On functionals of random convex hulls

The completeness of the induced distribution of the convex hull of a random set of points drawn uniformly from a convex region seems not to have been noticed before. The result generalizes a well-known result for dimension one. As a consequence, there exists a theory of best unbiased estimation for certain functionals of the convex region. For example, a best estimator of the centroid of the convex region can be constructed. This estimator is distinct from the centroid of the convex hull. Therefore another generalization of a property holding in dimension one, namely the unbiased-ness of the centroid of the convex hull, is seen to fail.     

On beta type-1 distributed random points in an n-ball

This article considers the problem of finding the exact density of the r-content of the simplicial convex hull of r+1 independent points in Rⁿ ” Consider r+1 independent and identically distributed points in a unit n–ball such that p of them are in the interior and r+l?p of them are on the surface of the unit n-ball., Consider the case when each point is type-1 beta distributed,, These points determine almost surely via their convex hull a unique r-simplex in Rⁿ Ihe problem of getting the exact density of the r-content of this random r-simplex is transformed into a problem in multivariate statistical analysis connected with the distribution theory of test statistics., Thus various representations for the exact density are given in this article.     

Maximum likelihood estimation in convex hull models

In this paper we consider models involving the convex hull operation of the parameter and the noise i.e. Y_i = CH(A, X_X). Then we generalize the basic models to ANOVA models; i.e. Y_ij=CH(A∪B_j,X_ij). In some cases the consistent estimators for the J U new parameters are derived. Assuming the existence of density forrandom convex sets, we derive the likelihood for the convex hull model. We then find the maximum Likelihood Estimators for the parameters. Examples for some random convex sets with finite dimensional distributions are derived to show how good these estimators are.     

Second‐order Accurate Confidence Regions Based on Members of the Generalized Power Divergence Family

Recently, a technique based on pseudo‐observations has been proposed to tackle the so‐called convex hull problem for the empirical likelihood statistic. The resulting adjusted empirical likelihood also achieves the high‐order precision of the Bartlett correction. Nevertheless, the technique induces an upper bound on the resulting statistic that may lead, in certain circumstances, to worthless confidence regions equal to the whole parameter space. In this paper, we show that suitable pseudo‐observations can be deployed to make each element of the generalized power divergence family Bartlett‐correctable and released from the convex hull problem. Our approach is conceived to achieve this goal by means of two distinct sets of pseudo‐observations with different tasks. An important effect of our formulation is to provide a solution that permits to overcome the problem of the upper bound. The proposal, which effectiveness is confirmed by simulation results, gives back attractiveness to a broad class of statistics that potentially contains good alternatives to the empirical likelihood.     

On uniformly distributed random points in an n-Ball

Consider a set of r+1 independently and identically and uniformly distributed random points X₀, X₁,…,X_r in RⁿThese points determine almost surely via their convex hull a unique r-simplex in R^e This article deals with the exact density of the r-content of this random r-simplex when the points are such that p of them are in the interior and r+l?p of them are on the surface of a unit n-ball. This problem is transformed into a distribution problem connected with multivariate test statistics. Various possible representations of the exact density in the general case, are also pointed out.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

Admissibility of exact conditional tests of stochastic order

Consider the problem of testing for independence against stochastic order in a 2 × J contingency table, with two treatments and J ordered categories. By conditioning on the margins, the null hypothesis becomes simple. Careful selection of the conditional alternative hypothesis then allows the problem to be formulated as one of a class of problems for which the minimal complete class of admissible tests is known. The exact versions of many common tests, such as t-tests and the Smirnov test, are shown to be inadmissible, and thus the non-randomized versions are overly conservative. The proportional hazards and proportional odds tests are shown to be admissible for a given data set and size. A new result allows a proof of the admissibility of convex hull and adaptive tests for all data sets and sizes.     

