Aaron Tenenbein

A Double Sampling Scheme for Estimating from Binomial Data with Misclassifications

Abstract Two measuring devices are available to classify units into one of two mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. To estimate p, the proportion of units which belong to one of the categories, a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimate of p is derived along with its asymptotic variance. Optimum values of n and N which minimize the measurement costs for a fixed variance of estimation and which minimize the precision for fixed cost are derived. This double sampling scheme is compared to the binomial sampling scheme in which only true measurements are obtained.

A Double Sampling Scheme for Estimating from Misclassified Multinomial Data with Applications to Sampling Inspection

In some situations, it is desired to estimate multinomial proportions from data which have been misclassified. One such area is the sampling inspection area of quality control. In this paper, it is assumed that two measuring devices are available to classify units into one of r mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. In order to estimate the proportions pi (i = 1,2, …, r) a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimates of the pi are derived along with their asymptotic variances. Optimum values of n and N which minimize the measurement costs for a fixed precision of estimation and which minimize the precision for fixed cost are derive...

A Bivariate Distribution Family with Specified Marginals

Abstract A systematic approach is given for constructing continuous bivariate distributions with specified marginals and ixed dependence measures. This approach is based on linear combinations of independent random variables and results in bivariate distributions that can attain the Frechet bounds. The dependence measures considered are Spearmans rho and Kendalls tau. Applications to testing for sensitivity in simulation models are discussed.

Misspecifications of the Normal Distribution

Abstract Incorrect statements about the normal distribution are discussed and illustrated with counterexamples.

The Racing Car Problem

The racing car problem can be stated as follows: A spectator at a race track is observing a car race in which the cars are numbered consecutively from one to some unknown number N. He wishes to estimate the number of cars on the race track after observing that M cars numbered Xi, X2, . . ., XM have passed. Each car is equally likely to hold a given position in the race at any given time. In this paper, we will find the minimum variance unbiased estimate and the maximum likelihood estimate of N. We will also compare these two estimates on the basis of relative efficiency. I like to give this problem in an introductory mathematical statistics course at the level of Mood and Graybill [1], because it illustrates all the points of the theory of minimum variance unbiased estimation without becoming too mathematically intractable.

Analysis of multivariate categorical data with misclassification errors by triple sampling schemes

Abstract Previous work has been carried out on the use of double-sampling schemes for inference from categorical data subject to misclassification. The double-sampling schemes utilize a sample of n units classified by both a fallible and true device and another sample of n2 units classified only by a fallible device. In actual applications, one often hasavailable a third sample of n1 units, which is classified only by the true device. In this article we develop techniques of fitting log-linear models under various misclassification structures for a general triple-sampling scheme. The estimation is by maximum likelihood and the fitted models are hierarchical. The methodology is illustrated by applying it to data in traffic safety research from a study on the effectiveness of belts in reducing injuries.

On triple sampling scremes for estimating from binomial data with misclassification errors

Previous work has been carried out on the use of double sampling schemes for inference from binomial data which are subject to misclassification. The double sampling scheme utilizes a sample of n units which are classified by both a fallible and a true device and another sample of n2 units which are classified only by a fallible device. A triple sampljng scheme incorporates an additional sample of nl units which are classified only by the true device. In this paper we apply this triple sampling to estimation from binomialdata. First estimation of a binomial proportion is discussed under different misclassification structures. Then, the problem of optimal allocation of sample sizes is discussed.

Number of matches and matched people in the birthday problem

The well known birthday problem asks for the probability of at least one match out of a group of n people. Also of interest are the number of matches and the number of matched people. In this paper the means and variances of the number of matches and matched people are obtained. A generalization of the use of these methods to computer storage analysis is discussed.

Probability computations for an extension of the geometric distribution

An extension of the stochastic process associated with the geometric distribution is presented. Combinatorial arguments are used to derive probabilities for various events of interest. Probabilities are approximated by evaluating truncated series. Bounds on the errors of approximation are developed. An example is presented and some additional applications are noted.

Expected number of passes in a binary search scheme

The binary search scheme is a method of finding a particular file from a set of ordered files stored in a computer. This scheme involves examining the middle file, and eliminating those files which do not contain the required file. This procedure is repeated until the required file is found. In this paper an exact expression for the expected number of passes required to find a file selected at random is derived. The expected number of passes is shown to be asymptotically equal to log2 k where k is the number of files. The asymptotic approximate values are compared to the exact values.