We consider a k-dimensional multiple-regime vector threshold autoregressive model, in which the regime-switching mechanism is governed by a bivariate threshold variable. Specifically, the regimes are induced by a partition of the threshold space by an unknown number of threshold lines. Within each regime, the process follows a specific vector autoregressive (VAR) model. We formulate the model selection and parameter estimation into a minimization problem based on the Minimum Description Length (MDL) principle and estimate the number of threshold lines, parametric forms of threshold lines and VAR model parameters in each regime simultaneously. Theoretically, we show that the MDL estimators of threshold lines are n-consistent and characterize their limiting distribution. This requires novel proving techniques of introducing a new functional space G for the local MDL difference functions and establishing weak convergence results therein. Finally, we conduct some empirical studies with simulated datasets and perform real data analyses on U.S. interest rates and U.S. GNP Data.
We study the fundamental statistical inference concerning the testing of independence between two random vectors. Existing asymptotic theories for test statistics based on distance covariance can only apply to either low-dimensional or high-dimensional settings and require stringent distributional assumptions. In this work we develop a new unified distributional theory of the sample generalized distance covariance that works for random vectors of arbitrary dimensions under fairly mild moment conditions. In particular, a Gaussian approximation result is established with a nonasymptotic error bound, and the asymptotic null distribution of the sample generalized distance covariance is shown to be distributed as a linear combination of independently and identically distributed chi-squared random variables. To estimate the asymptotic null distribution practically, we propose a half-permutation procedure and provide the theoretical justification for its validity. The exact asymptotic distribution of the resampling distribution is derived under general marginal moment conditions, and the proposed procedure is shown to be asymptotically equivalent to the oracle procedure with known marginal distributions.
This erratum contains a correction of the incorrect defined distribution on page 1109, last two lines, in Schneller (Ann. Statist. 17 (1989) 1103–1123). Additionally, some typographical errors are corrected.
We consider inference about a finite-dimensional parameter integrating samples from independent sources. A recently developed theory considers scenarios where sources align with subsets of the conditional distributions of a single factorization of the joint target distribution. While this theory applies in many settings, it falls short in important data fusion problems, such as two-sample instrumental variable analysis, settings that integrate data from epidemiological studies with diverse designs, and studies with mismeasured variables supplemented by external validation studies. In this paper, we derive a comprehensive theory that, in particular, covers these settings by allowing the integration of sources aligned with conditional distributions that do not correspond to a single factorization of the target distribution. We provide a universal characterization of the influence functions of regular and asymptotically linear estimators and the efficient influence function of a target parameter, irrespective of the parameter of interest or the statistical model for the target distribution, thus paving the way for a unified theory for machine-learning debiased, semiparametric efficient estimation.
