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Tightness of the sequence of empiric c.d.f. processes defined from regression fractiles. (English) Zbl 0568.62065

Robust and nonlinear time series analysis, Proc. Workshop, Heidelberg/Ger. 1983, Lect. Notes Stat., Springer-Verlag 26, 231-246 (1984).
[For the entire collection see Zbl 0553.00008.]
The linear regression model \(Y=(Y_ 1,...,Y_ n)'=X\beta +R\) is investigated where \(\beta '=(\beta_ 1\gamma ')\in {\mathbb{R}}^{p+1}\) is a parameter vector \((\beta_ 1\in {\mathbb{R}}^ 1\), \(\gamma \in {\mathbb{R}}^ p)\) and \(X=(w_ 1,...,w_ n)'\) the design matrix with \(w'_ i=(1,x'_ i)\), \(x_ i\in {\mathbb{R}}^ p\), \(\Sigma_ iX_ i=0\). The problem is to estimate the distribution function F(\(\cdot)\) of the components \(R_ i\) of the random error term \(R=(R_ 1,...,R_ n)'.\)
G. Basset and R. Koenker [see J. Am. Stat. Assoc. 77, 407-415 (1982; Zbl 0493.62047)] proposed to minimize the weighted sum of ”absolute” residuals \(\sum_{i}\{\theta (y_ i-w'_ i\beta)^++(1- \theta)(y-w'_ i\beta)^-\}^ w.\)r. to \(\beta\) (in principle: for all \(\theta\in (0,1))\), thus obtaining \({\hat \beta}(\theta)=({\hat \beta}_ 1(\theta),{\hat \gamma}(\theta))\) with an increasing function \({\hat \beta}{}_ 1(\theta)\) (normalized by continuity from the right). The ”quantile process” \({\hat \theta}(\beta_ 1):=\sup \{\theta \in [0,1]:{\hat \beta}_ 1(\theta)\leq \beta_ 1\}\) is an estimate for F(\(\cdot)\). It is a right continuous function with left limits (i.e. in \(D_ 0\) with Skorokhod topology) and by linear interpolation of \({\hat \beta}{}_ 1(\cdot)\), a continuous version \(\theta^*(\beta_ 1)\) may be defined as well belonging to \(C_ 0\) (with uniform convergence on compact sets).
The paper shows that \(\sqrt{n}({\hat \theta}(\beta_ 1)-F(\beta_ 1))\), \(\sqrt{n}(\theta^*(\beta_ 1)-F(\beta_ 1))\) are tight sequences in \(D_ 0\) resp. \(C_ 0\) under suitable conditions (e.g.: \(X'X/n\to \Sigma\) for \(n\to \infty\), \(| F''|\) bounded, \(\max_{ij}| x_{ij}| <\infty\) etc.). This implies the convergence to a transformed Brownian bridge with applications to goodness of fit tests for F, bootstrap methods, etc.
Reviewer: H.H.Bock

MSC:

62J05 Linear regression; mixed models
62G05 Nonparametric estimation
60F05 Central limit and other weak theorems