Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

Download Full-text

Density estimation for continuous time processes

Nonparametric Statistics for Stochastic Processes - Lecture Notes in Statistics ◽

10.1007/978-1-4684-0489-0_5 ◽

1996 ◽

pp. 95-128

Author(s):

D. Bosq

Keyword(s):

Density Estimation ◽

Continuous Time ◽

Continuous Time Processes

Download Full-text

Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

Journal of Function Spaces ◽

10.1155/2015/714745 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6 ◽

Cited By ~ 5

Author(s):

Caishi Wang ◽

Jinshu Chen

Keyword(s):

Brownian Motion ◽

Growth Condition ◽

Discrete Time ◽

Continuous Time ◽

Orthonormal Basis ◽

Continuous Time Processes ◽

Characterization Theorems ◽

Normal Martingale

We aim at characterizing generalized functionals of discrete-time normal martingales. LetM=(Mn)n∈Nbe a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofMwith an appropriate orthonormal basis forM’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.

Download Full-text