A Note on the Best Invariant Estimator of a Distribution Function Under the Kolmogorov-Smirnov Loss

For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss within the class of proper– distribution functions, it is proved that the sample distribution function is the best invariant estimator only for the sample size n = 1 and 2. Further it is shown that the best invariant estimator is minimax. Exact jumps of the best invariant estimator are derived for n 4.     

Random Logistic Maps. I

Let {C _i} ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n} ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1–X _n). It is shown here that: (i) E ln C ₁<0X _n0 w.p.1. (ii) E ln C ₁=0X _n0 in probability (iii) E ln C ₁>0, E |ln(4–C ₁)| such that (0, 1)=1 and is invariant for {X _n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C ₁>0 and – ln(1–x) (dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4–C ₁)|< and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀0.     

Componentwise estimation of ordered parameters ofk (≥2) exponential populations

We consider the estimation of ordered parameters ofk ( 2) exponential distributions by improving upon the usual estimators. TheBrewsterzidek technique is used to find sufficient conditions for an estimator of _i and/or _i (i=1,...,k), to be inadmissible with respect to the MSE criterion where _i and _i are the location and scale parameters respectively of thei-th exponential population. Using these sufficient conditions improved estimators of _i and/or _i (i=1,...,k) are obtained.     

On the abundance of chaotic behavior for generic one-parameter families of maps

In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that iff _a:I I, a P is a regular family satisfying some conditions described in the next section, then there exists a Borel set P of positive Lebesgue measure such that for everya ,f _a admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in [1] and [7], follows a similar line.Supported by the NSFC and the National 863 Project.     

The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension

We consider the sequence of the hyperspheres M _n , i.e., the homogeneous transitive spaces of the Cartan subgroup of the group and study the normalized limit of the corresponding sequence of invariant measures m _n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S ^n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S ^n-1). Our main result shows that the situation for noncompact groups, for example for the case , is completely different: the limit of the measures m _n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case. To my friend Dima Arnold     

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator , which satisfies , where is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property. The author thanks Laurens de Haan for motivating this work and giving helpful comments. The author also thanks two anonymous referees for their useful comments.     

Contributions to a Rigidity Conjecture

Let p and q be two relatively prime positive integers and a Borel probability measure invariant and ergodic by the semigroup generated by the action of both zp and zq. We analyse sufficient conditions to guarantee that is either the Lebesgue measure or supported on a periodic orbit. And extend the results for general expanding differentiable maps of the circle.     

Sequential estimation of a parameter of an exponential distribution

We consider the problem of minimum risk point estimation for the parameter =a+b of the exponential distribution with unknown location parameter and scale parameter when the loss function is squared error plus linear cost. In this paper, we propose a sequential estimator of and show that the associated risk is asymptotically one cost less than that given by Ghosh and Mukhopadhyay (1989,South African Statist. J.,23, 251–268).     

Robust and Efficient Estimation for the Generalized Pareto Distribution

In this article we implement the minimum density power divergence estimator (MDPDE) for the shape and scale parameters of the generalized Pareto distribution (GPD). The MDPDE is indexed by a constant 0 that controls the trade-off between robustness and efficiency. As increases, robustness increases and efficiency decreases. For = 0 the MDPDE is equivalent to the maximum likelihood estimator (MLE). We show that for > 0 the MDPDE for the GPD has a bounded influence function. For < 0.2 the MDPDE maintains good asymptotic relative efficiencies, usually above 90%. The results from a Monte Carlo study agree with these asymptotic calculations. The MDPDE is asymptotically normally distributed if the shape parameter is less than (1 + )/(2 + ), and estimators for standard errors are readily computed under this restriction. We compare the MDPDE, MLE, Dupuis optimally-biased robust estimator (OBRE), and Peng and Welshs Medians estimator for the parameters. The simulations indicate that the MLE has the highest efficiency under uncontaminated GPDs. However, for the GPD contaminated with gross errors OBRE and MDPDE are more efficient than the MLE. For all the simulated models that we studied the Medians estimator had poor performance.AMS 2000 Subject Classification. Primary—62F35, Secondary—62G35     

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on S ¹ . In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group exists if and only if the quotient group does not contain a free subgroup with two generators (here is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups of homeomorphisms of the line.     

