We want to reconstruct a signal based on inhomogeneous data (the amount of data can vary strongly), using the model of regression with a random design. Our aim is to understand the consequences of inhomogeneity on the accuracy of estimation within the minimax framework. Using the uniform metric weighted by a spatially-dependent rate as a benchmark for an estimator accuracy, we are able to capture the deformation of the usual minimax rate in situations with local lacks of data (modelled by a design density with vanishing points). In particular, we construct an estimator both design and smoothness adaptive, and a new criterion is developed to prove the optimality of these deformed rates.
We consider parameter estimation in finite hidden state space Markov models with time-dependent inhomogeneous noise, where the inhomogeneity vanishes sufficiently fast. Based on the concept of asymptotic mean stationary processes we prove that the maximum likelihood and a quasi-maximum likelihood estimator (QMLE) are strongly consistent. The computation of the QMLE ignores the inhomogeneity, hence, is much simpler and robust. The theory is motivated by an example from biophysics and applied to a Poisson- and linear Gaussian model.
We consider the nonparametric estimation problem of time-dependent multivariate functions observed in a presence of additive cylindrical Gaussian white noise of a small intensity. We derive minimax lower bounds for the $L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero, when the underlying unknown response function is assumed to belong to a ball of appropriately constructed inhomogeneous time-dependent multivariate functions, motivated by practical applications. Furthermore, we propose both non-adaptive linear and adaptive non-linear wavelet estimators that are asym…
Random uniform sampling has been studied in various statistical tasks but few of them have covered the Q-error metric for cardinality estimation (CE). In this paper, we analyze the confidence intervals of random uniform sampling with and without replacement for single-table CE. Results indicate that the upper Q-error bound depends on the sample size and true cardinality. Our bound gives a rule-of-thumb for how large a sample should be kept for single-table CE.
Modeling of longitudinal data often requires diffusion models that incorporate overall time-dependent, nonlinear dynamics of multiple components and provide sufficient flexibility for subject-specific modeling. This complexity challenges parameter inference and approximations are inevitable. We propose a method for approximate maximum-likelihood parameter estimation in multivariate time-inhomogeneous diffusions, where subject-specific flexibility is accounted for by incorporation of multidimensional mixed effects and covariates. We consider $N$ multidimensional independent diffusions $X^i = (X…
University of Pennsylvania（宾夕法尼亚大学）；Stanford University（斯坦福大学）
In network applications, it has become increasingly common to obtain datasets in the form of multiple networks observed on the same set of subjects, where each network is obtained in a related but different experiment condition or application scenario. Such datasets can be modeled by multilayer networks where each layer is a separate network itself while different layers are associated and share some common information. The present paper studies community detection in a stylized yet informative inhomogeneous multilayer network model. In our model, layers are generated by different stochastic b…
Parameter Estimation for Time-Scaled Inhomogeneous Phase-Type Distributions from Discrete Observations
Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Department of Statistics, University of Wisconsin, Madison（威斯康星大学麦迪逊分校统计学系）
We propose nonparametric estimation of divergence measures between continuous distributions. Our approach is based on a plug-in kernel- type estimators of density functions. We give the uniform in bandwidth consistency for the proposal estimators. As a consequence, their asymp- totic 100% confidence intervals are also provided.
We present two data-driven procedures to estimate the transition density of an homogeneous Markov chain. The first yields to a piecewise constant estimator on a suitable random partition. By using an Hellinger-type loss, we establish non-asymptotic risk bounds for our estimator when the square root of the transition density belongs to possibly inhomogeneous Besov spaces with possibly small regularity index. Some simulations are also provided. The second procedure is of theoretical interest and leads to a general model selection theorem from which we derive rates of convergence over a very wide…
We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known compactly supported density g with a finite number of well separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic minimax lower bounds for the global risk of an estimator of f and construct adaptive wavelet nonlinear th…
We explicitly quantify the empirically observed phenomenon that estimation under a stochastic block model (SBM) is hard if the model contains classes that are similar. More precisely, we consider estimation of certain functionals of random graphs generated by a SBM. The SBM may or may not be sparse, and the number of classes may be fixed or grow with the number of vertices. Minimax lower and upper bounds of estimation along specific submodels are derived. The results are nonasymptotic and imply that uniform estimation of a single connectivity parameter is much slower than the expected asymptot…
We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the $d$-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form $β+{θ}^{\top}z(u)$ where $z$ is a spatial covariate function and the focus is on estimating ${θ}$. The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We a…
The Ewens sampling formula is a distribution related to the random partition of a positive integer. In this study, we investigate the issue of non-existence solutions in parameter estimation under the distribution. As a result, the first and second moments matching estimators to the uniformly minimum variance unbiased estimator are derived using the Ewens sampling formula in asymptotic sense. A Monte Carlo simulation study is performed to evaluate the efficiency of the resulting estimators.
The class of inhomogeneous phase-type distributions (IPH) was recently introduced in Albrecher and Bladt (2019) as an extension of the classical phase-type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate phase-type class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for …
Owing to their capability of summarising interactions between elements of a system, networks have become a common type of data in many fields. As networks can be inhomogeneous, in that different regions of the network may exhibit different topologies, an important topic concerns their local properties. This paper focuses on the estimation of the local degree distribution of a vertex in an inhomogeneous network. The contributions are twofold: we propose an estimator based on local weighted averaging, and we set up a Monte Carlo cross-validation procedure to pick the parameters of this estimator…
