A remainder estimate for branched rough differential equations

Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths

Dimension-free Euler estimates of rough differential equations

We give a dimension-free Euler estimation of solution of rough differential equations in term of the driving rough path. In the meanwhile, we prove that, the solution of rough differential equation is close to the exponential of a Lie series, with a concrete error bound

Uniform factorial decay estimate for the remainder of rough Taylor expansion

We establish an uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations. Its proof requires a factorial decay estimate for controlled paths which is interesting in its own right

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox‐type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time‐transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co‐workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson‐type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple‐infection data taken from a clinical study on chronic granulomatous disease is also provided