## 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

### Citations:

Zbl 0553.00008; Zbl 0493.62047