Empirical‐process‐based specification tests for diffusion models

Joint and marginal specification tests for conditional mean and variance models

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in the context of topic models. The data consists of observed multinomial counts of p words across n independent documents. In topic models, the p×n expected word frequency matrix is assumed to be factorized as a p×K word-topic matrix A and a K×n topic-document matrix T. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of A are viewed as p-dimensional mixture components that are common to all documents while columns of T are viewed as the K-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, ℓ1-norm convergence rates for estimators of the mixture weights T when A is either known or unknown. For known A, we suggest MLE estimation of T. Our nonstandard analysis of the MLE not only establishes its ℓ1 convergence rate, but also reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of T. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When A is unknown, we estimate T by optimizing the likelihood function corresponding to a plug in, generic, estimator Aˆ of A. For any estimator Aˆ that satisfies carefully detailed conditions for proximity to A, we show that the resulting estimator of T retains the properties established for the MLE. Our theoretical results allow the ambient dimensions K and p to grow with the sample sizes. Our main application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between alternative probabilistic document representations, at the word and topic level, respectively. We derive finite sample bounds on the estimated proposed 1-Wasserstein distances. For word level document-distances, we provide contrast with existing rates on the 1-Wasserstein distance between standard empirical frequency estimates. The effectiveness of the proposed 1-Wasserstein distances is illustrated by an analysis of an IMDB movie reviews data set. Finally, our theoretical results are supported by extensive simulation studies.

This paper looks at some recent work on estimating quadratic variation using realized variance (RV)—that is, sums ofMsquared returns. This econometrics has been motivated by the advent of the common availability of high‐frequency financial return data. When the underlying process is a semimartingale we recall the fundamental result that RV is a consistent (asM→ ∞) estimator of quadratic variation (QV). We express concern that without additional assumptions it seems difficult to give any measure of uncertainty of the RV in this context. The position dramatically changes when we work with a rather general SV model—which is a special case of the semimartingale model. Then QV is integrated variance and we can derive the asymptotic distribution of the RV and its rate of convergence. These results do not require us to specify a model for either the drift or volatility functions, although we have to impose some weak regularity assumptions. We illustrate the use of the limit theory on some exchange rate data and some stock data. We show that even with large values ofMthe RV is sometimes a quite noisy estimator of integrated variance. Copyright © 2002 John Wiley & Sons, Ltd.

This article proposes goodness‐of‐fit tests for dynamic regression models, where regressors are allowed to be only weakly exogenous and arbitrarily correlated with past shocks. The null hypothesis is stated in terms of the lack of serial correlation of the errors of the model. The tests are based on a linear transformation of a Bartlett's Tp‐process of the residuals. This transformation approximates the martingale component of the process so that it converges weakly to the standard Brownian motion under the null hypothesis. One feature of our setup is that we do not require to specify the dynamic structure of the regressors. Due to this, the transformation employs a semi‐parametric correction that does not restrict the class of local alternatives that our tests can detect, in contrast with other works using smoothing techniques. A Monte Carlo study illustrates the finite sample performance of the tests.

Specification tests for time-varying coefficient models

Generalized spectral testing for multivariate continuous-time models

Specification Tests for Spatial Panel Data Models

f-divergence estimation is an important problem in the fields of information\ntheory, machine learning, and statistics. While several divergence estimators\nexist, relatively few of their convergence rates are known. We derive the MSE\nconvergence rate for a density plug-in estimator of f-divergence. Then by\napplying the theory of optimally weighted ensemble estimation, we derive a\ndivergence estimator with a convergence rate of O(1/T) that is simple to\nimplement and performs well in high dimensions. We validate our theoretical\nresults with experiments.\n

In some survival studies part of the population may be no longer subject to the event of interest. The called cure rate models take this fact into account. They have been extensively studied for several authors who have proposed extensions and applications in real lifetime data. Classic large sample tests are usually considered in these applications, especially the likelihood ratio. Recently a new test called \textit{gradient test} has been proposed. The gradient statistic shares the same asymptotic properties with the classic likelihood ratio and does not involve knowledge of the information matrix, which can be an advantage in survival models. Some simulation studies have been carried out to explore the behavior of the gradient test in finite samples and compare it with the classic tests in different models. However little is known about the properties of these large sample tests in finite sample for cure rate models. In this work we performed a simulation study based on the promotion time model with Weibull distribution, to assess the performance of likelihood ratio and gradient tests in finite samples. An application is presented to illustrate the results.

Specification Test for Panel Data Models with Interactive Fixed Effects

Nonparametric M-regression with free knot splines

Simple Tests for Models of Dependence Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rates

Additive modelling is known to be useful for multivariate nonparametric regression as it reduces the complexity of problem to the level of univariate regression. This usefulness could be compromised if the data set was contaminated by outliers whose detection and removal are particularly difficult to achieve in high dimension. We propose an estimation procedure for the additive component of the regression function , less sensitive to possible outliers in the sample. Our procedure is based on marginal integration of conditional R-estimators. In addition to univariate rate of convergence and asymptotic distribution, we also obtain robustness results for our estimator. All of our results are valid for a broad class of s mixing processes. Monte Carlo findings confirm the theoretical results in finite sample.

Whittle parameter estimation for vector ARMA models with heavy-tailed noises

Best Attainable Rates of Convergence for Estimators of the Stable Tail Dependence Function