http://people.stern.nyu.edu/wgreene/FrontierModeling/Reference-Papers/Aigner-Lovell-Schmidt-JE1977-ALS.pdf

Formulation and estimation of stochastic frontier production function models

The receiver operating characteristic (ROC) curve is used to evaluate a biomarker's ability for classifying disease status. The Youden Index (J), the maximum potential effectiveness of a biomarker, is a common summary measure of the ROC curve. In biomarker development, levels may be unquantifiable below a limit of detection (LOD) and missing from the overall dataset. Disregarding these observations may negatively bias the ROC curve and thus J. Several correction methods have been suggested for mean estimation and testing; however, little has been written about the ROC curve or its summary measures. We adapt non-parametric (empirical) and semi-parametric (ROC-GLM [generalized linear model]) methods and propose parametric methods (maximum likelihood (ML)) to estimate J and the optimal cut-point (c *) for a biomarker affected by a LOD. We develop unbiased estimators of J and c * via ML for normally and gamma distributed biomarkers. Alpha level confidence intervals are proposed using delta and bootstrap methods for the ML, semi-parametric, and non-parametric approaches respectively. Simulation studies are conducted over a range of distributional scenarios and sample sizes evaluating estimators' bias, root-mean square error, and coverage probability; the average bias was less than one percent for ML and GLM methods across scenarios and decreases with increased sample size. An example using polychlorinated biphenyl levels to classify women with and without endometriosis illustrates the potential benefits of these methods. We address the limitations and usefulness of each method in order to give researchers guidance in constructing appropriate estimates of biomarkers' true discriminating capabilities.

/pdf/youden-index-and-optimal-cut-point-estimated-from-1w3wz6ekkz.pdf

Youden Index and optimal cut-point estimated from observations affected by a lower limit of detection.

A model of category effects on reports from memory is presented. The model holds that stimuli are represented at 2 levels of detail: a fine-grain value and a category. When memory is inexact but people must report an exact value, they use estimation processes that combine the remembered stimulus value with category information. The proposed estimation processes include truncation at category boundaries and weighting with a central (prototypic) category value. These processes introduce bias in reporting even when memory is unbiased, but nevertheless may improve overall accuracy (by decreasing the variability of reports). Four experiments are presented in which people report the location of a dot in a circle. Subjects spontaneously impose horizontal and vertical boundaries that divide the circle into quadrants. They misplace dots toward a central (prototypic) location in each quadrant, as predicted by the model. The proposed model has broad implications; notably, it has the potential to explain biases of the sort described in psychophysics (contraction bias and the bias captured by Weber's law) as well as symmetries in similarity judgments, without positing distorted representations of physical scales.

Categories and particulars: prototype effects in estimating spatial location.

One result of increased scientific and public interest in water quality over the past few decades has been the gradual accumulation of reliable long-term water quality data records and an interest in examining these data for long-term trends. This paper summarizes and examines some of the major issues and choices involved in detecting and estimating the magnitude of temporal trends in measures of stream water quality. The first issue is the type of trend hypothesis to examine: step trends versus monotonic trend. The second relates to the general category of statistical methods to employ: parametric versus nonparametric. The third issue relates to the kind of data to analyze: concentration data versus flux data. The fourth relates to issues of data manipulation to achieve the best results from the trend analysis. These issues include the use of mathematical transformations of the data and the removal of natural sources of variability in water quality due to seasonal and stream discharge variations. The final issue relates to the choice of a trend technique for the analysis of data records with censored or “less than” values. The authors' experiences during the past decade with the development of several trend detection techniques and application of these techniques to a large number of water quality records provide insight into the issues related to a choice of a statistical test for trend in water quality.

Selection of methods for the detection and estimation of trends in water quality

n observations are taken from a mixture of K normal subpopulations, where the value of K is known. It is assumed that these n observations are given as N frequencies from equally spaced intervals. Initial guesses of the K means, K variances, and K − 1 proportions are made using the maximum likelihood estimates for a single truncated normal population as derived by Hald. Then an approximation to the likelihood function of the entire sample is used, and attempts to maximize this yield two iteration formulas. In practice, the method of steepest descent always converged, although the rate was not always fast. Special cases of equal variances and variances proportional to the square of the mean are also considered.

Estimation of parameters for a mixture of normal distributions

This paper is concerned with the problem of estimating the mean and variance of normal populations from singly and doubly truncated samples having known truncation points. Maximum likelihood estimating equations are derived which, with the aid of standard tables of areas and ordinates of the normal frequency function, can be readily solved by simple iterative processes. Asymptotic variances and covariances of these estimates are obtained from the information matrices. Numerical examples are given which illustrate the practical application of these results. In Sections 3 to 8 inclusive, the following cases of doubly truncated samples are considered: I, number of unmeasured observations unknown; II, number of unmeasured observations in each `tail' known; and III, total number of unmeasured observations known, but not the number in each `tail'. In Section 9, singly truncated samples are treated as special cases of I and II above.

/pdf/estimating-the-mean-and-variance-of-normal-populations-from-4ad9tvpuho.pdf

A. C. Cohen

Papers

Estimating the Mean and Variance of Normal Populations from Singly Truncated and Doubly Truncated Samples