Testing Hypotheses in Nonparametric Models of Production

Data envelopment analysis (DEA) and free disposal hull (FDH) estimators are widely used to estimate efficiency of production. Practitioners use DEA estimators far more frequently than FDH estimators, implicitly assuming that production sets are convex. Moreover, use of the constant returns to scale (CRS) version of the DEA estimator requires an assumption of CRS. Although bootstrap methods have been developed for making inference about the efficiencies of individual units, until now no methods exist for making consistent inference about differences in mean efficiency across groups of producers or for testing hypotheses about model structure such as returns to scale or convexity of the production set. We use central limit theorem results from our previous work to develop additional theoretical results permitting consistent tests of model structure and provide Monte Carlo evidence on the performance of the tests in terms of size and power. In addition, the variable returns to scale version of the DEA estimator is proved to attain the faster convergence rate of the CRS-DEA estimator under CRS. Using a sample of U.S. commercial banks, we test and reject convexity of the production set, calling into question results from numerous banking studies that have imposed convexity assumptions. Supplementary materials for this article are available online.     

Barnard Convex Sets

In calculating significance levels for statistical non inferiority tests, the critical regions that satisfy the Barnard convexity condition have a central role. According to a theorem proved by Röhmel and Mansmann (1999 Röhmel , J. , Mansmann , U. ( 1999 ). Unconditional nonasymptotic one sided tests for independent binomial proportions when the interest lies in showing noninferiority and or superiority . Biometr. J. 2 : 149 – 170 .[Crossref], [Web of Science ®] , [Google Scholar]), when the critical regions satisfy this condition, the significance level for non inferiority tests can be calculated much more efficiently. In this study, the sets that fulfil the Barnard convexity condition are called Barnard convex sets, and because of their relevance, we studied their properties independently of the context from which the sets originated. Among several results, we found that Barnard convex sets are a convex geometry and that each Barnard convex set has a unique basis. Also, we provide an algorithm for calculating the Barnard convex hull of any set. Finally, we present some applications of the concept of the Barnard convex hull of a set for non inferiority tests.     

Nonlinear filtering of convex sets of probability distributions

A solution is provided to the problem of computing a convex set of conditional probability distributions that characterize the state of a nonlinear dynamic system as it evolves in time. The estimator uses the Galerkin approximation to solve Kolmogorov's equation for the diffusion of a continuous-time nonlinear system with discrete-time measurement updates. Filtering of the state is accomplished for a convex set of distributions simultaneously, and closed-form representations of the resulting sets of means and covariances are generated.     

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.     

Adjusted empirical likelihood for value at risk and expected shortfall

Value at risk (VaR) and expected shortfall (ES) are widely used risk measures of the risk of loss on a specific portfolio of financial assets. Adjusted empirical likelihood (AEL) is an important non parametric likelihood method which is developed from empirical likelihood (EL). It can overcome the limitation of convex hull problems in EL. In this paper, we use AEL method to estimate confidence region for VaR and ES. Theoretically, we find that AEL has the same large sample statistical properties as EL, and guarantees solution to the estimating equations in EL. In addition, simulation results indicate that the coverage probabilities of the new confidence regions are higher than that of the original EL with the same level. These results show that the AEL estimation for VaR and ES deserves to recommend for the real applications.     

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

This paper develops an algorithm for uniform random generation over a constrained simplex, which is the intersection of a standard simplex and a given set. Uniform sampling from constrained simplexes has numerous applications in different fields, such as portfolio optimization, stochastic multi-criteria decision analysis, experimental design with mixtures and decision problems involving discrete joint distributions with imprecise probabilities. The proposed algorithm is developed by combining the acceptance–rejection and conditional methods along with the use of optimization tools. The acceptance rate of the algorithm is analytically compared to that of a crude acceptance–rejection algorithm, which generates points over the simplex and then rejects any points falling outside the intersecting set. Finally, using convex optimization, the setup phase of the algorithm is detailed for the special cases where the intersecting set is a general convex set, a convex set defined by a finite number of convex constraints or a polyhedron.     

