Nonlinear wavelet density estimation with censored dependent data

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α‐mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.     

Parameter derivatives of the generalized hypergeometric function

In this paper, the parameter derivatives to any order of the generalized hypergeometric function as well as for its various special cases are obtained.     

Improved estimation of the generalized precision under the entropy loss

1.IntroductionInthisarticleweconsiderthepointestimationofthegeneralizedprecisionofamultivariatenormaldistributionwithanunknownmeanvector.TObespecific,letXI,'?XubelidobservationfromNc(~,E)wherebothpERPandZ>0arecompletelyunknown.Insteadoftheoriginaldatasetonecanreducetheproblembysufficiencyandlookonlyatnn(X,S),whereX~n--1ZXiandS~Z(Xi--X)(Xi--X)'.ItiswellknownthatXisi=1i~1mutuallyindependentofSandX~Nc(~,n--'Z),S~Wb(n--1,Z).ThelossfunctionweconsiderinthispaperistheentropylossL(6,IZ…     

Asymptotics for a censored generalized linear model with unknown link function

For censored response variable against projected co-variable, a generalized linear model with an unknown link function can cover almost all existing models under censorship. Its special cases include the accelerated failure time model with censored data. Such a model in the uncensored case is called the single-index model in econometrics. In this paper, we systematically study the asymptotic properties. We derive the central limit theorem and the law of the iterated logarithm for an estimator of the direction parameter. We also obtain the optimal convergence rate of an estimator of the unknown link function in the model.      

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

Existence theorems of a maximum likelihood estimate from a generalized censored data sample

Summary A new type of random sample, called a generalized censored data sample, is defined. An approach to finding criteria for the existence of a maximum likelihood estiamte from a finite generalized censored data sample is presented. This approach, named the probability contents boundary analysis, gives systematically a number of practical criteria, each of which is effective for various kinds of typical distribution families in statistical analysis.     

分组删失数据下生存函数Turnbull估计的强收敛性

证明了分组删失情形下生存函数的Turnbull估计具有强极限，并给出极限值所满足的方程。     

Estimating generalized state density of near-extreme events and its applications in analyzing stock data

This paper studies the generalized state density (GDOS) of near-historical extreme events of a set of independent and identically distributed (i.i.d.) random variables. The generalized density of states is proposed which is defined as a probability density function (p.d.f.). For the underlying distribution in the domain of attraction of the three well-known extreme value distribution families, we show the approximate form of the mean GDOS. Estimates of the mean GDOS are presented when the underlying distribution is unknown and the sample size is sufficiently large. Some simulations have been performed, which are found to agree with the theoretical results. The closing price data of the Dow-Jones industrial index are used to illustrate the obtained results.     

Nonparametric estimation of a bivariate survivorship function with doubly censored data

The problem of nonparametric estimation of a bivariate survivorship function with doubly censored data is considered. A self-consistent estimator is developed by first “reducing” the problem to that of estimation in a singly and right censored situation. This estimator is shown to satisfy a likelihood equation, and its uniqueness is investigated. The results obtained naturally parallel those obtained by Campbell (1981) in the singly censored case.     

Comparative statics of the generalized maximum entropy estimator of the general linear model

The generalized maximum entropy method of information recovery requires that an analyst provides prior information in the form of finite bounds on the permissible values of the regression coefficients and error values for its implementation. Using a new development in the method of comparative statics, the sensitivity of the resulting coefficient and error estimates to the prior information is investigated. A negative semidefinite matrix reminiscent of the Slutsky-matrix of neoclassical microeconomic theory is shown to characterize the said sensitivity, and an upper bound for the rank of the matrix is derived.     

Inequalities and monotonicity of ratios for generalized hypergeometric function

We find two-sided inequalities for the generalized hypergeometric function of the form _q+1F_q(−x) with positive parameters restricted by certain additional conditions. Both lower and upper bounds agree with the value of _q+1F_q(−x) at the endpoints of positive semi-axis and are asymptotically precise at one of the endpoints. The inequalities are derived from a theorem asserting the monotony of the quotient of two generalized hypergeometric functions with shifted parameters. The proofs hinge on a generalized Stieltjes representation of the generalized hypergeometric function. This representation also provides yet another method to deduce the second Thomae relation for ₃F₂(1) and leads to an integral representations of ₄F₃(x) in terms of the Appell function F₃. In the last section of the paper we list some open questions and conjectures.     

Indefinite centroaffine surfaces with vanishing generalized Pick function

In this paper, we study indefinite centroaffine surfaces with vanishing generalized Pick function. We give a classification of such surfaces.     

