Maximum Likelihood Estimation of Variance Components: New Theoretical Developments

Abstract:

The computation of geodetic parameters using some data collected on satellites is usually done with a stochastic model. One of the most common models is as follows: the stochastic perturbation is gaussian with mean zero and the covariance matrix is written as an linear combination of known matrices whose coefficients are unknown. The problem is therefore to estimate both the parameters and the coefficients. The Maximum Likehood Estimator is one way to solve it. We study here this approach, we simplify it and we establish that it is consistent, meaning that the solution computed converges toward the parameters and the coefficients when the number of observables increases to infinity. We establish moreover that it is normally asymptotic, meaning that

$$

\sqrt{n}\left(\left(\hat{X}_{n},\wp_{n}\right)-\left(X,\wp\right)\right)\rigtarrow_{d} \mathcal{0,\Sigma\left(X,\wp\right)}

$$

where $\Sigma\left(X,\wp\right)$ is known. This result allows to compute formal errors and confidence interval. Eventually, we discuss about numerical tests.