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Register a new userNon-parametric threshold estimation for classical risk process perturbed by diffusion
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Added by
Chunhao Cai
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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In this paper,we consider a macro approximation of the flow of a risk reserve, The process is observed at discrete time points. Because we cannot directly observe each jump time and size then we will make use of a technique for identifying the times when jumps larger than a suitably defined threshold occurred. We estimate the jump size and survival probability of our risk process from discrete observations.
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We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback--Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.
We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of the Path Estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and diffusion coefficients in the effective dynamics based on a semi-parametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates.
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.
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.
In this paper, we prove almost surely consistency of a Survival Analysis model, which puts a Gaussian process, mapped to the unit interval, as a prior on the so-called hazard function. We assume our data is given by survival lifetimes $T$ belonging to $mathbb{R}^{+}$, and covariates on $[0,1]^d$, where $d$ is an arbitrary dimension. We define an appropriate metric for survival functions and prove posterior consistency with respect to this metric. Our proof is based on an extension of the theorem of Schwartz (1965), which gives general conditions for proving almost surely consistency in the setting of non i.i.d random variables. Due to the nature of our data, several results for Gaussian processes on $mathbb{R}^+$ are proved which may be of independent interest.
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