thinned data obtained from the high frequency data, adaptive estimators of the coefficient parameters including the volatility parameter of a parabolic linear SPDE model are proposed. Moreover, we give some examples and simulation results of the adaptive estimators of the SPDE model based on the high frequency data.
We propose a two-step estimating procedure for generalized additive partially linear models with clustered data using estimating equations. Our proposed method applies to the case that the number of observations per cluster is allowed to increase with the number of independent subjects. We establish oracle properties for the two-step estimator of each function component such that it performs as well as the univariate function estimator by assuming that the parametric vector and all other function components are known. Asymptotic distributions and consistency properties of the estimators are ob…
Multiple matrix sampling is a survey methodology technique that randomly chooses a relatively small subset of items to be presented to survey respondents for the purpose of reducing respondent burden. The data produced are missing completely at random (MCAR), and special missing data techniques should be used in linear regression and other multivariate statistical analysis. We derive asymptotic variances of regression parameter estimates that allow us to conduct power analysis for linear regression models fit to the data obtained via a multiple matrix sampling design. The ideas are demonstrate…
Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion…
The Generalized Additive Model (GAM) is a powerful tool and has been well studied. This model class helps to identify additive regression structure. Via available test procedures one may identify the regression structure even sharper if some component functions have parametric form. The Generalized Additive Partially Linear Models (GAPLM) enjoy the simplicity of the GLM and the flexibility of the GAM because they combine both parametric and nonparametric components. We use the hybrid spline-backfitted kernel estimation method, which combines the best features of both spline and kernel methods …
The linear coefficient in a partially linear model with confounding variables can be estimated using double machine learning (DML). However, this DML estimator has a two-stage least squares (TSLS) interpretation and may produce overly wide confidence intervals. To address this issue, we propose a regularization and selection scheme, regsDML, which leads to narrower confidence intervals. It selects either the TSLS DML estimator or a regularization-only estimator depending on whose estimated variance is smaller. The regularization-only estimator is tailored to have a low mean squared error. The …
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
We propose a new unsupervised and non-parametric method to detect change points in intricate quasi-periodic signals. The detection relies on optimal transport theory combined with topological analysis and the bootstrap procedure. The algorithm is designed to detect changes in virtually any harmonic or a partially harmonic signal and is verified on three different sources of physiological data streams. We successfully find abnormal or irregular cardiac cycles in the waveforms for the six of the most frequent types of clinical arrhythmias using a single algorithm. The validation and the efficien…
The Fisher information matrix (FIM) is a foundational concept in statistical signal processing. The FIM depends on the probability distribution, assumed to belong to a smooth parametric family. Traditional approaches to estimating the FIM require estimating the probability distribution function (PDF), or its parameters, along with its gradient or Hessian. However, in many practical situations the PDF of the data is not known but the statistician has access to an observation sample for any parameter value. Here we propose a method of estimating the FIM directly from sampled data that does not r…
Non-parametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing proble…
Oops, I Sampled it Again: Reinterpreting Confidence Intervals in Few-Shot Learning
IMT Atlantique（法国大西洋高等矿业电信学院）；Australian Institute for Machine Learning（澳大利亚机器学习研究所）；University of Adelaide（阿德莱德大学）；CNRS（法国国家科学研究中心）；Univ Brest（布雷斯特大学）；University of Münster（明斯特大学）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）
University of California, Irvine（加州大学欧文分校）；The George Washington University（乔治华盛顿大学）
We study the problem of off-policy evaluation (OPE) for episodic Partially Observable Markov Decision Processes (POMDPs) with continuous states. Motivated by the recently proposed proximal causal inference framework, we develop a non-parametric identification result for estimating the policy value via a sequence of so-called V-bridge functions with the help of time-dependent proxy variables. We then develop a fitted-Q-evaluation-type algorithm to estimate V-bridge functions recursively, where a non-parametric instrumental variable (NPIV) problem is solved at each step. By analyzing this challe…
We provide a comprehensive semi-parametric study of Bayesian partially identified econometric models. While the existing literature on Bayesian partial identification has mostly focused on the structural parameter, our primary focus is on Bayesian credible sets (BCS's) of the unknown identified set and the posterior distribution of its support function. We construct a (two-sided) BCS based on the support function of the identified set. We prove the Bernstein-von Mises theorem for the posterior distribution of the support function. This powerful result in turn infers that, while the BCS and the…
We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observa…
We consider non-parametric estimation and inference of conditional moment models in high dimensions. We show that even when the dimension $D$ of the conditioning variable is larger than the sample size $n$, estimation and inference is feasible as long as the distribution of the conditioning variable has small intrinsic dimension $d$, as measured by locally low doubling measures. Our estimation is based on a sub-sampled ensemble of the $k$-nearest neighbors ($k$-NN) $Z$-estimator. We show that if the intrinsic dimension of the covariate distribution is equal to $d$, then the finite sample estim…
