Sharp non-asymptotic Concentration Inequalities for the Approximation of the Invariant Measure of a Diffusion

For an ergodic Brownian diffusion with invariant measure $\nu$, we consider a sequence of empirical distributions ($\nu$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($\gamma$n) n$\ge$1 along an adapted regular enough class of test functions f such that f --$\nu$(f) is a coboundary of the infinitesimal generator A. Denote by $\sigma$ the diffusion coefficient and $\Phi$ the solution of the Poisson equation A$\Phi$ = f -- $\nu$(f). When the square norm of |$\sigma$ * $\Phi$| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizations of $\nu$n(f) -- $\nu$(f). Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pag{\`e}s in [LP02], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. We provide as well a Slutsky like Theorem, for practical applications, where the deviation bounds are also asymptotically independent of the corresponding Poisson problem. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting

Bounds on Parameters in Dynamic Discrete Choice Models

Identification of dynamic nonlinear panel data models is an important and delicate problem in econometrics. In this paper we provide insights that shed light on the identification of parameters of some commonly used models. Using this insight, we are able to show through simple calculations that point identification often fails in these models. On the other hand, these calculations also suggest that the model restricts the parameter to lie in a region that is very small in many cases, and the failure of point identification may therefore be of little practical importance in those cases. Although the emphasis is on identification, our techniques are constructive in that they can easily form the basis for consistent estimates of the identified sets.

Estimation of Discrete Time Duration Models with Grouped Data

Dynamic discrete choice panel data models have received a great deal of attention. In those models, the dynamics is usually handled by including the lagged outcome as an explanatory variable. In this paper we consider an alternative model in which the dynamics is handled by using the duration in the current state as a covariate. We propose estimators that allow for group specific effect in parametric and semiparametric versions of the model. The proposed method is illustrated by an empirical analysis of child mortality allowing for family specific effects.Panel Data; Discrete Choice; Duration Models

Estimation of a transformation model with truncation, interval observation and time-varying covariates

Abrevaya (1999b) considered estimation of a transformation model in the presence of left-truncation. This paper observes that a cross-sectional version of the statistical model considered in Frederiksen, Honoré, and Hu (2007) is a generalization of the model considered by Abrevaya (1999b) and the generalized model can be estimated by a pairwise comparison version of one of the estimators in Frederiksen, Honoré, and Hu (2007). Specifically, our generalization will allow for discretized observations of the dependent variable and for piecewise constant time- varying explanatory variables.

Estimation of a Transformation Model with Truncation, Interval Observation and Time–Varying Covariates

Abrevaya (1999b) considered estimation of a transformation model in the presence of left–truncation. This paper observes that a cross–sectional version of the statistical model considered in Frederiksen, Honoré, and Hu (2007) is a generalization of the model considered by Abrevaya (1999b) and the generalized model can be estimated by a pairwise comparison version of one of the estimators in Frederiksen, Honoré, and Hu (2007). Specifically, our generalization will allow for discretized observations of the dependent variable and for piecewise constant time–varying explanatory variables.

Bounds in Competing Risks Models and the War on Cancer

Competing risks models are fundamentally unidentified. This paper derives bounds for aspects of the underlying distributions under a number of different assumptions. These bounds are then applied to mortality data from the US. We find that trends in cancer show much larger improvements than was previously estimated.Bounds; Competing Risks; Cancer

Non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions

This paper is a revised version of the original paper of same title--published in Applied Mathematics Letters 89--containing some corrections and clarifications to the original text. We derive non-singular Green's functions for the unbounded Poisson equation in one, two and three dimensions, using a cut-off function in the Fourier domain to impose a smallest length scale when deriving the Green's function. The resulting non-singular Green's functions are relevant to applications which are restricted to a minimum resolved length scale (e.g. a mesh size h) and thus cannot handle the singular Green's function of the continuous Poisson equation. We furthermore derive the gradient vector of the non-singular Green's function, as this is useful in applications where the Poisson equation represents potential functions of a vector field