Exponential inequalities in linear calibration problem

ABSTRACT

Calibration, also called inverse regression, is a classical problem which appears often in a regression setup under fixed design. The aim of this article is to propose a stochastic method which gives an estimated solution for a linear calibration problem. We establish exponential inequalities of Bernstein–Frechet type for the probability of the distance between the approximate solutions and the exact one. Furthermore, we build a confidence domain for the so-mentioned exact solution. To check the validity of our results, a numerical example is proposed.     

关于元朝的国号、年代与疆域问题

蒙古原是金朝属部,成吉思汗受金官封“属部统领”,乘金朝衰败,称帝建国,接着出兵攻金,逐鹿中原。其继承者先后灭金、灭南宋,实现了对全国的统治。其国号本为“大蒙古国”（在中原汉地称“大朝”）,第五代大汗忽必烈建新国号“大元”以示承继中原历朝统绪,同时保留原国号,遂称“大元大蒙古国”。元人以太祖成吉思皇帝为本朝开国君主,明初所修《元史》同此。所以元朝起始年代应为成吉思汗建国的1206年。成吉思汗及其继承者征服了直到小亚、东欧的西域许多国家,实行皇族领地分封,形成西北四个汗国,称“宗藩之国”。元朝皇帝是大蒙古国大汗,各宗藩奉为宗主,名义上统有包括各宗藩国的辽阔版图,实际上各宗藩国均自主统治本国;被宗藩称为“大汗之国”的元朝廷直接管辖之地为中书省和岭北、辽阳、河南、陕西、四川、甘肃、云南、湖广、江浙、江西十行省,以及吐蕃三道（算一个行省）,这就是元朝的疆域。     

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Asymptotic normality of kernel type regression estimators for random fields

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