Observing how long a dynamical system takes to return to some state is one of the most simple ways to model and quantify its dynamics from data series. This work proposes two formulas to estimate the KS entropy and a lower bound of it, a sort of Shannon's entropy per unit of time, from the recurrence times of chaotic systems. One formula provides the KS entropy and is more theoretically oriented since one has to measure also the low probable very long returns. The other provides a lower bound for the KS entropy and is more experimentally oriented since one has to measure only the high probable short returns. These formulas are a consequence of the fact that the series of returns do contain the same information of the trajectory that generated it. That suggests that recurrence times might be valuable when making models of complex systems.
Let Z 1 , Z 2 , ... be i.i.d. random variables with tail behaviour, where r is a regularly varying function at infinity and R is a positive constant. We consider the problem of estimating the exponential tail coefficient R, by methods mainly based on least squares considerations. Using a geometrical reasoning, we introduce a consistent estimator, whose values lay between the least squares estimates proposed by Schultze and Steinebach (1996). We investigate here the weak asymptotic properties of this geometrictype estimator, showing in particular that, under general conditions, its distribution is asymptotically normal. The results are applied to the related problem of estimating the adjustment coefficient in risk theory (Csörgő and Steinebach (1991)) and a simulation study is performed in order to illustrate the finite sample behaviour of this estimator.
Neuroendocrine tumors (NETs) are rare, representing 0.5% of all newly diagnosed malignancies. Rectal and anal canal (AC) NETs account for less than 1% of all rectal and AC cancers. Review our institutional experience on NET of the rectum and AC, with emphasis on demographic, histological and treatment features and oncologic outcomes. The study group was identified from the Portuguese Regional South Oncological Registry. From 2000 to 2014, 22 patients with rectal or AC NETs were treated at our institution. Medical records were retrospectively reviewed. There were 12 males (54.5%) and 10 females (45.5%) and the median age at diagnosis was 59.5 years. The majority had rectal NET (81.8%). All 4 patients with AC NETs had neuroendocrine carcinoid (NEC) tumors. Of the patients with rectal NETs, 3 had NEC and 15 had NET, mainly G1. Different approaches to treatment were made according to histological and staging features. After an average follow-up of 39.1 months, 16 patients were alive and only one with evidence of disease. The median time to progression was 12.4 months and the liver was the most frequent site of metastasis. The European and North American Neuroendocrine Societies offer guidelines for the treatment of rectal NETs. However, for AC NETs there are only small series and not prospective studies due to their rarity, hence the importance to report institutional experience. Our practice demonstrated that primary excisional treatment, regardless the histology, provides a favorable prognosis and long survival.
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