This chapter describes some procedures that are commonly referred to as distribution-free or nonparametric methods. It first focuses on the problem of unbiased (nonparametric) estimation. The theory of U-statistics is developed since many estimators and test statistics may be viewed as U-statistics. The chapter then deals with some common hypotheses testing problems and investigates applications of order statistics in nonparametric methods, mainly focusing on tolerance intervals for distributions, coverages, and confidence interval estimates for quantiles and location parameters. The chapter finally considers underlying assumptions in some common parametric problems and the effect of relaxing these assumptions.
In this article we prove a strong law of large numbers for Borel measurable nonseparably valued random elements in the case of generalized random sets.
Linear models in which the unobserved error constitutes a realization of some stationary ARMA process or, equivalently, ARMA processes with a linear regression trend, are considered under unspecified innovation densities. Due to serial dependence among the observations, the classical rank-based techniques, which have been developed for linear models with independent observations and unspecified error densities, do not apply; nor do the existing rank-based procedures for serial dependence problems, where the observations are assumed to be stationary in the mean (or the median). Moreover, all problems of practical interest (testing the significance of a subset of regression coefficients, identifying the orders p and q of the ARMA (p, q) dependence, overall diagnostic checking of the model,...) involve nuisance parameters. Typically, one is interested either in the regression trend, and the serial dependence parameters are nuisance parameters; or the serial dependence structure is the main concern, and trend somehow has to be removed. A rank-based approach to such problems thus not only requires extending the classical Hajek-type theory of rank tests to serially dependent situations, it also requires a generalized theory of aligned rank tests. This is the purpose of the present paper. The key result is a local asymptotic normality (LAN) result involving a rank-measurable central sequence; depending on the model considered (with symmetric or totally unspecified innovation densities), the ranks to be used are either signed or unsigned. This LAN result, along with a particular local asymptotic linearity property, implies the local asymptotic sufficiency of (aligned) ranks for a broad class of testing problems-mainly, testing linear restrictions on the parameters of the model. Asymptotically invariant aligned rank tests which are locally asymptotically most stringent similarly are derived. Unlike former results on aligned rank tests for linear models with independent obser(This abstract was borrowed from another version of this item.)
Let g(λ) be the spectral density of a stationary process and let fθ(λ), θ ∈ Θ, be a fitted spectral model for g(λ). A minimum contrast estimator of θ is defined that minimizes a distance between , where is a nonparametric spectral density estimator based on n observations. It is known that is asymptotically Gaussian efficient if g(λ) = fθ(λ). Because there are infinitely many candidates for the distance function , this paper discusses higher order asymptotic theory for in relation to the choice of D. First, the second-order Edgeworth expansion for is derived. Then it is shown that the bias-adjusted version of is not second-order asymptotically efficient in general. This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically second-order asymptotically efficient after a suitable bias adjustment (e.g., Ghosh, 1994, Higher Order Asymptotics, p. 57). The paper establishes therefore that for semiparametric estimation it does not hold in general that “first-order efficiency implies second-order efficiency.” The paper develops verifiable conditions on D that imply second-order efficiency.This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor. The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly improved the paper.
The problem of testing a given autoregressive moving average (ARMA) model (in which the density of the generating white noise is unspecified) against other ARMA models is considered.A distribution-free asymptotically most powerful test, based on a generalized linear serial rank statistic, is provided against contiguous ARMA alternatives with specified coefficients.In the case in which the ARMA model in the alternative has unspecified coefficients, the asymptotic suffi- ciency (in the sense of Hajek) of a finite-dimensional vector of rank statistics is established.This asymptotic sufficiency is used to derive an asymptotically maximin most powerful test, based on a generalized quadratic serial rank statistic.The as- ymptotically maximin optimal test statistic can be interpreted as a rank-based, weighted version of the classical Box-Pierce portmanteau statistic, to which it reduces, in some particular problems, under gaussian assumptions.Section 1.
