Author Contact: David E. Giles, Dept. of Economics, University of Victoria, P.O. Box 1700, STN CSC, Victoria, B.C., Canada V8W 2Y2; e-mail: [email protected]; Phone: (250) 721-8540; FAX: (250) 721-6214 Abstract We show that the full asymptotic null distribution for Watson’s 2 N U statistic, modified for discrete data, can be computed simply and exactly by standard methods. Previous approximate qua...

and u is the total variation (or just variation) of σ. For example, the total variation of the word σ = 1121415221 ∈ [5] is u(σ) = 0 + 1 + 1 + 3 + 3 + 4 + 3 + 0 + 1 = 16. Several authors have studied the statistic u(σ) in different contexts. For example, Curtis [5] discussed the problem of finding a word σ = σ1σ2 · · ·σn over the alphabet {a1, a2, . . . , ak} which maximizes the risk function ∑...

This paper introduces nonparametric U-statistics for testing the curvature (e.g., linearity, concavity, or convexity) of an unknown regression function. The U-statistics are based on pairwise slopes in the data. No choice of smoothing parameters is required for the global version of the statistics or their (asymptotic) standard errors, making them simple to compute. The proposed tests are analo...

The problem of ranking/ordering instances, instead of simply classifying them, has recently gained much attention in machine learning. In this paper we formulate the ranking problem in a rigorous statistical framework. The goal is to learn a ranking rule for deciding, among two instances, which one is “better,” with minimum ranking risk. Since the natural estimates of the risk are of the form o...

Consider n players whose ‘‘scores’’ are independent and identically distributed values fXigi 1⁄4 1 from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p-0 and (ii) F is uniform on f1;2; . . . ;Ng; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the U-statistic W whic...

A single bootstrap test is to be based on a statistic τ in asymptotic P value form. Rejection by an asymptotic test at level α is thus the event τ < α. Rejection by the bootstrap test is the event τ < Q(α, μ∗), where μ∗ is the bootstrap DGP, and Q(α, μ∗) is the (random) α--quantile of the distribution of the statistic τ as generated by μ∗. We define two random variables that are deterministic f...

in some quality control applications, the quality of a process or a product is described by the relationship between a response variable and one or more explanatory variables, called a profile. moreover, in most practical applications, the qualitative characteristic of a product/service is vague, uncertain and linguistic and cannot be precisely stated. the purpose of this paper is to propose a ...