Contiguous alternatives which preserve Cramér-type large deviations for a general class of statistics

Let P _N and Q _N , N1, be two possible probability distributions of a random vector X _N =(X_N1,...,X_NN), whose components are independent. Suppose P _N and Q _N have respective densities % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0YaaCbi% aeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6eaaeqaaO% Gaaiykaaaa!4DEC!\[p_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \mathop \theta \limits^\_ _N )\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamyCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0IaeqiU% de3aaSbaaSqaaiaad6eacaWGPbaabeaakiaacMcaaaa!4DA5!\[q_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \theta _{Ni} )\], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6ea% aeqaaOGaeyypa0JaamOtamaaCaaaleqabaGaeyOeI0IaaGymaaaakm% aaqahabaGaeqiUde3aaSbaaSqaaiaad6eacaWGPbaabeaaaeaacaWG% PbGaeyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa!4C75!\[\mathop \theta \limits^\_ _N = N^{ - 1} \sum\limits_{i = 1}^N {\theta _{Ni} } \], such that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaacaqGTbGaaeyyaiaabIhaaSqaaiaaigdacqGHKjYOcaWG% PbGaeyizImQaamOtaaqabaGccaGG8bGaeqiUde3aaSbaaSqaaiaad6% eacaWGPbaabeaakiabgkHiTmaaxacabaGaeqiUdehaleqabaGaai4x% aaaakmaaBaaaleaacaWGobaabeaakiaacYhacqGH9aqpcaWGpbGaai% ikaiaad6eadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaGOmaaaa% kiaacMcaaaa!5647!\[\mathop {{\rm{max}}}\limits_{1 \le i \le N} |\theta _{Ni} - \mathop \theta \limits^\_ _N | = O(N^{ - 1/2} )\], f(x)>0 for almost every real x, f is absolutely continuous, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaaciGGZbGaaiyDaiaacchaaSqaaiabeI7aXjaad+gacqGH% KjYOcqaH4oqCcqGHKjYOcqaH4oqCcaWGVbaabeaakmaapedabaGaai% 4waiaadAgaaSqaaiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaai4j% aiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykamaaCaaaleqabaGaaG% Omaaaakiaac+cacaWGMbGaaiikaiaadIhacaGGPaGaamizaiaadIha% cqGH8aapcqGHEisPaaa!5ECE!\[\mathop {\sup }\limits_{\theta o \le \theta \le \theta o} \int_\infty ^\infty {[f} '(x - \theta )^2 /f(x)dx < \infty \] for some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaeqiUde3aaSbaaSqaaiaaicdaaeqaaOGaeyOpa4JaaGimaaaa!3FD4!\[\theta _0 > 0\]. The contiguity of {q _N } to {p _N } is well known. In this paper it is proven that under these conditions {Q _N } preserves C.-T.L.D. (Cramér-type large deviation) from {P _N } for a general class of statistics which includes R-, U- and L-statistics as members. That means, for any {S _N =S_N(X_N)} from , a C.-T.L.D. theorem with range Cxo(N) (any C0), 0<4^-1, holds for {S _N } under {P _N }, implying that the same theorem holds for {S _N } under {Q _N }. It also provides a quick and simple way to establish C.-T.L.D. results for statistics under {Q _N }.Research supported in part by grant VE87080 from the National Science Council, Republic of China.Part of the research was done while the author was visiting the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan.     

Higher order asymptotics in estimation for two-sided Weibull type distributions

We consider the estimation problem of a location parameter on a sample of size n from a two-sided Weibull type density % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOzaiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykaiabg2da9iaa% doeacaGGOaGaeqySdeMaaiykaiGacwgacaGG4bGaaiiCaiaacIcacq% GHsislcaGG8bGaamiEaiabgkHiTiabeI7aXjaacYhadaahaaWcbeqa% aiabeg7aHbaakiaacMcaaaa!52AD!\[f(x - \theta ) = C(\alpha )\exp ( - |x - \theta |^\alpha )\] for –<x<, –<< and 1<a<3/2, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaam4qaiaacIcacqaHXoqycaGGPaGaeyypa0JaeqySdeMaai4laiaa% cUhacaaIYaGaeu4KdCKaaiikaiaaigdacaGGVaGaeqySdeMaaiykai% aac2haaaa!4B0E!\[C(\alpha ) = \alpha /\{ 2\Gamma (1/\alpha )\} \]. Then the bound for the distribution of asymptotically median unbiased estimators is obtained up to the 2a-th order, i.e., the order % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOBamaaCaaaleqabaGaeyOeI0IaaiikaiaaikdacqaHXoqycqGH% sislcaaIXaGaaiykaiaac+cacaaIYaaaaaaa!4444!\[n^{ - (2\alpha - 1)/2} \]. The asymptotic distribution of a maximum likelihood estimator (MLE) is also calculated up to the 2a-th order. It is shown that the MLE is not 2a-th order asymptotically efficient. The amount of the loss of asymptotic information of the MLE is given.     

