Performance analysis of the Least-Squares estimator in Astrometry

We characterize the performance of the widely-used least-squares estimator in astrometry in terms of a comparison with the Cramer-Rao lower variance bound. In this inference context the performance of the least-squares estimator does not offer a closed-form expression, but a new result is presented (Theorem 1) where both the bias and the mean-square-error of the least-squares estimator are bounded and approximated analytically, in the latter case in terms of a nominal value and an interval around it. From the predicted nominal value we analyze how efficient is the least-squares estimator in comparison with the minimum variance Cramer-Rao bound. Based on our results, we show that, for the high signal-to-noise ratio regime, the performance of the least-squares estimator is significantly poorer than the Cramer-Rao bound, and we characterize this gap analytically. On the positive side, we show that for the challenging low signal-to-noise regime (attributed to either a weak astronomical signal or a noise-dominated condition) the least-squares estimator is near optimal, as its performance asymptotically approaches the Cramer-Rao bound. However, we also demonstrate that, in general, there is no unbiased estimator for the astrometric position that can precisely reach the Cramer-Rao bound. We validate our theoretical analysis through simulated digital-detector observations under typical observing conditions. We show that the nominal value for the mean-square-error of the least-squares estimator (obtained from our theorem) can be used as a benchmark indicator of the expected statistical performance of the least-squares method under a wide range of conditions. Our results are valid for an idealized linear (one-dimensional) array detector where intra-pixel response changes are neglected, and where flat-fielding is achieved with very high accuracy.Comment: 35 pages, 8 figures. Accepted for publication by PAS

A smoothed least squares estimator for threshold regression models

We propose a smoothed least squares estimator of the parameters of a threshold regression model. Our model generalizes that considered in Hansen (2000) to allow the thresholding to depend on a linear index of observed regressors, thus allowing discrete variables to enter. We also do not assume that the threshold e¤ect is vanishingly small. Our estimator is shown to be consistent and asymptotically normal thus facilitating standard inference techniques based on estimated standard errors or standard bootstrap for the threshold parameters themselves. We compare our con dence intervals with those of Hansen (2000) in a simulation study and show that our methods outperform his for large values of the threshold. We also include an application to cross-country growth regressions

Bootstrapping unit root AR(1) models.

We propose abootstrap resampling scheme for the least squares estimator of the parameter of an unstable first-order autoregressive model and we prove its asymptotic validity. This method is alternative to the invalid one studied by Basawa et al. (1991).Autoregressive processes; Bootstrapping least squares estimator; Unit root; Bootstrap invariance principle;

Testing stability in a spatial unilateral autoregressive model

Least squares estimator of the stability parameter $\varrho := |\alpha| + |\beta|$ for a spatial unilateral autoregressive process $X_{k,\ell}=\alpha X_{k-1,\ell}+\beta X_{k,\ell-1}+\varepsilon_{k,\ell}$ is investigated. Asymptotic normality with a scaling factor $n^{5/4}$ is shown in the unstable case, i.e., when $\varrho = 1$, in contrast to the AR(p) model $X_k=\alpha_1 X_{k-1}+... +\alpha_p X_{k-p}+ \varepsilon_k$, where the least squares estimator of the stability parameter $\varrho :=\alpha_1 + ... + \alpha_p$ is not asymptotically normal in the unstable, i.e., in the unit root case

Estimating the intercept in an orthogonally blocked experiment when the block effects are random.

Abstract: For an orthogonally blocked experiment, Khuri (1992) has shown that the ordinary least squares estimator and the generalized least squares estimator of the factor effects in a response surface model with random block effects coincide. However, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. When the block sizes are homogeneous, ordinary and generalized least squares provide an identical estimate for the intercept.Effects;