We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the sample mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the studentized sample mean, to reflect econometric interest in autocorrelation-consistent or long-run variance estimation. Our main focus is on stationary Gaussian series, though we discuss relaxation of the Gaussianity assumption. Only smoothness conditions on the spectral density that are local to the frequency of interest are imposed. We deduce empirical expansions from our Edgeworth expansions designed to improve on the normal approximation in practice and also deduce a feasible rule of bandwidth choice.

/pdf/edgeworth-expansions-for-spectral-density-estimates-and-2nj2uf79vw.pdf

Higher Order Asymptotic Theory for Time Series Analysis

Edgeworth expansions for spectral density estimates and studentized sample mean

The asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the Malliavin calculus. From this the second order efficiency of the maximum likelihood estimator is proved.

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Asymptotic expansion for statistics related to small diffusions

The three asymptotic tests, Neyman and Pearson Likelihood Ratio (LR), Wald’s statistic (W) and Rao’s score (RS)are referred to in statistical literature on testing of hypotheses as the Holy Trinity. All these tests are equivalent to the first-order of asymptotics, but differ to some extent in the second-order properties. Some of the merits and defects of these tests are presented.

Score Test: Historical Review and Recent Developments

We present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale. The emphasis is put on the use of the Malliavin calculus; the uniform nondegeneracy of the Malliavin covariance under certain truncation plays an essential role as the Cramer condition did in the case of independent observations. Applications to statistics are presented.

methods

/pdf/malliavin-calculus-and-asymptotic-expansion-for-martingales-2jp4k80wmk.pdf

Higher Order Asymptotic Theory for Time Series Analysis

Citations

Edgeworth expansions for spectral density estimates and studentized sample mean

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Asymptotic expansion for statistics related to small diffusions

Score Test: Historical Review and Recent Developments

Malliavin calculus and asymptotic expansion for martingales

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