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.     

Distribution and density functions of the feasible generalized ridge regression estimator

In this paper, we derive the distribution and density functions of the feasible generalized ridge regression (GRR) estimator. It is shown that when the absolute value of a regression coefficient is close to zero, the distribution of the feasible GRR estimator is bimodal and has thinner tails than that of the OLS estimator.     

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.     

Robust Likelihood Cross-Validation for Kernel Density Estimation

Likelihood cross-validation for kernel density estimation is known to be sensitive to extreme observations and heavy-tailed distributions. We propose a robust likelihood-based cross-validation method to select bandwidths in multivariate density estimations. We derive this bandwidth selector within the framework of robust maximum likelihood estimation. This method establishes a smooth transition from likelihood cross-validation for nonextreme observations to least squares cross-validation for extreme observations, thereby combining the efficiency of likelihood cross-validation and the robustness of least-squares cross-validation. We also suggest a simple rule to select the transition threshold. We demonstrate the finite sample performance and practical usefulness of the proposed method via Monte Carlo simulations and a real data application on Chinese air pollution.     

Bayesian Approaches to Nonparametric Estimation of Densities on the Unit Interval

This paper investigates nonparametric estimation of density on [0, 1]. The kernel estimator of density on [0, 1] has been found to be sensitive to both bandwidth and kernel. This paper proposes a unified Bayesian framework for choosing both the bandwidth and kernel function. In a simulation study, the Bayesian bandwidth estimator performed better than others, and kernel estimators were sensitive to the choice of the kernel and the shapes of the population densities on [0, 1]. The simulation and empirical results demonstrate that the methods proposed in this paper can improve the way the probability densities on [0, 1] are presently estimated.     

SERIES REPRESENTATION OF DOUBLY NONCENTRAL t DISTRIBUTION BY USE OF THE MELLIN INTEGRAL TRANSFORM

ABSTRACT

The Mellin integral transform is widely used to find the distributions of products and quotients of independent random variables defined over the positive domain. But it is hardly used to derive the distributions defined over both positive and negative values of the random variables. In this paper, the Mellin integral transform is applied to obtain the doubly noncentral t density and its distribution function in convergent series forms.     

A class of distributions with the linear mean residual quantile function and it’s generalizations

In the present paper, we introduce and study a class of distributions that has the linear mean residual quantile function. Various distributional properties and reliability characteristics of the class are studied. Some characterizations of the class of distributions are presented. We then present generalizations of this class of distributions using the relationship between various quantile based reliability measures. The method of L-moments is employed to estimate parameters of the class of distributions. Finally, we apply the proposed class of distributions to a real data set.     

Multivariate regression analysis of panel data with binary outcomes applied to unemployment data

Summary  In panel studies binary outcome measures together with time stationary and time varying explanatory variables are collected over time on the same individual. Therefore, a regression analysis for this type of data must allow for the correlation among the outcomes of an individual. The multivariate probit model of Ashford and Sowden (1970) was the first regression model for multivariate binary responses. However, a likelihood analysis of the multivariate probit model with general correlation structure for higher dimensions is intractable due to the maximization over high dimensional integrals thus severely restricting ist applicability so far. Czado (1996) developed a Markov Chain Monte Carlo (MCMC) algorithm to overcome this difficulty. In this paper we present an application of this algorithm to unemployment data from the Panel Study of Income Dynamics involving 11 waves of the panel study. In addition we adapt Bayesian model checking techniques based on the posterior predictive distribution (see for example Gelman et al. (1996)) for the multivariate probit model. These help to identify mean and correlation specification which fit the data well. C. Czado was supported by research grant OGP0089858 of the Natural Sciences and Engineering Research Council of Canada.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Estimating the number of clusters

Hartigan (1975) defines the number q of clusters in a d ‐variate statistical population as the number of connected components of the set {f > c}, where f denotes the underlying density function on R^d and c is a given constant. Some usual cluster algorithms treat q as an input which must be given in advance. The authors propose a method for estimating this parameter which is based on the computation of the number of connected components of an estimate of {f > c}. This set estimator is constructed as a union of balls with centres at an appropriate subsample which is selected via a nonparametric density estimator of f. The asymptotic behaviour of the proposed method is analyzed. A simulation study and an example with real data are also included.