Balanced augmented empirical likelihood for regression models

This paper studies the problem of convex hull constraint in conventional empirical likelihood. Specifically, in the framework of regression, a balanced augmented empirical likelihood (BAEL) procedure through adding two synthetic data points is proposed. It can be used to resolve the under-coverage issue, especially in small-sample or high-dimension setting. Furthermore, some asymptotic properties for proposed BAEL ratio statistic are established under mild conditions. The proposed approach performs robust to different random errors by choosing a robust loss function. Extensive simulation studies and a real example are carried out to support our results.     

Estimation of regression parameters in the presence of outliers in the response

In this paper, we consider a regression model with non-spherical covariance structure and outliers in the response. The generalized least squares estimator obtained from the full data set is generally not used in the presence of outliers and an estimator based on only the non-outlying observations is preferred. Here we propose as an estimator a convex combination of the full set and the deleted set estimators and compare its performance with the other two.     

A NOTE ON THE DEMOGRAPHIC IMPACT OF CONTRACEPTION IN A MATHEMATICAL MODEL OF HUMAN REPRODUCTION

The application of mathematical models of human reproduction to the study of reproductive behaviour as a function of contraceptive behaviour was pioneered by Perrin and Sheps (1963). More recent work in this area continues to focus on an examination of the birth rate as the principal dependent variable. This note suggests an alternative procedure for studying the demographic impact of contraception through the analysis of birth intervals. A mathematical model is formulated for the waiting time between successive live births, and a procedure is described for incorporating into the model certain contraceptive parameters. A controlled experiment is then performed to determine the effect of these parameters on expected child spacing patterns.     

Extending the empirical likelihood by domain expansion

We extend the empirical likelihood beyond its domain by expanding its contours nested inside the domain with a similarity transformation. The extended empirical likelihood achieves two objectives at the same time: escaping the “convex hull constraint” on the empirical likelihood and improving the coverage accuracy of the empirical likelihood ratio confidence region to $O(n^{-2})$ . The latter is accomplished through a special transformation which matches the extended empirical likelihood with the Bartlett corrected empirical likelihood. The extended empirical likelihood ratio confidence region retains the shape of the original empirical likelihood ratio confidence region. It also accommodates adjustments for dimension and small sample size, giving it good coverage accuracy in large and small sample situations. The Canadian Journal of Statistics 41: 257–274; 2013 © 2013 Statistical Society of Canada     

Improved estimation for Poisson INAR(1) models

We consider the first-order Poisson autoregressive model proposed by McKenzie [Some simple models for discrete variate time series. Water Resour Bull. 1985;21:645–650] and Al-Osh and Alzaid [First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275], which may be suitable in situations where the time series data are non-negative and integer valued. We derive the second-order bias of the squared difference estimator [Weiß. Process capability analysis for serially dependent processes of Poisson counts. J Stat Comput Simul. 2012;82:383–404] for one of the parameters and show that this bias can be used to define a bias-reduced estimator. The behaviour of a modified conditional least-squares estimator is also studied. Furthermore, we access the asymptotic properties of the estimators here discussed. We present numerical evidence, based upon Monte Carlo simulation studies, showing that the here proposed bias-adjusted estimator outperforms the other estimators in small samples. We also present an application to a real data set.     

Depth and outliers for samples of sets and random sets distributions

We suggest several constructions suitable to define the depth of set-valued observations with respect to a sample of convex sets or with respect to the distribution of a random closed convex set. With the concept of a depth, it is possible to determine if a given convex set should be regarded an outlier with respect to a sample of convex closed sets. Some of our constructions are motivated by the known concepts of half-space depth and band depth for function-valued data. A novel construction derives the depth from a family of non-linear expectations of random sets. Furthermore, we address the role of positions of sets for evaluation of their depth. Two case studies concern interval regression for Greek wine data and detection of outliers in a sample of particles.