Parametric estimation for planar random flights observed at discrete times

Abstract

We deal with a planar random flight $\{(X(t),Y(t)),0<t\le T\}$ observed at $n+1$ equidistant times $t_i=i{\Delta }_n,i=0,1,...,n$. The aim of this paper is to estimate the unknown value of the parameter $\lambda$, the underlying rate of the Poisson process. The planar random flights are not markovian, then we use an alternative argument to derive a pseudo-maximum likelihood estimator $\hat{\lambda }$ of the parameter $\lambda$. We consider two different types of asymptotic schemes and show the consistency, the asymptotic normality and efficiency of the estimator proposed. A Monte Carlo analysis for small sample size $n$ permits us to analyze the empirical performance of $\hat{\lambda }$. A different approach permits us to introduce an alternative estimator of $\lambda$ which is consistent, asymptotically normal and asymptotically efficient without the request of other assumptions.