Parametric estimation for planar random flights observed at discrete times

We deal with a planar random flight {(X(t), Y (t)), 0 < t ≤ T } observed at n + 1 equidistant times ti = i∆n, i = 0, 1, ..., n. The aim of this paper is to estimate the unknown value of the parameter λ, 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 λ̂ of the parameter λ. 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 λ̂. A different approach permits us to introduce an alternative estimator of λ which is consistent, asymptotically normal and asymptotically efficient without the request of other assumptions.