| Popis: |
An important challenge in statistical analysis concerns the control of the finite sample bias of estimators. This problem is magnified in high dimensional settings where the number of variables p diverge with the sample size n. However, it is difficult to establish whether an estimator θˆ of θ0 is unbiased and the asymptotic order of E[θˆ] − θ0 is commonly used instead. We introduce a new property to assess the bias, called phase transition unbiasedness, which is weaker than unbiasedness but stronger than asymptotic results. An estimator satisfying this property is such that E[θˆ] − θ0 2 = 0, for all n greater than a finite sample size n ∗ . We propose a phase transition unbiased estimator by matching an initial estimator computed on the sample and on simulated data. It is computed using an algorithm which is shown to converge exponentially fast. The initial estimator is not required to be consistent and thus may be conveniently chosen for computational efficiency or for other properties. We demonstrate the consistency and the limiting distribution of the estimator in high dimension. Finally, we develop new estimators for logistic regression models, with and without random effects, that enjoy additional properties such as robustness to data contamination and to the problem of separability. |
