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Motivated by the fact that circular or spherical data are often much concentrated around a location $\pmb\theta$, we consider inference about $\pmb\theta$ under "high concentration" asymptotic scenarios for which the probability of any fixed spherical cap centered at $\pmb\theta$ converges to one as the sample size $n$ diverges to infinity. Rather than restricting to Fisher-von Mises-Langevin distributions, we consider a much broader, semiparametric, class of rotationally symmetric distributions indexed by the location parameter $\pmb\theta$, a scalar concentration parameter $\kappa$ and a functional nuisance $f$. We determine the class of distributions for which high concentration is obtained as $\kappa$ diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on $\pmb\theta$ in asymptotic scenarios where $\kappa_n$ diverges to infinity at an arbitrary rate with the sample size $n$. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on $\pmb\theta$ show consistency rates that depend on $f$. Using asymptotics "\`a la Le Cam", we show that the spherical mean is, at any $f$, a parametrically super-efficient estimator of $\pmb\theta$ and that the Watson and Wald tests for $\mathcal{H}_0:{\pmb\theta}={\pmb\theta}_0$ enjoy similar, non-standard, optimality properties. We illustrate our results through simulations and treat a real data example. On a technical point of view, our asymptotic derivations require challenging expansions of rotationally symmetric functionals for large arguments of $f$., Comment: 51 pages, 5 figures

We propose a strategy for computing estimators in some non-standard M-estimation problems, where the data are distributed across different servers and the observations across servers, though independent, can come from heterogeneous sub-populations, thereby violating the identically distributed assumption. Our strategy fixes the super-efficiency phenomenon observed in prior work on distributed computing in (i) the isotonic regression framework, where averaging several isotonic estimates (each computed at a local server) on a central server produces super-efficient estimates that do not replicate the properties of the global isotonic estimator, i.e. the isotonic estimate that would be constructed by transferring all the data to a single server, and (ii) certain types of M-estimation problems involving optimization of discontinuous criterion functions where M-estimates converge at the cube-root rate. The new estimators proposed in this paper work by smoothing the data on each local server, communicating the smoothed summaries to the central server, and then solving a non-linear optimization problem at the central server. They are shown to replicate the asymptotic properties of the corresponding global estimators, and also overcome the super-efficiency phenomenon exhibited by existing estimators.

We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with $m_{0}$ components, the optimal local minimax rate of estimation of a mixing distribution with $m$ components is $n^{-1/(4(m-m_{0})+2)}$. This corrects a previous paper by Chen [Ann. Statist. 23 (1995) 221–233]. ¶ By contrast, it turns out that there are estimators with a (nonuniform) pointwise rate of estimation of $n^{-1/2}$ for all mixing distributions with a finite number of components.

Recent approaches in causal inference have proposed estimating average causal effects that are local to some subpopulation, often for reasons of efficiency. These inferential targets are sometimes data-adaptive, in that they are dependent on the empirical distribution of the data. In this short note, we show that if researchers are willing to adapt the inferential target on the basis of efficiency, then extraordinary gains in precision can potentially be obtained. Specifically, when causal effects are heterogeneous, any asymptotically normal and root- n $n$ consistent estimator of the population average causal effect is superefficient for a data-adaptive local average causal effect.

In finite mixture models, apart from underlying mixing measure, true kernel density function of each subpopulation in the data is, in many scenarios, unknown. Perhaps the most popular approach is to choose some kernel functions that we empirically believe our data are generated from and use these kernels to fit our models. Nevertheless, as long as the chosen kernel and the true kernel are different, statistical inference of mixing measure under this setting will be highly unstable. To overcome this challenge, we propose flexible and efficient robust estimators of the mixing measure in these models, which are inspired by the idea of minimum Hellinger distance estimator, model selection criteria, and superefficiency phenomenon. We demonstrate that our estimators consistently recover the true number of components and achieve the optimal convergence rates of parameter estimation under both the well- and mis-specified kernel settings for any fixed bandwidth. These desirable asymptotic properties are illustrated via careful simulation studies with both synthetic and real data., 41 pages, 5 figures

The connections between information pooling and adaptability as well as superefficiency are considered. Separable rules, which figure prominently in wavelet and other orthogonal series methods, are shown to lack adaptability; they are necessarily not rate-adaptive. A sharp lower bound on the cost of adaptation for separable rules is obtained. We show that adaptability is achieved through information pooling. A tight lower bound on the amount of information pooling required for achieving rate-optimal adaptation is given. Furthermore, in a sharp contrast to the separable rules, it is shown that adaptive non-separable estimators can be superefficient at every point in the parameter spaces. The results demonstrate that information pooling is the key to increasing estimation precision as well as achieving adaptability and even superefficiency.

Copyright © 2013 Springer Netherlands. Superefficiency is an important extension of DEA that overcomes some limitations of the traditional models, specifically allowing ranking of efficient units and a unique set of weights for those units. Weights restriction is a well-known technique in the DEA field. When those techniques are applied, weights cluster around its new limits, making its evaluation dependent of its levels. This chapter introduces a new approach to weights adjustment by goal programming techniques, avoiding the imposition of hard restrictions that can even lead to unfeasibility. This method results in models that are more flexible.

In some estimation problems, especially in applications dealing with information theory, signal processing and biology, theory provides us with additional information allowing us to restrict the parameter space to a finite number of points. In this case, we speak of discrete parameter models. Even though the problem is quite old and has interesting connections with testing and model selection, asymptotic theory for these models has hardly ever been studied. Therefore, we discuss consistency, asymptotic distribution theory, information inequalities and their relations with efficiency and superefficiency for a general class of $m$-estimators., Comment: Published in at http://dx.doi.org/10.1214/11-STS371 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In 1952 Lucien Le Cam announced his celebrated result that, for regular univariate statistical models, sets of points of superefficiency have Lebesgue measure zero. After reviewing the turbulent history of early studies of superefficiency, I suggest using the notion of computability as a tool for exploring the phenomenon of superefficiency. It turns out that only computable parameter points can be points of superefficiency for computable estimators. This algorithmic version of Le Cam's result implies, in particular, that sets of points of superefficiency not only have Lebesgue measure zero but are even countable., Published in at http://dx.doi.org/10.1214/09-STS279 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This paper extends Imbens and Manski's (2004) analysis of confidence intervals for interval identified parameters. For their final result, Imbens and Manski implicitly assume superefficient estimation of a nuisance parameter. This appears to have gone unnoticed before, and it limits the result's applicability. I re-analyze the problem both with assumptions that merely weaken the superefficiency condition and with assumptions that remove it altogether. Imbens and Manski's confidence region is found to be valid under weaker assumptions than theirs, yet superefficiency is required. I also provide a different confidence interval that is valid under superefficiency but can be adapted to the general case, in which case it embeds a specification test for nonemptiness of the identified set. A methodological contribution is to notice that the difficulty of inference comes from a boundary problem regarding a nuisance parameter, clarifying the connection to other work on partial identification.

At a superficial level, the idea of maximum likelihood must be prehistoric: early hunters and gatherers may not have used the words ``method of maximum likelihood'' to describe their choice of where and how to hunt and gather, but it is hard to believe they would have been surprised if their method had been described in those terms. It seems a simple, even unassailable idea: Who would rise to argue in favor of a method of minimum likelihood, or even mediocre likelihood? And yet the mathematical history of the topic shows this ``simple idea'' is really anything but simple. Joseph Louis Lagrange, Daniel Bernoulli, Leonard Euler, Pierre Simon Laplace and Carl Friedrich Gauss are only some of those who explored the topic, not always in ways we would sanction today. In this article, that history is reviewed from back well before Fisher to the time of Lucien Le Cam's dissertation. In the process Fisher's unpublished 1930 characterization of conditions for the consistency and efficiency of maximum likelihood estimates is presented, and the mathematical basis of his three proofs discussed. In particular, Fisher's derivation of the information inequality is seen to be derived from his work on the analysis of variance, and his later approach via estimating functions was derived from Euler's Relation for homogeneous functions. The reaction to Fisher's work is reviewed, and some lessons drawn., Comment: Published in at http://dx.doi.org/10.1214/07-STS249 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The problem of loss adaptation is investigated: given a fixed parameter, the goal is to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. Given the class of estimator sequences that achieve the minimax rate, over a fixed Besov space, for estimating the entire function a lower bound is given on the performance for estimating the function at each point. This bound is larger by a logarithmic factor than the usual minimax rate for estimation at a point when the global and local minimax rates of convergence differ. A lower bound for the maximum global risk is given for estimators that achieve optimal minimax rates of convergence at every point. An inequality concerning estimation in a two-parameter statistical problem plays a key role in the proof. It can be considered as a generalization of an inequality due to Brown and Low. This may be of independent interest. A particular wavelet estimator is constructed which is globally optimal and which attains the lower bound for the local risk provided by our inequality.

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd-valued processes and to N-valued processes. In the particular case where squareintegrable local time does exist, it is shown that our estimator is strictly better than the local time estimator over F0.

Dieses Discussion Paper beschäftigt sich mit der Analyse der technischen und Skaleneffizienz russischer Agrarunternehmen im Transformationsprozess. Insbesondere wird hierbei der Einfluss der Faktoren Betriebsgröße, Rechtsform und Spezialisierung auf die Effizienz untersucht. Um die Auswirkungen der Reformen und die Veränderungen in den Agrarunternehmen zu erfassen, werden die Jahre 1993 und 2000 miteinander verglichen. Als methodisches Instrument wurde ein input-orientiertes Modell der Data Envelopment Analysis (DEA) zur Effizienzanalyse sowie DEA-Supereffizienzmodell zur Identifikation von Ausreißern und Überprüfung der Stabilität der Daten gewählt. Analysiert wurden im Jahr 1993 373 und im Jahr 2000 367 Nachfolgebetriebe ehemaliger Kolchosen und Sowchosen in Oblast Moskau. Die Ergebnisse der DEA-Modellrechnungen zeigen eine Verbesserung der durchschnittlichen Effizienz im Vergleich von 1993 zu 2000. Bei der Betrachtung der Einflussfaktoren auf die technische Effizienz hat die Wahl der Rechtsform keinen Einfluss. Hingegen beeinflussen Betriebsgröße und Spezialisierung die technische und Skaleneffizienz in beiden Jahren. This Discussion Paper has been occupied with the analysis of the technical and scale efficiency of Russian farms during the transformation process. Especially the influence of the factors farm size, legal form and specialization is analysed. In order to list the effects of reforms and the changes in farms, the observation years 1993 and 2000 are compared. As a methodical instrument has been chosen an input-oriented model of Data Envelopment Analysis (DEA) for efficiency measurement as well as the DEA-Superefficiency model for identification of outliers and inspection of data stability. We analysed in year 1993 373 and the year of 2000 367 successors of former kolkhozes and state farms in Oblast Moscow. The results of the DEA model calculations show an improvement in average efficiency from 1993 to 2000. In consideration of the performance-influencing factors to technical efficiency, the choice of the legal form does not have any influence. On the other hand farm size and specialization influence the technical and scale efficiency in both years.

Some models are presented in this paper which extend the concept of measuring superefficiency to the useful case of variable returns-to-scales (VRS), thus enabling the ranking of efficient as well as inefficient units. Two models, namely the Universal Radial Model and the Universal Additive Model, are presented that also have strong invariance properties (units and translation invariance). For both of these models a method for normalising the efficiency scores on a (0-1+) scale is presented. These models have been implemented in a software package and applied to the ranking of units in an industrial context.

The asymptotic risk of efficient estimators with Kullback–Leibler loss in smoothly parametrized statistical models is $k/2_n$, where $k$ is the parameter dimension and $n$ is the sample size. Under fairly general conditions, we given a simple information-theoretic proof that the set of parameter values where any arbitrary estimator is superefficient is negligible. The proof is based on a result of Rissanen that codes have asymptotic redundancy not smaller than $(k/2)\log n$, except in a set of measure 0.

Fixed parameter asymptotic statements are often used in the context of nonparametric curve estimation problems (e.g., nonparametric density or regression estimation). In this context several forms of superefficiency can occur. In contrast to what can happen in regular parametric problems, here every parameter point (e.g., unknown density or regression function) can be a point of superefficiency. ¶ We begin with an example which shows how fixed parameter asymptotic statements have often appeared in the study of adaptive kernel estimators, and how superefficiency can occur in this context. We then carry out a more systematic study of such fixed parameter statements. It is shown in four general settings how the degree of superefficiency attainable depends on the structural assumptions in each case.

International audience; [...]linear markovian model for sériés of correlated exponential random variables is [con]sidered. This is analytically tractable and easy to generate, and the conditional [p]robability density of tbe observations is discontinuons. An estimât or is proposed [...]r the parameters, and is shown to be strongly consistent and asymptotically effient. Its relationship with the maximum likelihood estimator is given. For one of [...]e parameters the estimator is "superefficient" and for the others it is asymptoti [...]llly normal.; [On] considère un modèle linéaire markovien pour des suites de variables aléatoires [ex]ponentielles corrélées, qui est analytiquement traitable et facile à simuler. La [den]sité conditionnelle des observations est discontinue. On propose un estimateur des [pa]ramètres dont on démontre la consistance forte et l'efficacité asymptotique. On [d]onne également sa relation avec l'estimateur de maximum de vraisemblance. Pour un [d]es paramètres l'estimateur est "superefficace" et pour les autres il est asymptoti[...]ement normal.

For a particular pseudoloss function, local asymptotic minimaxity and admissibility in the sense of Hajek and Le Cam are studied when probability measures are replaced by certain capacities ($\epsilon$-contamination, total variation). A minimax bound for arbitrary estimator sequences is established, admissibility of minimax estimators is proved, and it is shown that minimax estimators must necessarily have an asymptotic expansion in terms of a truncated logarithmic derivative.