A bayesian approach to the probability density estimation

Summary A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution. The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's Bayesian Information Criterion) is minimized. The basic procedure is developed under the assumption that the density is defined on a bounded interval. The handling of the general case where the support of the density function is not necessarily bounded is also discussed. The practical usefulness of the procedure is demonstrated by numerical examples. The Institute of Statistical Mathematics     

Nonparametric bayesian estimation of a survival curve with dependent censoring mechanism

Summary Let (X ₁,Y ₁),..., (X _N ,Y _N ) be a random sample from a bivariate distribution functionF. Based on observing only (δ _i ,Z _i ) whereδ _i =1 ifX _i ≦X _i and =0 otherwise andZ _i =min{X _i ,Y _i } fori=1,...,n, we obtain the Bayes estimator ofF whenF is a Dirichlet process under the usual integrated squared error loss function. It should be pointed out here thatX _i andY _i neednot be independent which is the usual assumption in survival analysis models. The effect of this dependence can be seen clearly in the estimators obtained and also in the given example which illustrates the estimator when Freunds' bivariate exponential distribution is taken as the parameter of the Dirichlet process. The research of this author is supported in part by an NSF Grant MCS80-03244. The research of this author is supported by the NIH Grant NO: 1 R01 GM 28405.     

A bayesian approach to nonparametric test problems

We propose an alternative approach to the classical nonparametric test problems, such as the goodness of fit test and the two-sample nonparametric test. In this approach, those problems are reviewed from the viewpoint of the estimation of the underlying population distributions and are formulated as the problem of model selection between Bayesian models which were recently proposed by the present authors. The model selection can be easily realized by choosing a model with the smallest ABIC, Akaike Bayesian information criterion. The approach provides the estimates of the density of the underlying population distribution(s) of any shape as well as the evaluation of the goodness of fit or the check of homogeneity of distributions. The practical utility of the present procedure is demonstrated by numerical examples. The difference in behavior between the present procedure and a density estimator GALTHY proposed by Akaike and Arahata is also briefly discussed.This paper was originally read at the Conference on Graphical Models to Analyze Structures (Organizer: N. Wermuth, Johannes Gutenberg University), June 30-July 2, 1986, Wiesbaden, West Germany.     

A bayesian approach to inference for monotone failure rates

In reliability theory, the notion of monotone failure rates plays a central role. When prior information indicates that such monotonicity is meaningful, it must be incorporated into the prior distribution whenever inference about the failure rates needs to be made. In this paper we show how this can be done in a straightforward and intuitively pleasing manner. The time interval is partitioned into subintervals of equal width and the number of failures and censoring in each interval is recorded. By defining a Dirichlet as the joint prior distribution for the forward or the backward differences of the conditional probabilities of survival in each interval, we find that the monotonicity is presenved in the posterior estimate of the failure rates. A posterior estimate of the survival function can also be obtained. We illustrate our method by applying it to some real life medical data.     

Minimax and minimax adaptive estimation in multiplicative regression: locally bayesian approach

This paper deals with the non-parametric estimation in the regression with the multiplicative noise. Using the local polynomial fitting and the bayesian approach, we construct the minimax on isotropic H?lder class estimator. Next, applying Lepski’s method we propose an estimator which is optimally adaptive over the collection of isotropic H?lder classes. To prove the optimality of the proposed procedure, we establish in particular the exponential inequality for the deviation of locally bayesian estimator since the parameter estimated. These theoretical results are illustrated by simulation study.     

A Bayesian approach for quantile and response probability estimation with applications to reliability

In this paper we propose a Bayesian approach for the estimation of a potency curve which is assumed to be nondecreasing and concave or convex. This is done by assigning the Dirichlet as a prior distribution for transformations of some unknown parameters. We motivate our choice of the prior and investigate several aspects of the problem, including the numerical implementation of the suggested scheme. An approach for estimating the quantiles is also given. By casting the problem in a more general context, we argue that distributions which are IHR or IHRA can also be estimated via the suggested procedure. A problem from a government laboratory serves as an example to illustrate the use of our procedure in a realistic scenario.     

A combinatorial approach to binary positional number systems

Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional number system (binary radix system) with base equal to the golden ratio \((1+\sqrt{5}\,)/2\) is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system, can be constructed in this way.     

A family of elliptic curve pseudorandom binary sequences

In this paper we give a new family of elliptic curve pseudorandom binary sequences, and study the well-distribution, correlation, linear complexity, collision and avalanche effect of the family of sequences, by using the estimate of exponential sums over elliptic curves.     

