Confidence Bands for Convex Median Curves using Sign-Tests

Suppose that one observes pairs (x1,Y1), (x2,Y2), ..., (xn,Yn), where x1 < x2 < ... < xn are fixed numbers while Y1, Y2, ..., Yn are independent random variables with unknown distributions. The only assumption is that Median(Yi) = f(xi) for some unknown convex or concave function f. We present a confidence band for this regression function f using suitable multiscale sign tests. While the exact computation of this band seems to require O(n4) steps, good approximations can be obtained in O(n2) steps. In addition the confidence band is shown to have desirable asymptotic properties as the sample size n tends to infinity.

Marshall's Lemma for Convex Density Estimation

Marshall's (1970) lemma is an analytical result which implies root-n-consistency of the distribution function corresponding to the Grenander (1956) estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on [0,\infty).

Large deviations for heavy-tailed random elements in convex cones

We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.

Fractionated stereotactic radiotherapy with static field conformal and non coplanar arcs for pediatric patients with craniopharyngioma: analysis of long term visual outcome and endocrine toxicity

We assessed the efficacy and the toxicity for pediatric craniopharyngioma patients of fractionated stereotactic radiotherapy (FSRT). Between May 2000 and May 2009, 9 patients (male to female ratio, 5:4) with craniopharyngiomas underwent FSRT (median dose, 54 Gy). Among the 9 patients, 6 received radiation therapy (RT) for recurrent tumors and 3 for residual disease as adjuvant therapy after incomplete surgery. Median tumor 3 volume was 2.3 cm (range, 0.1-5.8). The median target coverage was 93.7% (range 79.3-99.8%). The median conformity index was 0.94 (range, 0.6-1.4). Dose to the hippocampal region was assessed for all patients. After a median follow-up of 62.5 months (range, 32-127)the treated volume decreased in size in four of eight patients (50%). One patient was lost to follow-up. Local control and survival rates at 3 years were 100% and there were no marginal relapses. One patient, with a chronic bilateral papillary oedema after surgery, visual defect deteriorated after FSRT to a complete hemianopsia. One male patient with normal pituitary function before FSRT presented with precocious puberty at the age of 7.4 years, 24 months after FSRT. Four patients (50%) were severely obese at their last visit. FSRT is a safe treatment option for craniopharyngioma after incomplete resection.