Estimation of entropy and other functionals of a multivariate density

For a multivariate density f with respect to Lebesgue measure , the estimation of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGkbGaaiikaiaadAgacaGGPaGaamOzaiaadsgacqaH% 8oqBaSqabeqaniabgUIiYdaaaa!4404!\[\int {J(f)fd\mu } \], and in particular % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGMbWaaWbaaSqabeaacaaIYaaaaOGaamizaiabeY7a% TbWcbeqab0Gaey4kIipaaaa!41E4!\[\int {f^2 d\mu } \] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGMbGaciiBaiaac+gacaGGNbGaamOzaiaadsgacqaH% 8oqBaSqabeqaniabgUIiYdaaaa!44AC!\[\int {f\log fd\mu } \], is studied. These two particular functionals are important in a number of contexts. Asymptotic bias and variance terms are obtained for the estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaacaWGkbGaaiikamaawagabeWcbeqaaiabgEIizdqdbaGaamOzaa% aakiaacMcacaWGKbGaamOramaaBaaaleaacaWGobaabeaaaeqabeqd% cqGHRiI8aaaa!4994!\[\mathop I\limits^ \wedge = \int {J(\mathop f\limits^ \wedge )dF_N } \] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaeSipIOdaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaacaWGkbGaaiikamaawagabeWcbeqaaiabgEIizdqdbaGaamOzaa% aakiaacMcadaGfGbqabSqabeaacqGHNis2a0qaaiaadAgaaaGccaWG% KbGaeqiVd0galeqabeqdcqGHRiI8aaaa!4C40!\[\mathop I\limits^ \sim = \int {J(\mathop f\limits^ \wedge )\mathop f\limits^ \wedge d\mu } \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGMbaaaaaa!3E9C!\[{\mathop f\limits^ \wedge }\] is a kernel density estimate of f and F _n is the empirical distribution function based on the random sample X ₁ ,..., X _n from f. For the two functionalsmentioned above, a first order bias term for Î can be made zero by appropriate choices of non-unimodal kernels. Suggestions for the choice of bandwidth are given; for % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaadaGfGbqabSqabeaacqGHNis2a0qaaiaadAgaaaGccaWGKbGaam% OramaaBaaaleaacaWGobaabeaaaeqabeqdcqGHRiI8aaaa!476C!\[\mathop I\limits^ \wedge = \int {\mathop f\limits^ \wedge dF_N } \], a study of optimal bandwidth is possible.This research was supported by an NSERC Grant and a UBC Killam Research Fellowship.     

Optimal convergence properties of kernel density estimators without differentiability conditions

Let X ₁, X ₂, ..., X _n be independent observations from an (unknown) absolutely continuous univariate distribution with density f and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmOzayaajaGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad6ga% caWGObGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqadaba% Gaam4saiaacUfadaWcgaqaaiaacIcacaWG4bGaeyOeI0Iaamiwamaa% BaaaleaacaWGPbaabeaakiaacMcaaeaacaWGObGaaiyxaaaaaSqaai% aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa!5356!\[\hat f(x) = (nh)^{ - 1} \sum\nolimits_{i = 1}^n {K[{{(x - X_i )} \mathord{\left/ {\vphantom {{(x - X_i )} {h]}}} \right. \kern-\nulldelimiterspace} {h]}}} \] be a kernel estimator of f(x) at the point x, \s-<x<, with h=h _n (h _n O and nh _n , as n) the bandwidth and K a kernel function of order r. Optimal rates of convergence to zero for the bias and mean square error of such estimators have been studied and established by several authors under varying conditions on K and f. These conditions, however, have invariably included the assumption of existence of the r-th order derivative for f at the point x. It is shown in this paper that these rates of convergence remain valid without any differentiability assumptions on f at x. Instead some simple regularity conditions are imposed on the density f at the point of interest. Our methods are based on certain results in the theory of semi-groups of linear operators and the notions and relations of calculus of finite differences.This research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the University of Alberta Central Research Fund.     

On an optimum test of the equality of two covariance matrices

Let X: p × 1, Y: p × 1 be independently and normally distributed p-vectors with unknown means ₁, ₂ and unknown covariance matrices ₁, ₂ (>0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H ₀: ₁=₂ against the alternative H ₁: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeiDaiaabkhacaqGOaWaaabmaeaadaaeqaqaaiabgkHiTiaadMea% caGGPaGaaiiiaiabg2da9iaacccacqaHdpWCcaGGGaGaeyOpa4Jaai% iiaiaaicdaaSqaaiaaigdaaeqaniabggHiLdaaleaacaqGYaaabaGa% aeylaiaabgdaa0GaeyyeIuoaaaa!4E3F!\[{\rm{tr(}}\sum\nolimits_{\rm{2}}^{{\rm{ - 1}}} {\sum\nolimits_1 { - I) = \sigma > 0} }\]as 0. However this test is not of type D among G-invariant tests.Research supported by the Canadian N.S.E.R.C. Grant.     

