Rank-based inference for linear models: asymmetric errors

A comparison of confidence intervals fromR-estimators in regression

“…Under mild regularity conditions, Koul, Sievers and McKean (1987) showed that their estimate is uniformly consistent under either symmetric or skewed errors. For a similar estimate of y see Aubuchon and Hettmansperger (1989). George and Osborne (George, 1993) propose an estimator of z derived from the asymptotic linearity result for rank statistics of Jureckova (1969).…”

On Hadamard Differentiability and Its Application to R-Estimation in Linear Models

“…Note that Çn is a diagonal matrix with diagonal elements to be the norm of the column vector of (D n -D n ). Hence, our Ca(ßR-ß) ' s more general than {ä(ß R -ß) considered by Jurecková (1971) and Aubuchon (1982), and the advantage of considering Çn(0R -ß) is that its asymptotic normality clearly implies the convergence rate in probability for each component of ß R is just the norm of the column vector of (D n -D n ), respectively.…”

“…We notice that M n (u) is a linear functional of ?n(t,u) = EjLibnilCYi^F-^tJ+b^u), te[0,l], ueR p , viz., M n (y) = J v(F l (0) dï*(t,u) = Φ(?£(·,μ)), o and E n (y) is (equivalent to) a functional of ( §£(·,v), FQ( ·,ν)) (see Theorem 4.3), viz., E a (y) J S^F^^y), y) d*(t) = *(?£(·,y), F£(-,y)), o where for te[0,l], veR P , The essential difficulty in the study of the asymptotic properties of M-and Restimators is the nonlinearity of the estimators, and often it is the case that the uniform asymptotic linear approximation of the estimators (Jurecková, 1971(Jurecková, , 1977 The asymptotic normality of R-estimators was studied by Jurecková (1971) general scores with technical assumptions on the structure of the design matrix D n and by Aubuchon (1982) (a proof can be found in Hettmansperger's (1984)) for a specific score. Applying an analogue of the Convexity Lemma (Pollard, 1991), which was established to derive the limit distribution using a technique analogous to the method by Jurecková (1977), Heiler and Willers (1988) relaxed Jurecková's (1971) assumptions on D n .…”

A general method for estimating standard errors

“…Finally, we report the results of a small Monte Carlo study which compares the performance of the translation method standard error estimate of ,8 with that of kernel density based estimate of Aubuchon and Hettmansperger (1989). The rank estimate ,8 and the latter standard error estimate are available via the rregress command in Minitab.…”