Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOptimal convergence rates for the invariant density estimation of jump-diffusion processes
208
0
0.0
(
0
)
Added by
Chiara Amorino
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some Holder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $frac{log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates ${frac{1}{T},frac{log T}{T}}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
rate research
Read More
In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on an unknown parameter $theta$. We suppose that the process is discretely observed at the instants (t n i)i=0,...,n with $Delta$n = sup i=0,...,n--1 (t n i+1 -- t n i) $rightarrow$ 0. We introduce an estimator of $theta$, based on a contrast function, which is efficient without requiring any conditions on the rate at which $Delta$n $rightarrow$ 0, and where we allow the observed process to have non summable jumps. This extends earlier results where the condition n$Delta$ 3 n $rightarrow$ 0 was needed (see [10],[24]) and where the process was supposed to have summable jumps. Moreover, in the case of a finite jump activity, we propose explicit approximations of the contrast function, such that the efficient estimation of $theta$ is feasible under the condition that n$Delta$ k n $rightarrow$ 0 where k > 0 can be arbitrarily large. This extends the results obtained by Kessler [15] in the case of continuous processes. L{e}vy-driven SDE, efficient drift estimation, high frequency data, ergodic properties, thresholding methods.
We undertake a precise study of the non-asymptotic properties of vanilla generative adversarial networks (GANs) and derive theoretical guarantees in the problem of estimating an unknown $d$-dimensional density $p^*$ under a proper choice of the class of generators and discriminators. We prove that the resulting density estimate converges to $p^*$ in terms of Jensen-Shannon (JS) divergence at the rate $(log n/n)^{2beta/(2beta+d)}$ where $n$ is the sample size and $beta$ determines the smoothness of $p^*.$ This is the first result in the literature on density estimation using vanilla GANs with JS rates faster than $n^{-1/2}$ in the regime $beta>d/2.$
We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for distributions with $beta$-Holder smooth densities where $beta in (0, 2)$, we observe polynomially faster minimax rates of estimation when, additionally, the total positivity condition is imposed. Moreover, we demonstrate fast algorithms to compute the proposed estimators and corroborate the theoretical rates of estimation by simulation studies.
We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.
We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter $theta_1$ in a non-degenerate diffusion coefficient and a parameter $theta_2$ in the drift term. The second component has a drift term parameterized by $theta_3$ and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for $theta_3$ with some initial estimators for ($theta_1$ , $theta_2$), an adaptive one-step estimator for ($theta_1$ , $theta_2$ , $theta_3$) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for ($theta_1$ , $theta_2$ , $theta_3$) without any initial estimator. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for $theta_1$ is smaller than the standard one based only on the first component. The convergence of the estimators for $theta_3$ is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