A class of consistent tests for exponentiality based on the empirical Laplace transform

The Laplace transform (t=E[exp(–tX)]) of a random variable with exponential density exp(–x), x0, satisfies the differential equation (+t)(t)+(t=0, t0). We study the behaviour of a class of consistent (omnibus) tests for exponentiality based on a suitably weighted integral of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGBbGaaiikai% qbeU7aSzaajaWaaSbaaSqaaGqaciaa-5gaaeqaaOGaey4kaSIaamiD% aiaacMcacqaHipqEcaWFNaWaaSbaaSqaaiaad6gaaeqaaOGaaiikai% aadshacaGGPaGaey4kaSIaeqiYdK3aaSbaaSqaaiaad6gaaeqaaOGa% aiikaiaadshacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaaaaa!4C69!\[[(\hat \lambda _n + t)\psi '_n (t) + \psi _n (t)]^2 \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqcam% aaBaaaleaaieGacaWFUbaabeaaaaa!3A66!\[\hat \lambda _n \] is the maximum-likelihood-estimate of and _n is the empirical Laplace transform, each based on an i.i.d. sample X ₁,...,X _n .     

Noise reduction by gradual memorization

Zusammenfassung Der vorliegende Beitrag beschreibt ein einfaches, nichtlineares, abgetastetes Filter, das «Gradueller Speicher» genannt wird. Das Filter besteht aus einem Speicher, welches neue Eingangswerte nur graduell, d. h. schrittweise, speichert: Bei jedem neuen Eingangswert wird er um einen konstanten Schritt ±s in Richtung des letzteren nachgestellt. Es wird gezeigt, dass eine solche «Graduelle Speicherung» eine Verminderung von Gaussschem Rauschen um einen Faktor gestattet. (=/s, d. h. ist gleich der Streuung des Eingangsrauschens geteilt durch den Schritt s.) Die Wahrscheinlichkeitsfunktion des Speicherausganges wird berechnet und durch eine Gauss-Funktion angenähert. eine Formel wird hergeleitet, welche die richtige Wahl von s in praktischen Fällen gestattet. Im letzten Abschnitt wird das dynamische Verhalten eines graduellen Speichers betrachtet, die Reaktionszeitt _r , welche der Speicher braucht, um einer plötzlichen Änderung U_o im wahren Eingangssignal nachzufolgen, wird berechnet.t _r wird umso grösser, je grösser die Streuung des Eingangsrauschens ist.

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List of Symbols Real valueX=U ₀+U Input signal (true value + noise) - Real valueU ₀ Normalized valueu ₀=U ₀/s True input signal - Real valueU Normalized valueu=U/s Input noise - Real value s Step of memory - Real valueY=U ₀+R Output signal (true value + noise) - Real valueR Normalized valuer=R/s Output noise - Real value Normalized value Average momentary value of output signal - Real value Normalized value =/s Standard deviation of Gaussian noise at input - Normalized value Standard deviation of Gaussian noise at output - Normalized valueb Parameter of logistical distribution - Real value U ₀ Normalized valuea=U ₀/s Step of true input value - Real valuet Normalized value =t/t Time - Real value t=T Normalized value 1 Sampling interval - Real value Normalized value Average momentary velocity of output signal - Real valueU _tol Prescribed tolerance interval for output signal - Real value Prescribed probability (that the output signal remains within the tolerance interval) - Real valuew(U) Normalized valuew(u) Probability density function (input) - Real value (R) Normalized value (r) Probability (density) function (output) - Normalized valueq _r =1-P _r Cumulative distribution function - Normalized valueK=r/ Momentary value of output noise, expressed in multiples of the standard deviation - Real valueQ Normalized valueQ=/ Noise reduction factor - Real valuet _r Normalized value _r =t _r /t Time of reaction - Normalized valuez Increase in reaction time due to input noise     

Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function

Let X=(X ₁, X ₂,..., X _d ) ^t be a random vector of positive entries, such that for some =(₁,₂,..., _d ) ^t , the vector X ⁽⁾ defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\[X_i^{(\lambda _{i)} } = ({{X_i^{\lambda _i } - 1)} \mathord{\left/ {\vphantom {{X_i^{\lambda _i } - 1)} {\lambda _{i,} i = 1 \ldots ,d}}} \right. \kern-\nulldelimiterspace} {\lambda _{i,} i = 1 \ldots ,d}}\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the _i's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.Adolfo Quiroz and Miguel Nakamura's research was partially supported by CONACYT (Mexico) grants numbers 1858E9219 and 4224E9405, while Dr. Quiroz was visiting Centro de Investigación en Matemáticas at Guanajuato, Mexico.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

Estimation of a common mean of several univariate inverse Gaussian populations

The problem of estimating the common mean of k independent and univariate inverse Gaussian populations IG(, _i ), i=1,..., k with unknown and unequal 's is considered. The difficulty with the maximum likelihood estimator of is pointed out, and a natural estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] of along the lines of Graybill and Deal is proposed. Various finite sample properties and some decision-theoretic properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] are discussed.This research was partially supported by research grants #A3661 and #A3450 from NSERC of Canada.     

On the long time behaviour of a generalized KdV equation

We consider the Cauchy problem for the generalized Korteweg-de Vries equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkGi2oaaBaaaleaacaaIXaaabeaakiaadwhacqGHRaWkcqGH% ciITdaWgaaWcbaGaamiEaaqabaGccaGGOaGaeyOeI0IaeyOaIy7aa0% baaSqaaiaadIhaaeaacaaIYaaaaOGaaiykamaaCaaaleqabaGaeqyS% degaaOGaamyDaiabgUcaRiabgkGi2oaaBaaaleaacaWG4baabeaakm% aabmGabaWaaSaaaeaacaWG1bWaaWbaaSqabeaacqaH7oaBaaaakeaa% cqaH7oaBaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa!56D5!\[\partial _1 u + \partial _x ( - \partial _x^2 )^\alpha u + \partial _x \left( {\frac{{u^\lambda }}{\lambda }} \right) = 0\]where is a positive real and and integer larger than 1. We obtain the detailed large distance behaviour of the fundamental solution of the linear problem and show that for 1/2 and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeU7aSjabg6da+iabeg7aHjabgUcaRmaalaaabaGaaG4maaqa% aiaaikdaaaGaey4kaSYaaeWaceaacqaHXoqydaahaaWcbeqaaiaaik% daaaGccqGHRaWkcaaIZaGaeqySdeMaey4kaSYaaSaaaeaacaaI1aaa% baGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVa% GaaGOmaaaaaaa!4FF7!\[\lambda > \alpha + \frac{3}{2} + \left( {\alpha ^2 + 3\alpha + \frac{5}{4}} \right)^{1/2} \], solutions of the nonlinear equation with small initial conditions are smooth in the large and asymptotic when t± to solutions of the linear problem.     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Bounds for Non-Gaussian Approximations of U-Statistics

Let X, ,X ₁,...,X _n be i.i.d. random variables taking values in a measurable space ( ). Consider U-statistics of degree two

Die Dupinschen Zykliden im pseudoeuklidischen Raum

In the pseudoeuclidean space¹ E ³ a surface of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus such that a (pseudoeuclidean) spherical inversion maps upon . If the axis of is respectively space-like, isotropic or time-like is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide is the envelope of two families of spheres, the caustic surfaces of degenerate in caustic curves and the lines of curvature of constitute two (orthogonal) families of circles.     

Pasting of R 2-planes

The notion pasting sum (P _i , _i ) of two R ²-planes (or Salzmann planes) (P _i , _i ) is developed. Necessary and sufficient conditions for it to be an R ²-plane again are given. The notion is applied to classify all flat projective planes whose collineation group contains a sub-group with (isomorphism type of , fixed element configuration)=(², x).     

Shape-preserving Kolmogorov widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ₀ and let ₊ ^s be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t ₀,..., t _s I, the corresponding divided differences [x; t ₀,...,t _s ] of order s are nonnegative. Let ₊ ^s B _{p +} ^s B _p, 1 p where B _p is a unit ball in the space L _p, and let ₊ ^s L _{q +} ^s L _q, 1 q . For every s 3 and 1 q p , we determine the exact orders of the shape-preserving Kolmogorov widths {x - y} \right\ L_q , $$]]>, where M ⁿ is the collection of all affine linear manifolds M ⁿ in L _q such that dim M ⁿ n and M ⁿ ₊ ^s L _q .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.     

On a property of N-functions

We consider three classes of N-functions: ('), the class of functions satisfying the ' condition, (₂), the class of functions satisfying the ₂ condition, and (M), the class of functions M(u) satisfying the condition: ln M (u) / ln u = p < . We establish the connection between the class of powers and the class of N-functions M(u) which belong to the class (') together with their complementary functions and we also establish the connections between the classes ('), (M) and (₂).Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 281–290, 1968.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.