Qualitative robustness of statistical functionals under strong mixing

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

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Qualitative Robustness of S*‐estimators of Multivariate Location and Dispersion

Statistica Neerlandica ◽

10.1111/1467-9574.00054 ◽

1997 ◽

Vol 51 (3) ◽

pp. 257-268 ◽

Cited By ~ 2

Author(s):

X. He ◽

G. Wang

Keyword(s):

Qualitative Robustness ◽

Multivariate Location

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TESTING HOMOGENEITY IN PANEL DATA MODELS WITH INTERACTIVE FIXED EFFECTS

Econometric Theory ◽

10.1017/s0266466613000017 ◽

2013 ◽

Vol 29 (6) ◽

pp. 1079-1135 ◽

Cited By ~ 47

Author(s):

Liangjun Su ◽

Qihui Chen

Keyword(s):

Panel Data ◽

Fixed Effects ◽

Data Models ◽

Development Economic ◽

Strong Mixing ◽

Panel Data Models ◽

Finite Sample ◽

Test Statistic ◽

Lm Test ◽

Interactive Fixed Effects

This paper proposes a residual-based Lagrange Multiplier (LM) test for slope homogeneity in large-dimensional panel data models with interactive fixed effects. We first run the panel regression under the null to obtain the restricted residuals and then use them to construct our LM test statistic. We show that after being appropriately centered and scaled, our test statistic is asymptotically normally distributed under the null and a sequence of Pitman local alternatives. The asymptotic distributional theories are established under fairly general conditions that allow for both lagged dependent variables and conditional heteroskedasticity of unknown form by relying on the concept of conditional strong mixing. To improve the finite-sample performance of the test, we also propose a bootstrap procedure to obtain the bootstrap p-values and justify its validity. Monte Carlo simulations suggest that the test has correct size and satisfactory power. We apply our test to study the Organization for Economic Cooperation and Development economic growth model.

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ON THE SPECTRAL MEASURES OF SOME WEAKLY STATIONARY SEQUENCES INVOLVING RANDOMLY SPACED OBSERVATIONS

Stochastics and Dynamics ◽

10.1142/s0219493712003535 ◽

2012 ◽

Vol 12 (01) ◽

pp. 1150004

Author(s):

RICHARD C. BRADLEY

Keyword(s):

Spectral Density ◽

Density Function ◽

Spectral Density Function ◽

Strong Mixing ◽

Spectral Measures ◽

Mixing Conditions ◽

Stationary Sequences ◽

Second Moments ◽

The Central Limit Theorem ◽

Complex Valued

In an earlier paper by the author, as part of a construction of a counterexample to the central limit theorem under certain strong mixing conditions, a formula is given that shows, for strictly stationary sequences with mean zero and finite second moments and a continuous spectral density function, how that spectral density function changes if the observations in that strictly stationary sequence are "randomly spread out" in a particular way, with independent "nonnegative geometric" numbers of zeros inserted in between. In this paper, that formula will be generalized to the class of weakly stationary, mean zero, complex-valued random sequences, with arbitrary spectral measure.

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Asymptotic Distributions of M-Estimates for Parameters of Multivariate Time Series with Strong Mixing Property

Engineering Proceedings ◽

10.3390/engproc2021005019 ◽

2021 ◽

Vol 5 (1) ◽

pp. 19

Author(s):

Alexander Kushnir ◽

Alexander Varypaev

Keyword(s):

Time Series ◽

Asymptotic Normality ◽

Multivariate Time Series ◽

Asymptotic Properties ◽

Asymptotic Distributions ◽

Stationary Time Series ◽

Strong Mixing ◽

Statistical Estimates ◽

Distribution Parameters ◽

M Estimates

The publication is devoted to studying asymptotic properties of statistical estimates of the distribution parameters u∈Rq of a multidimensional random stationary time series zt∈Rm, t∈ℤ satisfying the strong mixing conditions. We consider estimates u^nδ(z¯n), z¯n=(z1T,…,znT)T∈Rmn that provide in asymptotic n→∞ the maximum values for some objective functions Qn(z¯n;u), which have properties similar to the well-known property of local asymptotic normality. These estimates are constructed by solving the equations δn(z¯n;u)=0, where δn(z¯n;u) are arbitrary functions for which δn(z¯n;u)−gradhQn(z¯n;u+n−1/2h)→0(n→∞) in Pn,u(z¯n)-probability uniformly on u∈U, were U is compact in Rq. In many cases, the estimates u^nδ(z¯n) have the same asymptotic properties as well-known M-estimates defined by equations u^nQ(z¯n)=arg maxu∈UQn(z¯n;u) but often can be much simpler computationally. We consider an algorithmic method for constructing estimates u^nδ(z¯n), which is similar to the accumulation method first proposed by R. Fischer and rigorously developed by L. Le Cam. The main theoretical result of the article is the proof of the theorem, in which conditions of the asymptotic normality of estimates u^nδ(z¯n) are formulated, and the expression is proposed for their matrix of asymptotic mean-square deviations limn→∞nEn,u{(u^δ(z¯n)−u)(u^δ(z¯n)−u)T}.

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