A data dependent approach to density estimation

Summary A data dependent approach to density estimation is proposed here. The proposed method requires boundedness and some weak integrability condition on the unknown density, but not any assumption of smoothness. Applications to histogram, kernel, spline and orthogonal series methods are discussed.     

A filtered polynomial approach to density estimation

In this paper, a little known computational approach to density estimation based on filtered polynomial approximation is investigated. It is accompanied by the first online available density estimation computer program based on a filtered polynomial approach. The approximation yields the unknown distribution and density as the product of a monotonic increasing polynomial and a filter. The filter may be considered as a target distribution which gets fixed prior to the estimation. The filtered polynomial approach then provides coefficient estimates for (close) algebraic approximations to (a) the unknown density function and (b) the unknown cumulative distribution function as well as (c) a transformation (e.g., normalization) from the unknown data distribution to the filter. This approach provides a high degree of smoothness in its estimates for univariate as well as for multivariate settings. The nice properties as the high degree of smoothness and the ability to select from different target distributions are suited especially in MCMC simulations. Two applications in Sects. 1 and 7 will show the advantages of the filtered polynomial approach over the commonly used kernel estimation method.      

A new infinitesimal approach to robust estimation

Summary The infinitesimal robustness of the asymptotic variance of location M-estimators is investigated by means of the change-of-variance curve (CVC), which bears some resemblance to the influence curve (IC). It is proved that this CVC leads to a more stringent robustness property than the IC and that the Huber estimators are still optimal in this new sense.     

A quantile-copula approach to conditional density estimation

A new kernel-type estimator of the conditional density is proposed. It is based on an efficient quantile transformation of the data. The proposed estimator, which is based on the copula representation, turns out to have a remarkable product form. Its large-sample properties are considered and comparisons in terms of bias and variance are made with competitors based on nonparametric regression. A comparative simulation study is also provided.     

Independence issues in imprecise data models: A bayesian approach

A non-precise data problem is defined as a two-stage hierarchical model. The question considered in this Note is to explore some theoretical and practical problems involving independence and/or conditional independence relations among the parameters and the data. We present some results in this direction.     

A nonparametric estimation of the infection curve

Predicting the future course of an epidemic depends on being able to estimate the current numbers of infected individuals.However,while back-projection techniques allow reliable estimation of the numbers of infected individuals in the more distant past,they are less reliable in the recent past.We propose two new nonparametric methods to estimate the unobserved numbers of infected individuals in the recent past in an epidemic.The proposed methods are noniterative,easily computed and asymptotically normal wit...     

Generalized curve approach to elementary particles

The theory developed in Ref. 1 is applied in a simplified form to a relativistic one-particle universe, and boundary symmetry conditions are imposed. The existence, for the particle, of a series of resonance states and a series of gauge states is derived. For the proton and the electron, a selection of such states accounts for the elementary particles observed to date.Dedicated to L. CesariThis research was sponsored by the United States Army under Contracts Nos. DAAG29-75-C-0024 and DAAG29-80-C-0041.This work was carried out at IMPA while the first author was visiting there in November 1981.     

A new genetic approach to construct near-optimal binary search trees

Many definitive and approximate methods have been so far proposed for the construction of an optimal binary search tree. One such method is the use of evolutionary algorithms with satisfactorily improved cost efficiencies. This paper will propose a new genetic algorithm for making a near-optimal binary search tree. In this algorithm, a new greedy method is used for the crossover of chromosomes while a new way is also developed for inducing mutation in them. Practical results show a rapid and desirable convergence towards the near-optimal solution. The use of a heuristic to create not so costly chromosomes as the first offspring, the greediness of the crossover, and the application of elitism in the selection of future generation chromosomes are the most important factors leading to near-optimal solutions by the algorithm at desirably high speeds. Due to the practical results, increasing problem size does not cause any considerable difference between the solution obtained from the algorithm and exact solution.     

A martingale approach to state estimation in delay-differential systems

A rigorous derivation of filtering arid smoothing equations for linear stochastic systems with time delay is presented. The estimation equations are obtained in term of the innovation process of the problem under consideration. The method used is based on a representation theorem on Gaussian martingales.     

A level set reduced basis approach to parameter estimation

We introduce an efficient level set framework to parameter estimation problems governed by parametrized partial differential equations. The main ingredients are: (i) an “admissible region” approach to parameter estimation; (ii) the certified reduced basis method for efficient and reliable solution of parametrized partial differential equations; and (iii) a parameter-space level set method for construction of the admissible region. The method can handle nonconvex and multiply connected regions. Numerical results for two examples in design and inverse problems illustrate the versatility of the approach.