Minimax kernels for density estimation with biased data

This paper considers the asymptotic properties of two kernel estimates % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\], which have been proposed by Bhattacharyya et al. (1988, Comm. Statist. Theory Methods, A17, 3629–3644) and Jones (1991, Biometrika, 78, 511–519), respectively, for estimating the underlying density f at a point under a general selection biased model. The asymptotic optimality of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]is measured by the corresponding asymptotic minimax mean squared errors under a compactly supported Lipschitz continuous family of the underlying densities. It is shown that, in general, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is a superior local estimate than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]in the sense that the asymptotic minimax risk of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is lower than that of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]. The minimax kernels and bandwidths of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]are computed explicity and shown to have simple forms and depend on the weight functions of the model.     

Bootstrapping a Bayes estimator of a survival function with censored data

The large-sample frequentist property of a frequentist bootstrap for a posterior mean with respect to a Dirichlet prior of the survival function for a randomly censored data is given. The weak convergence of a bootstrap version of the Susarla-Van Ryzin estimator is established on the whole real line. An illustration of the technique and some Monte Carlo studies are also given.     

A note on estimation of the mean residual life function with left-truncated and right-censored data

This note focuses on estimating the mean residual life function with left-truncated and right-censored data. We show that the proposed estimator converges weakly to a Gaussian process. The performances of the estimator and its pointwise confidence intervals are illustrated through simulation studies.     

Lambert function and a new non-extensive form of entropy

We propose a new way of defining entropy of a system, whichgives a general form that is non-extensive like Tsallis entropy,but is linearly dependent on component entropies, like Renyientropy, which is extensive. This entropy has a conceptuallynovel but simple origin and is mathematically easy to defineby a very simple expression involving a derivative. It leadsto a probability distribution function involving the Lambertfunction resulting from optimizing the entropy, which has hithertonever appeared in this context, and is somewhat more complexthan the Shannon or Boltzmann form, but is nevertheless mathematicallyquite tractable. We have compared it numerically with the Tsallisand Shannon entropies. We have also considered constraints onthe energy spectra imposed by the properties of the Lambertfunction, which are absent in the Shannon form. It may turnout to be a more appropriate candidate in a physical situationwhere the probability distribution does not suit any of thepreviously defined forms, especially when the probability densityfunction sought is expected to be stiffer than that resultingfrom maximizing the other entropies. We consider the problemof defining free energy and other thermodynamic functions whenthe entropy is given as a general function of the probabilitydistribution, including that for non-extensive forms. We thenfind that the free energy, which is central to the determinationof all other quantities of interest in a thermodynamic context,can be obtained uniquely, at least numerically, even when itis the root of a transcendental equation. In particular, weexamine the cases of the Tsallis form and the new form proposedby us. We compare the free energy, the internal energy and thespecific heat of a simple system of two energy states for eachof these forms and find significant departures for some quantities,while some others are less sensitive to the parametrization.     

求解含平衡约束数学规划的熵函数法

提出求解含平衡约束数学规划问题(简记为MPEC问题)的熵函数法，在将原问题等价改写为单层非光滑优化问题的基础上，通过熵函数逼近，给出求解MPEC问题的序列光滑优化方法，证明了熵函数逼近问题解的存在性和算法的全局收敛性，数值算例表明了算法的有效性。     

E-Bayesian estimation for the Lomax distribution based on type-II censored data

This paper is concerned with using the E-Bayesian method for computing estimates of the unknown parameter and some survival time parameters e.g. reliability and hazard functions of Lomax distribution based on type-II censored data. These estimates are derived based on a conjugate prior for the parameter under the balanced squared error loss function. A comparison between the new method and the corresponding Bayes and maximum likelihood techniques is conducted using the Monte Carlo simulation.     

Fourier transform and distributional representation of the generalized gamma function with some applications

We present a Fourier transform representation of the generalized gamma functions, which leads to a distributional representation for them as a series of Dirac-delta functions. Applications of these representations are shown in evaluation of the integrals of products of the generalized gamma function with other functions. The results for Euler’s gamma function are deduced as special cases. The relation of the generalized gamma function with the Macdonald function is exploited to deduce new identities for it.     

Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

In [H. Brézis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pure Appl. (9) (1983) 73–97.] Brézis and Friedman prove that certain nonlinear parabolic equations, with the δ-measure as initial data, have no solution. However in [J.F. Colombeau, M. Langlais, Generalized solutions of nonlinear parabolic equations with distributions as initial conditions, J. Math. Anal. Appl (1990) 186–196.] Colombeau and Langlais prove that these equations have a unique solution even if the δ-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais’ result proving that we may take any generalized function as the initial data. Our approach relies on recent algebraic and topological developments of the theory of Colombeau generalized functions and results from [J. Aragona, Colombeau generalized functions on quasi-regular sets, Publ. Math. Debrecen (2006) 371–399.].