Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

Estimation of the multi-criteria worths of the alternatives in a hierarchical structure of comparisons

The aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty (1980). A multiplicative model is used for the paired comparisons data which are collected in a ratio scale in this set-up in any level of this hierarchy. An iterative scheme is found for the maximum likelihood estimation of the worth parameters in this multiplicative model. The iterative values are shown to be convergent monotonically to the estimates. We also obtain the asymptotic dispersion matrix of the maximum likelihood estimates of the relative worths of the alternatives according to a single criterion as well as those according to the over-all suitability when compared under several criteria. A numerical example is presented to illustrate the method developed in this paper. Simulation techniques are employed to find the average number of iterations required for the convergence of the above iterative scheme.     

Combined estimate of a location parameter

In this paper, we revisit the problem of combining estimates of location considered by Cohen (1976). Our results unify and strengthen the results of Cohen (1976), Bhattacharya (1981) and Akai (1982).     

Cross-Validated Assessments of Test Scoring Models

On a multiple choice test in which each item has r alternative options, a given number c of which are correct, various scoring models have been proposed. In one case the test-taker is allowed to choose any size solution subset and he/she is graded according to whether the subset is small and according to how many correct answers the subset contains. In a second case the test-taker is allowed to select only solution subsets of a prespecified maximum size and is graded as above. The first case is analogous to the situation where the test-taker is given a set of r options with each question; each question calls for a solution which consists of selecting that subset of the r responses which he/she believes to be correct. In the second case, when the prespecified solution subset is restricted to be of size at most one, the resulting scoring model corresponds to the usual model, referred to below as standard. The number c of correct options per item is usually known to the test-taker in this case.

Scoring models are evaluated according to how well they correctly identify the total scores of the individuals in the class of test-takers. Loss functions are constructed which penalize scoring models resulting in student scores which are not associated with the students true (or average) total score on the exam. Scoring models are compared on the basis of cross-validated assessments of the loss incurred by using each of the given models. It is shown that in many cases the assessment of the loss for scoring models which allow students the opportunity to choose more than one option for each question are smaller than the assessment of the loss for the standard scoring model.     

A plug-in technique in nonparametric regression with dependence

The problem addressed is that of smoothing parameter selection in kernel nonparametric regression in the fixed design regression model with dependent noise. An asymptotic expression of the optimum bandwidth parameter has been obtained in recent studies, where this takes the form h = C ₀ n ^?1/5. This paper proposes to use a plug-in methodology, in order to obtain an optimum estimation of the bandwidth parameter, through preliminary estimation of the unknown value of C ₀.     

Estimating a frequency distribution when the sampling is biased

A simple random sample on a random variable A allows its density to be consistently estimated, by a histogram or preferably a kernel density estimate. When the sampling is biased towards certain x-values these methods instead estimate a weighted version of the density function. This article proposes a method for estimating both the density and the sampling bias simultaneously. The technique requires two independent samples and utilises ideas from mark-recapture experiments. An estimator of the size of the sampled population also follows simply from this density estimate.     

Smoothing techniques via the bezier curve

Although the Bezier curve is very popular in the area of computational graphics it has rarely been used by statisticians. In this paper we develop methods and techniques for use of the Bezier curve in estimation of density and regression function. Also, asymptotic mean integrated square error for both estimators are derived. Comparisons with kernel estimator are conducted using simulation.