State space models (SSMs) are now ubiquitous in many fields and increasingly complicated with observed and unobserved variables often interacting in non-linear fashions. The crucial task of validating model assumptions thus becomes difficult, particularly since some assumptions are formulated about unobserved states and thus cannot be checked with data. Motivated by the complex SSMs used for the assessment of fish stocks, we introduce a robust estimation method for SSMs. We prove the Fisher consistency of our estimator and propose an implementation based on automatic differentiation and the Laplace approximation of integrals which yields fast computations. Simulation studies demonstrate that our robust procedure performs well both with and without deviations from model assumptions. Applying it to the stock assessment model for pollock in the North Sea highlights the ability of our procedure to identify years with atypical observations.
robustloggamma is an R package for robust estimation and inference in the generalized loggamma model. We briefly introduce the model, the estimation procedures and the computational algorithms. Then, we illustrate the use of the package with the help of a real data set.
In this article we consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises observed in discrete time moments. An adaptive model selection procedure is proposed. A sharp non-asymptotic oracle inequality for the robust risks is obtained. We obtain sufficient conditions on the frequency observations under which the robust efficiency is shown. It turns out that for the semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-ε$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $α$-locally differentially private mechanism. The remaining $εn $ batches are an adver…
Non-parametric estimation of a convex discrete distribution may be of interest in several applications, such as the estimation of species abundance distribution in ecology. In this paper we study the least squares estimator of a discrete distribution under the constraint of convexity. We show that this estimator exists and is unique, and that it always outperforms the classical empirical estimator in terms of the $\ell_{2}$-distance. We provide an algorithm for its computation, based on the support reduction algorithm. We compare its performance to those of the empirical estimator, on the basi…
Imposing some flexible sampling scheme we provide some discretization of continuous time discrete scale invariant (DSI) processes which is a subsidiary discrete time DSI process. Then by introducing some simple random measure we provide a second continuous time DSI process which provides a proper approximation of the first one. This enables us to provide a bilateral relation between covariance functions of the subsidiary process and the new continuous time processes. The time varying spectral representation of such continuous time DSI process is characterized, and its spectrum is estimated. Al…
We develop here several goodness-of-fit tests for testing the k-monotonicity of a discrete density, based on the empirical distribution of the observations. Our tests are non-parametric, easy to implement and are proved to be asymptotically of the desired level and consistent. We propose an estimator of the degree of k-monotonicity of the distribution based on the non-parametric goodness-of-fit tests. We apply our work to the estimation of the total number of classes in a population. A large simulation study allows to assess the performances of our procedures.
Estimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M-type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are su…
In some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points. In this case, we speak of discrete parameter models. Even though the problem is quite old and has interesting connections with testing and model selection, asymptotic theory for these models has hardly ever been studied. Therefore, we discuss consistency, asymptotic distribution theory, information inequalities and their relations with efficiency and supere…
We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of poorly identifiable parameters. Moreover, ODEs are generally approximations of the true process and the influence of misspecification on inference is often neglected. Methods based on control theory have emerged to regularize the ill posed problem of parameter estimation in this context. However, they are computationally intensive and rely on a nonparametric …
This note considers the problem of statistical inference of the parameters of the input process to a queue from periodic workload observations. The main focus is the open problem of constructing statistically efficient estimators for a given observation scheme, in the sense of minimizing the asymptotic variance of the estimation error.
The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead…
Time-varying covariates are often available in survival studies and estimation of the hazard function needs to be updated as new information becomes available. In this paper, we investigate several different easy-to-implement ways that random forests can be used for dynamic estimation of the survival or hazard function from discrete-time survival data. The results from a simulation study indicate that all methods can perform well, and that none dominates the others. In general, situations that are more difficult from an estimation point of view (such as weaker signals and less data) favour a g…
A method of estimating the joint probability mass function of a pair of discrete random variables is described. This estimator is used to construct the conditional Shannon-Réyni-Tsallis entropies estimates. From there almost sure rates of convergence and asymptotic normality are established. The theorical results are validated by simulations.
In this article, a discrete analogue of continuous Teissier distribution is presented. Its several important distributional characteristics have been derived. The estimation of the unknown parameter has been done using the method of maximum likelihood and the method of moment. Two real data applications have been presented to show the applicability of the proposed model.
We consider parametric estimation for ergodic diffusion processes with noisy sampled data based on the hybrid method, that is, the multi-step estimation with the initial Bayes type estimators. In order to select proper initial values for optimisation of the quasi likelihood function of ergodic diffusion processes with noisy observations, we construct the initial Bayes type estimator based on the local means of the noisy observations. The asymptotic properties of the initial Bayes type estimators and the hybrid multi-step estimators with the initial Bayes type estimators are shown, and a concre…
In this paper, different strands of literature are combined in order to obtain algorithms for semi-parametric estimation of discrete choice models that include the modelling of unobserved heterogeneity by using mixing distributions for the parameters defining the preferences. The models use the theory on non-parametric maximum likelihood estimation (NP-MLE) that has been developed for general mixing models. The expectation-maximization (EM) techniques used in the NP-MLE literature are combined with strategies for choosing appropriate approximating models using adaptive grid techniques. \\ Join…
Fields like public health, public policy, and social science often want to quantify the degree of dependence between variables whose relationships take on unknown functional forms. Typically, in fact, researchers in these fields are attempting to evaluate causal theories, and so want to quantify dependence after conditioning on other variables that might explain, mediate or confound causal relations. One reason conditional mutual information is not more widely used for these tasks is the lack of estimators which can handle combinations of continuous and discrete random variables, common in app…
In various practical situations, we encounter data from stochastic processes which can be efficiently modelled by an appropriate parametric model for subsequent statistical analyses. Unfortunately, the most common estimation and inference methods based on the maximum likelihood (ML) principle are susceptible to minor deviations from assumed model or data contamination due to their well known lack of robustness. Since the alternative non-parametric procedures often lose significant efficiency, in this paper, we develop a robust parameter estimation procedure for discretely observed data from a …
The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity $+ v$ or $-v$. The changes of direction are governed by an homogeneous Poisson process with rate $λ>0.$ In this paper, we consider a change point estimation problem for the rate of the underlying Poisson process by means of least squares method. The consistency and the rate of convergence for the change point estimator are obtained and its asymptotic dist…
