Variance estimation using judgment post-stratification

We consider the problem of estimating the variance of a population using judgment post-stratification. By conditioning on the observed vector of ordered in-stratum sample sizes, we develop a conditionally unbiased nonparametric estimator that outperforms the sample variance except when the rankings are very poor. This estimator also outperforms the standard unbiased nonparametric variance estimator from unbalanced ranked-set sampling.     

Constrained estimation: Consistency and asymptotics

We review some of the recent results obtained for constrained estimation, involving possibly non-differentiable criterion functions. New tools are required to push consistency and asymptotic results beyond those that can be reached by classical means.     

A note on Constrained Total Least-Squares estimation

It is shown here how - similarly to the unconstrained case - the Constrained Total Least Squares Estimate (CTLSE) can be generated by solving a certain sequence of eigenvalue problems iteratively. For this, the normal matrix from the constrained (standard) least-squares approach has to be suitably augmented by one row and one column. Further modification of the augmented row and column allows the treatment of “fiducial constraints” for which the RHS vector is affected by random errors, but not the constraining matrix itself.     

A continuous form of post-stratification

Summary The problem of estimating a given integral of a regression function is considered. The proposed estimate may be viewed as continuous analog of the post-stratified mean of sample survey theory. The asymptotic distribution of the estimate is derived, under regularity conditions, and an estimate of its variance suggested.     

Constrained nonparametric estimation of the mean and the CDF using ranked-set sampling with a covariate

Ranked-set sampling (RSS) and judgment post-stratification (JPS) are related schemes in which more efficient statistical inference is obtained by creating a stratification based on ranking information. The rankings may be completely subjective, or they may be based on values of a covariate. Recent work has shown that regardless of how the rankings are done, the in-stratum cumulative distribution functions (CDFs) must satisfy certain constraints, and we show here that if the rankings are done according to a covariate, then tighter constraints must hold. We also show that under a mild stochastic ordering assumption, still tighter constraints must hold. Taking advantage of these new constraints leads to improved small-sample estimates of the in-stratum CDFs in all RSS and JPS settings. For JPS, the new constraints also lead to improved estimates of the overall CDF and the population mean.     

Constrained optimization problems using multiplier methods

A modified multiplier method for optimization problems with equality constraints is suggested and its application to constrained optimal control problems described. For optimal control problems with free terminal time, a gradient descent technique for updating control functions as well as the terminal time is developed. The modified multiplier method with the simplified conjugate gradient method is used to compute the solution of a time-optimal control problem for a V/STOL aircraft.     

Constrained optimization using multiple objective programming

In practical applications of mathematical programming it is frequently observed that the decision maker prefers apparently suboptimal solutions. A natural explanation for this phenomenon is that the applied mathematical model was not sufficiently realistic and did not fully represent all the decision makers criteria and constraints. Since multicriteria optimization approaches are specifically designed to incorporate such complex preference structures, they gain more and more importance in application areas as, for example, engineering design and capital budgeting. The aim of this paper is to analyze optimization problems both from a constrained programming and a multicriteria programming perspective. It is shown that both formulations share important properties, and that many classical solution approaches have correspondences in the respective models. The analysis naturally leads to a discussion of the applicability of some recent approximation techniques for multicriteria programming problems for the approximation of optimal solutions and of Lagrange multipliers in convex constrained programming. Convergence results are proven for convex and nonconvex problems.     

Tests of perfect judgment ranking using pseudo-samples

Ranked set sampling (RSS) is a sampling approach that can produce improved statistical inference when the ranking process is perfect. While some inferential RSS methods are robust to imperfect rankings, other methods may fail entirely or provide less efficiency. We develop a nonparametric procedure to assess whether the rankings of a given RSS are perfect. We generate pseudo-samples with a known ranking and use them to compare with the ranking of the given RSS sample. This is a general approach that can accommodate any type of raking, including perfect ranking. To generate pseudo-samples, we consider the given sample as the population and generate a perfect RSS. The test statistics can easily be implemented for balanced and unbalanced RSS. The proposed tests are compared using Monte Carlo simulation under different distributions and applied to a real data set.     

Constrained l1 estimation via geometric programming

The l₁ and constrained l₁ estimation problems are viewed in the light of extended geometric programming. As a result of this point of view we are able to establish an equivalence between geometric and linear programming duality results for these classes of problems. In addition, the duality results provide some useful insights into the properties of the l₁ estimation problems. Finally we establish an equivalence between the l_∞ norm problem and a class of constrained l₁ estimation problems.     

Constrained interpolation using rational Cubic Spline with linear denominators

In this paper, a rational cubic interpolant spline with linear denominator has been constructed, and it is used to constrain interpolation curves to be bounded in the given region. Necessary and sufficient conditions for the interpolant to satisfy the constraint have been developed. The existence conditions are computationally efficient and easy to apply. Finally, the approximation properties have been studied.     

