In this article we propose two new Multiplicative Bias Correction (MBC) techniques for nonparametric multivariate density estimation. We deal with positively supported data but our results can easily be extended to the case of mixtures of bounded and unbounded supports. Both methods improve the optimal rate of convergence of the mean squared error up to O(n −8/(8+d) ), where d is the dimension of the underlying data set. Moreover, they overcome the boundary effect near the origin and their values are always non-negative. We investigate asymptotic properties like bias and variance as well as the performance of our estimators in Monte Carlo Simulations and in a real data example.
We present limit theorems for locally stationary processes that have a one sided time-varying moving average representation. In particular, we prove a central limit theorem (CLT), a weak and a strong law of large numbers (WLLN, SLLN) and a law of the iterated logarithm (LIL) under mild assumptions using a time-varying Beveridge–Nelson decomposition.
Locally stationary processes are characterised by spectral densities that are functions of rescaled time. We study the asymptotic properties of spectral density estimators in the locally stationary framework. In particular, we show that for a locally stationary process with time-varying spectral density function f (u, λ) standard spectral density estimators consistently estimate the time-averaged spectral density 1 0 f (u, λ) du. This result is complemented by some illustrative examples and applications including HAC-inference in the multiple linear regression model and a simple visual tool for the detection of unconditional heteroskedasticity.
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