Christian Léger

Bio: Christian Léger is an academic researcher from Université de Montréal. The author has contributed to research in topics: Estimator & Asymptotically optimal algorithm. The author has an hindex of 13, co-authored 25 publications receiving 679 citations. Previous affiliations of Christian Léger include University of Paris.

Bootstrap technology and applications

Abstract: Bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis. Although these methods depend on the availability of fast, inexpensive computing, they offer the potential for highly accurate methods of inference. Moreover, they can even eliminate the need to impose a convenient statistical model that does not have a strong scientific basis. In this article, we review some bootstrap methods, emphasizing applications through illustrations with some real data. Special attention is given to regression, problems with dependent data, and choosing tuning parameters for optimal performance.

CT and MR imaging findings in adults with cerebellar medulloblastoma: comparison with findings in children

Abstract: The goal of this study was to determine if certain imaging features suggest the diagnosis of cerebellar medulloblastoma in adults and to determine how often the classic CT appearance seen in children is present in adults. The study included 28 adult patients with proved cerebellar medulloblastoma. The tumor was located in the cerebellar vermis in 14 patients and in a cerebellar hemisphere in 14 patients. Thirteen patients had unenhanced CT of the brain, all patients had contrast-enhanced CT, and eight patients had unenhanced MR imaging. The imaging features in adults were compared with those in children, as described in the literature. In our adult patients, all tumors were hyperdense compared with gray matter on unenhanced CT and showed a slight to moderate increase in density after injection of contrast medium. Thirteen lesions had well-defined margins, and 15 had poorly defined margins. Low-density areas consistent with cystic and necrotic degeneration were detected in 23 (82%) of the 28 tumors. By comparison, in children, medulloblastoma usually originates in the vermis. As in adults, the mass is hyperdense on unenhanced CT, but enhances markedly and homogeneously after injection of contrast medium. Usually no evidence of cyst formation or necrosis is seen, and the tumor margins are well defined. This classic CT appearance of medulloblastoma in children was identified in only three (11%) of the 28 adult patients. Medulloblastoma has a variable MR appearance in both children and adults. On T2-weighted images, lesions are hypo-, iso-, or hyperintense compared with normal gray matter. The CT findings of medulloblastoma in adults usually differ from those of medulloblastoma in children. The tumor has a variable and nonspecific appearance in adults and should always be considered in the differential diagnosis of a mass in the posterior fossa.

A survey of bootstrap methods in finite population sampling

Abstract: We review bootstrap methods in the context of survey data where the effect of the sampling design on the variability of estimators has to be taken into account. We present the methods in a unified way by classifying them in three classes: pseudo-population, direct, and survey weights methods. We cover variance estimation and the construction of confidence intervals for stratified simple random sampling as well as some unequal probability sampling designs. We also address the problem of variance estimation in presence of imputation to compensate for item non-response.

Bootstrap choice of tuning parameters

Abstract: Consider the problem of estimating θ=θ(P) based on datax n from an unknown distributionP. Given a family of estimatorsT n, β of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT n, β . In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β, $$\hat \beta _n$$ , is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on $$T_{n,\hat \beta _n }$$ , are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

The stationary bootstrap

Abstract: This article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations. Previously, a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m-dimensional joint distribution of m consecutive observations, where m is fixed. This procedure has been generalized by constructing a “blocks of blocks” resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole (i.e., infinite-dimensional) joint distribution of the stationary sequence of observations. These methods share the construction of resampling blocks of observations to form a pseudo-time series, so that the statistic of interest may be recalculated based on the resampled data set. But in the context of applying this method to stationary data, it is natural...

The knowledge complexity of interactive proof-systems

Abstract: Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

Entity authentication and key distribution

Abstract: We provide the first formal treatment of entity authentication and authenticated key distribution appropriate to the distributed environment. Addressed in detail are the problems of mutual authentication and authenticated key exchange for the symmetric, two-party setting. For each we present a definition, protocol, and proof that the protocol meets its goal, assuming only the existence of a pseudorandom function.