...algorithm the computation of confidence bands for the median function M whose running time is of order O([n.sup.2]). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.

Key Words: Computational complexity; Distribution-free; Monotonicity; Rademacber variables; Signs of residuals.

1. INTRODUCTION

Consider a pair ([X.sub.o], [Y.sub.o]) consisting of a deterministic or random covariable [X.sub.o] [member of] R and a random response [Y.sub.o] [member of] R. In many applications the conditional distribution of [Y.sub.o], given [X.sub.o], is known (or assumed) to depend isotonically on [X.sub.o]. One example are growth curves in medicine with [X.sub.o] and [Y.sub.o] being the age and body height of a newborn, respectively. Another example are so-called Engel curves in econometrics, where [X.sub.o] stands for the annual income of a randomly chosen household, while [Y.sub.o] denotes its expenditure for certain consumer goods. This monotone dependence implies that the median function M,

M(x) := Median([Y.sub.o] | [X.sub.o] = x),

is isotonic.

It is possible to obtain meaningful confidence sets for the function M without further regularity conditions. More precisely, suppose that we observe independent random pairs ([X.sub.1], [Y.sub.1]), ([X.sub.2], [Y.sub.2]), ..., ([X.sub.n], [Y.sub.n]) such that Median([Y.sub.i] | [X.sub.i] = x) = M(x). Our goal is to compute a confidence band (L, U) for M; that is, a pair of functions L, U : R [right arrow] [??] depending on the data ([X.sub.i], [Y.sub.i]) such that

(1.1) P{L,(x) [less than or equal to] M(x) [less than or equal to] U (x) for all x [member of] R}} [greater than or equal to] 1 - [alpha]

for some given [alpha] [member of]]0, 1[Here [??] stands for [-[infinity], [infinity]]. In order to get a first impression of the method developed subsequently, Figure 1 shows n = 500 simulated data pairs ([X.sub.i], [Y.sub.i]) together with the underlying median function M and a 95%-confidence band for it. Note that there is no point estimator for M involved; details are provided later.

[FIGURE 1 OMITTED]

To achieve (1.1), we condition on the values [X.sub.i] and rearrange the data such that [X.sub.i] [less than or equal to] [X.sub.2] [less than or equal to] ... [less than or equal to] [X.sub.n]. Then we focus on the median vector

m = [([m.sub.i].sup.n.sub.i=1] := [(M([X.sub.i])).sup.n.sub.i=1].

We only utilize that P{[Y.sub.i] < [m.sub.i]} [less than or equal to] 1/2 [less than or equal to] P{[Y.sub.i] [less than or equal to] [m.sub.i]} for all i, and that [m.sub.1] [less than or equal to] [m.sub.2] [less than or equal to] ... [less than or equal to] [m.sub.n]. Our primary goal is to compute a confidence band (L, U) for m; that is, a pair of vectors L, U [member of] [R.sup.n] depending on Y = [([Y.sub.i]).sup.n.sub.i=1] such that

(1.2) P{[L.sub.i] [less than or equal to] [m.sub.i] [less than or equal to] for l [less than or equal to] i [less than or equal to] n} [greater than or equal to] 1 - [alpha].

This goal can be achieved by means of sign tests. Given any candidate g [member of] [[??].sup.n] for m, we want to judge whether the signs of the corresponding residuals [Y.sub.i] - [g.sub.i] look like random signs. For that purpose we choose some test statistic To : [[??].sup.n] [right arrow] R such that

[T.sub.o]([v.sub.i], ..., [v.sub.n]) - [T.sub.o](]sign.bar]([v.sub.1], ..., [sign.bar]([v.sub.n]))

for arbitrary v [member of] [R.sup.n], where [sign.bar](r) := l{r > 0} - l{r [less than or equal to] 0}. Moreover, we require that [T.sub.o](v) [less than or equal to] [T.sub.o] (w) whenever v [less than or equal to] w component-wise. Large values of [T.sub.o](Y - g) indicate that some components of [g.sub.i] are too small, whereas [T.sub.o] (g - Y) is used for the opposite purpose. Thus, we compute the test statistic T(Y - g) given by

T(v) := max ([T.sub.o](v), [T.sub.o](-v)).

This value T(Y - g) is compared with T([zeta]), where [zeta] [member of] [{- l, 1}.sup.[eta]] is a Rademacher vector; that is, a random vector with independent components such that P {[[zeta].sub.i] = [+...

NOTE: All illustrations and photos have been removed from this article.



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