Can we define a best estimator in simple one-dimensional cases?

International audienceWhat is the best estimator for assessing a parameter of a probability distribution from a small number of measurements? Is the same answer valid for a location parameter like the mean as for a scale parameter like the variance? It is sometimes argued that it is better to use a biased estimator with low dispersion than an unbiased estimator with a higher dispersion. In which cases is this assertion correct? To answer these questions, we will compare, on a simple example, the determination of a location parameter and a scale parameter with three "optimal" estimators: the minimum-variance unbiased estimator, the minimum square error estimator, and the a posteriori mean

Stability Variances: A filter Approach

We analyze the Allan Variance estimator as the combination of Discrete-Time linear filters. We apply this analysis to the different variants of the Allan variance: the Overlapping Allan Variance, the Modified Allan variance, the Hadamard Variance and the Overlapping Hadamard variance. Based on this analysis we present a new method to compute a new estimator of the Allan Variance and its variants in the frequency domain. We show that the proposed frequency domain equations are equivalent to extending the data by periodization in the time domain. Like the Total Variance \cite{totvar}, which is based on extending the data manually in the time domain, our frequency domain variances estimators have better statistics than the estimators of the classical variances in the time domain. We demonstrate that the previous well-know equation that relates the Allan Variance to the Power Spectrum Density (PSD) of continuous-time signals is not valid for real world discrete-time measurements and we propose a new equation that relates the Allan Variance to the PSD of the discrete-time signals and that allows to compute the Allan variance and its different variants in the frequency domain

Frequency Stability Measurement of Cryogenic Sapphire Oscillators with a Multichannel Tracking DDS and the Two-Sample Covariance

open6sìThis article shows the first measurement of three 100 MHz signals exhibiting fluctuations from 2×10-16 to parts in 10-15 for integration time τ between 1 s and 1 day. Such stable signals are provided by three Cryogenic Sapphire Oscillators (CSOs) operating at about 10 GHz, also delivering the 100 MHz output via a dedicated synthesizer. The measurement is made possible by a 6-channel Tracking DDS (TDDS) and the two-sample covariance tool, used to estimate the Allan variance. The use of two TDDS channels per CSO enables high rejection of the instrument background noise. The covariance outperforms the Three-Cornered Hat (TCH) method in that the background converges to zero "out of the box", with no need of the hypothesis that the instrument channels are equally noisy, nor of more sophisticated techniques to estimate the background noise of each channel. Thanks to correlation and averaging, the instrument background (AVAR) rolls off with a slope 1/√m, the number of measurements, down to 10-18 at τ=104 s. For consistency check, we compare the results to the traditional TCH method beating the 10 GHz outputs down to the MHz region. Given the flexibility of the TDDS, our methods find immediate application to the measurement of the 250 MHz output of the FS combs.openCalosso, Claudio E; Vernotte, Francois; Giordano, Vincent; Fluhr, Christophe; Dubois, Benoit; Rubiola, EnricoCalosso, Claudio E; Vernotte, Francois; Giordano, Vincent; Fluhr, Christophe; Dubois, Benoit; Rubiola, Enric

La mesure du temps à l’observatoire de Besançon : évolutions et tendances

La mesure du temps a toujours été une des missions essentielles de l’observatoire de Besançon. Ainsi, lorsque, rompant avec la tradition des échelles de temps astronomiques, le temps atomique international est devenu la référence officielle pour dater les événements (19675), l’observatoire s’était déjà doté d’horloges atomiques afin de contribuer à l’établissement du temps atomique international. Aujourd’hui encore, les recherches menées à l’observatoire dans ce domaine concernent des enjeux d’actualité, qu’il s’agisse des nouvelles techniques de transfert de temps que permettra le système Galileo ou l’étude de nouvelles échelles de temps utilisant les pulsars milliseconde. Plus généralement, nous nous intéresserons aussi au cours de l’article aux impacts sociétaux et aux retombées grand public qui résultent des avancées réalisées dans ce domaine.Vernotte François, Meyer François. La mesure du temps à l’observatoire de Besançon : évolutions et tendances. In: Pratique et mesure du temps. Actes du 129e Congrès national des sociétés historiques et scientifiques, « Le temps », Besançon, 2004. Paris : Editions du CTHS, 2011. pp. 9-18. (Actes des congrès nationaux des sociétés historiques et scientifiques, 129-5

SIDGET: A first composite clock prototype

International audienc

Can we define a best estimator in simple one-dimensional cases?

International audienceWhat is the best estimator for assessing a parameter of a probability distribution from a small number of measurements? Is the same answer valid for a location parameter like the mean as for a scale parameter like the variance? It is sometimes argued that it is better to use a biased estimator with low dispersion than an unbiased estimator with a higher dispersion. In which cases is this assertion correct? To answer these questions, we will compare, on a simple example, the determination of a location parameter and a scale parameter with three “optimal” estimators: the minimum-variance unbiased estimator, the minimum square error estimator, and the a posteriori mean

Three-Cornered Hat and Groslambert Covariance: A First Attempt to Assess the Uncertainty Domains

International audienceThe three-cornered hat method and the Groslambert covariance are very often used to estimate the frequency stability of each individual oscillator in a set of three oscillators by comparing them in pairs. However, no rigorous method to assess the uncertainties over their estimates has yet been formulated. In order to overcome this lack, this paper will first study the direct problem, i.e., the calculation of the statistics of the clock stability estimates by assuming known values of the true clock stabilities and then will propose a first attempt to solve the inverse problem, i.e., the assessment of a confidence interval over the true clock stabilities by assuming known values of the clock stability estimates. We show that this method is reliable from 5 equivalent degrees of freedom (EDF) and beyond

Three-Cornered Hat and Groslambert Covariance: A first attempt to assess the uncertainty domains

The three-cornered hat method and the Groslambert Covariance are very often used to estimate the frequency stability of each individual oscillator in a set of three oscillators by comparing them in pairs. However, no rigorous method to assess the uncertainties over their estimates has yet been formulated. In order to overcome this lack, this paper will first study the direct problem, i.e. the calculation of the statistics of the clock stability estimates by assuming known values of the true clock stabilities and then will propose a first attempt to solve the inverse problem, i.e. the assessment of a confidence interval over the true clock stabilities by assuming known values of the clock stability estimates. We show that this method is reliable from 5 Equivalent Degrees of Freedom (EDF) and beyond