Density estimation using game theory

In this note we show that the mathematical tools of cooperative game theory allow a successful approach to the statistical problem of estimating a density function. Specifically, any random sample of an absolutely continuous random variable determines a transferable utility game, the Shapley value of which proves to be an estimator of the density function of binned kernel and WARPing types, with good computational and statistical properties.Authors acknowledge the financial support of Spanish Ministry for Science and Technology and FEDER through projects BFM2002-03213 and BEC2002-04102-C02-02 and of Xunta de Galicia through projects PGIDT00PXI20104PR and PGIDT03PXIC20701PN. They also thank the comments of two anonymous referees.     

Constrained least squares estimation of payment lag distributions in the United States' Medical Assistance Program

^** Email: nathbear{at}copper.net^*** Email: jpowell{at}usd.edu The financial effectiveness of the United States' Medical AssistanceProgram (Medicaid) depends on sound financial management atall political levels. States receive payments from the FederalGovernment based on the number, kinds and payment rates of servicesprovided to eligible patients. Because payments can lag servicesby a considerable time and because claims may be denied, adjustedor re-processed, the state cannot easily predict its total paymentsfor services recently delivered. As a matter of fact, some claimslinger as long as 24 months (Arnett & Foster, Pllc., 2002,unpublished data) in the system. This paper presents a procedureto estimate the percentages of payments due in the months followingthe service delivery month. The procedure relies on the mostrecent 24 months of services and payments. Service months fromthe recent past generate fewer monthly payments than those moredistant and can have large, unknown outstanding balances. Througha transformation of the available data, our procedure takesinto account the dwindling amounts of data available. The procedure,optimal in the sense of constrained least squares, results inestimates of the percentages paid as a function of time elapsedsince service. A dynamic programming procedure stabilizes tailarea percentages and a hold-out sample provides evidence ofgood fit. Real-world data illustrate the process.     

Bayesian estimation ofg-and-k distributions using MCMC

Summary  In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of theg-and-k distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach. The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.     

Nonparametric quantile estimation using importance sampling

Nonparametric estimation of a quantile of a random variable m(X) is considered, where \(m: \mathbb {R}^d\rightarrow \mathbb {R}\) is a function which is costly to compute and X is a \(\mathbb {R}^d\)-valued random variable with a given density. An importance sampling quantile estimate of m(X), which is based on a suitable estimate \(m_n\) of m, is defined, and it is shown that this estimate achieves a rate of convergence of order \(\log ^{1.5}(n)/n\). The finite sample size behavior of the estimate is illustrated by simulated data.     

Second derivative estimation using harmonic analysis

Simulation sensitivity analysis is an important problem for simulation practitioners analyzing complex systems. The significance of this problem has resulted in the development of various gradient estimators that can be used to address this issue. Although higher derivative estimators have been discussed concurrently, less attention has been given to assess the efficiency and feasibility of computing such estimators. In this paper, two second derivative estimators are presented. The first estimators, called the HFD estimators, combine harmonic gradient estimators with finite differences second derivative estimators. The resulting hybrid estimators requireO(p) fewer simulation runs to implement compared to the straightforward finite differences approach, wherep is the number of input parameters in the simulation model. The second estimators, called the HA estimators, incorporate harmonic analysis directly, requiring one or two simulation runs to implement, depending on whether a control variate simulation run is made. Expressions for the bias and the variance of the HFD and the HA estimators (with and without variance reduction techniques) are derived. Optimal mean squared error convergence rates are also discussed. In particular, the convergence rates for both these estimators are shown to be the same, though the computational performance of the HFD estimators is better than that for the HA estimators on anM/M/1 queue simulation model. Computational results for the HFD estimators on an (s, S) inventory system simulation model are also included.     

Matrix-free preconditioning using partial matrix estimation

We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph coloring problems for partial matrix estimation. Numerical experiments utilizing one of these formulations demonstrate the viability of this approach. AMS subject classification (2000) 65F10, 65F50, 49M37, 90C06     

Cpk index estimation using fuzzy numbers

Process capacity indices (PCIs) were developed and have been successfully used by companies to compete in and dominate the high-profit markets by improving the quality and the productivity since the past two decades. There is an essential assumption, in the conventional application, wherein the output process measurements are precise and distributed as normal random variables. Since the assumption of normal distribution is untenable, errors can occur if the C_pk index is computed using non-normal data. In the present study, we address the situation that the output of data from measurement of the quality of a product is insufficiently precise or scarce. This is possible when the quality measurement refers to the decision-maker’s subjective determination. In such a situation, the linguistic variable that is easier to capture the decision-maker’s subjective perception is applied to construct the PCI C_pk. The present approach can mitigate the effect when the normal assumption is inappropriate and extends the application of C_pk index.     

Pseudo-empirical likelihood estimation using nonparametric regression

Pseudo-empirical likelihood estimation of the population mean is considered. A nonparametric regression theory is proposed, to provide the fitted values on which to calibrate, and the common model misspecification problem is therefore addressed. Results derived from empirical studies show that the proposed estimator for the population mean can perform better than alternative estimators.     

Constrained Consequence

There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reasoning demands (beyond a\modelsb{\alpha\models\beta}) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence ·a{\bullet\alpha} with b\models·a{\beta\models\bullet\alpha}. Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies.