No abstract

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., is time variation in the data. Thus, the recovery of à`s mooth'' regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, non-stationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g. that of a time-varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near-optimal rate for the L 2 -risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coef®cient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non-stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear non-stationarity in its variance.

Abstract:The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoffs estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chemoffs estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1 ! A propos du bootstrap pour des estimateurs convergeant a la vitesse racine cubique R h m C : Les auteurs Btudient I'application du bootstrap h une classe d'estimateurs qui convergent une vitesse et vers une loi non standard. 11s prksentent un cadre thBorique pour I'Btude de son comportement asymptotique. Une simulation dtmontre que dans le cas d'un estimateur du mode de Chernoff, la probabilitB de couverture de I'intervalle de confiance bootstrap de base est grandement infkrieure au niveau prescrit, alors que celle des intervalles de type percentile dBpasse le niveau prescrit. C'est un rare cas oii les intervalles de confiance de base et percentile ont un comportement si diffBrent malgrB des longueurs identiques. Dans le cas de I'estimateur de Chernoff, si la distribution est symBtrique, il est possible d'appliquer le bootstrap h partir d'un estimateur lisse et symBtrique de la distribution qui menera h des intervalles bootstrap de base dont la probabilitd de couverture asymptotique sera la bonne, alors que celle de I'intervalle percentile convergera vers 1 !

No abstract

The proven optimality properties of empirical Bayes estimators and their documented successful performance in practice have made them popular. Although many statisticians have used these estimators since the landmark paper of James and Stein (1961), relatively few have proposed techniques for protecting them from the effects of outlying observations or outlying parameters. One notable series of studies in protection against outlying parameters was conducted by Efron and Morris (1971, 1972, 1975). In the fully Bayesian case, a general discussion on robust procedures can be found in Berger (1984, 1985). Here we implement and evaluate a different approach for outlier protection in a random‐effects model which is based on appropriate specification of the prior distribution. When unusual parameters are present, we estimate the prior as a step function, as suggested by Laird and Louis (1987). This procedure is evaluated empirically, using a number of simulated data sets to compare the effects of the step‐function prior with those of the normal and Laplace priors on the prediction of small‐area proportions.