NON ASYMPTOTIC EFFICIENCY OF A MAXIMUM LIKELIHOOD ESTIMATOR AT FINITE NUMBER OF SAMPLES

International audienceIn estimation theory, the asymptotic (in the number of samples) efficiency of the Maximum Likelihood (ML) estimator is a well known result [1]. Nevertheless, in some scenarios, the number of snapshots may be small. We recently investigated the asymptotic behavior of the Stochastic ML (SML) estimator at high Signal to Noise Ratio (SNR) and finite number of samples [2] in the array processing framework: we proved the non-Gaussiannity of the SML estimator and we obtained the analytical expression of the variance for the single source case. In this paper, we generalize these results to multiple sources, and we obtain variance expressions which demonstrate the non-efficiency of SML estimates

Gini estimation under infinite variance

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index $\alpha\in(1,2)$). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of $\alpha$. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution

Endogenous Sampling and Matching Method in Duration Models

Endogenous sampling with matching (also called gmixed samplingh) occurs when the statistician samples from the non-right- censored subset at a predetermined proportion and matches on one or more exogenous variables when sampling from the right-censored subset. This is widely applied in the duration analysis of firm failures, loan defaults, insurer insolvencies, and so on, due to the low frequency of observing non-right-censored samples (bankrupt, default, and insolvent observations in respective examples). However, the common practice of using estimation procedures intended for random sampling or for the qualitative response model will yield either an inconsistent or inefficient estimator. This paper proposes a consistent and efficient estimator and investigates its asymptotic properties. In addition, this paper evaluates the magnitude of asymptotic bias when the model is estimated as if it were a random sample or an endogenous sample without matching. This paper also compares the relative efficiency of other commonly used estimators and provides a general guideline for optimally choosing sample designs. The Monte Carlo study with a simple example shows that random sampling yields an estimator of poor finite sample properties when the population is extremely unbalanced in terms of default and non-default cases while endogenous sampling and mixed sampling are robust in this situation.Duration models; Endogenous sampling with matching; Maximum likelihood estimator; Manski-Lerman estimator; Asymptotic distribution

Smooth tail index estimation

Both parametric distribution functions appearing in extreme value theory - the generalized extreme value distribution and the generalized Pareto distribution - have log-concave densities if the extreme value index gamma is in [-1,0]. Replacing the order statistics in tail index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density leads to novel smooth quantile and tail index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean squared error.Comment: 17 pages, 5 figures. Slightly changed Pickand's estimator, added some more introduction and discussio

Instrumental Variables Estimation of Heteroskedastic Linear Models Using All Lags of Instruments

We propose and evaluate a technique for instrumental variables estimation of linear models with conditional heteroskedasticity. The technique uses approximating parametric models for the projection of right hand side variables onto the instrument space, and for conditional heteroskedasticity and serial correlation of the disturbance. Use of parametric models allows one to exploit information in all lags of instruments, unconstrained by degrees of freedom limitations. Analytical calculations and simulations indicate that there sometimes are large asymptotic and finite sample efficiency gains relative to conventional estimators (Hansen (1982)), and modest gains or losses depending on data generating process and sample size relative to quasi-maximum likelihood. These results are robust to minor misspecification of the parametric models used by our estimator.