New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

Invariant σ-fields for a class of diffusions

Summary The invariant -field for a diffusion gives all bounded harmonic functions for the infinitesimal generator of that diffusion. We specify the invariant -field for a class of two dimensional diffusions and thereby obtain a representation for all bounded harmonic functions for the process. When the infinitesimal generator is radially symmetric we obtain the Martin boundary. This is used to find the invariant -field for the corresponding process.     

On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average

Suppose {f ₁,...,f _m } is a set of Lipschitz maps of ^d . We form the iterated function system (IFS) by independently choosing the maps so that the map f _i is chosen with probability p _i ( ^m _i=1 p _i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ^d and as a corollary show that the measure will be singular if the modulus of the entropy _i p _i log p _i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of .     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006     

The set of smooth metrics in the torus without continuous invariant graphs is open and dense in the <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> topology

We show that the set of C metrics in the two dimensional torus with no continuous invariant graphs of the geodesic flow is open and dense in the C ¹ topology. The generic nonexistence of invariant graphs with rational rotation numbers was known in the C topology for metrics, and in general the generic nonexistence in the C topology of invariant graphs with Liouville rotation numbers is known for twist maps and Hamiltonian flows in the torus. The main idea of the proof is that small C ¹ bumps are enough to prevent the existence of invariant graphs.Partially supported by CNPq, FAPERJ, TWAS     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Fuzzy weighted scaled coefficients in semi-parametric model

In general, the regressor variables are stochastic, Duan and Li (1987, J. Econometrics, 35, 25–35), Li and Duan (1989, Ann. Statist., 17, 1009–1052) have been shown that under very general design conditions, the least squares method can still be useful in estimating the scaled regression coefficients of the semi-parametric model Y _i =Q ₁(+X _i ; _i , i+ 1,2,...,n. Here is a constant, is a 1×p row vector, X _i is a p×1 column vector of explanatory variables, _i is an unobserved random error and Q ₁ is an arbitrary unknown function. When the data set (X _i , Y _i ),i=1, 2, ..., n, contains one or several outliers, the least squares method can not provide a consistent estimator of the scaled coefficients . Therefore, we suggest the fuzzy weighted least squares method to estimate the scaled coefficients for the data set with one or several outliers. It will be shown that the proposed fuzzy weighted least squares estimators are % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaakaaabaGaamOBaaWcbeaaaaa!3D3C!\[\sqrt n \] and asymptotically normal under very general design condition. Consistent measurement of the precision for the estimator is also given. Moreover, a limited Monte Carlo simulation and an example are used to study the practical performance of the procedures.This research partially supported by the National Science Council, R.O.C.     

Moderate Deviations and Large Deviations for Kernel Density Estimators

Let f _n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ^d . It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) 0 as |x| , then the moderate deviation principle and large deviation principle for hold.     

Parametric ranked set sampling

Ranked set sampling was introduced by McIntyre (1952,Australian Journal of Agricultural Research,3, 385–390) as a cost-effective method of selecting data if observations are much more cheaply ranked than measured. He proposed its use for estimating the population mean when the distribution of the data was unknown. In this paper, we examine the advantage, if any, that this method of sampling has if the distribution is known, for a specific family of distributions. Specifically, we consider estimation of and for the family of random variables with cdf's of the formF(x–/). We find that the ranked set sample does provide more information about both and than a random sample of the same number of observations. We examine both maximum likelihood and best linear unbiased estimation of and , as well as methods for modifying the ranked set sampling procedure to provide even better estimation.This paper has been prepared with partial support from the United States Environmental Protection Agency under Cooperative Agreement Number CR821801-01-0. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.