Rudy A. Gideon

A Rank Correlation Coefficient Resistant to Outliers

Abstract In this article, a nonparametric correlation coefficient is defined that is based on the principle of maximum deviations. This new correlation coefficient, Rg , is easy to compute by hand for small to medium sample sizes. In comparing it with existing correlation coefficients, it was found to be superior in a sampling situation that we call “biased outliers,” and hence appears to be more resistant to outliers than the Pearson, Spearman, and Kendall correlation coefficients. In a correlational study not included in this article of some social data consisting of five variables for each of 51 observations, Rg was compared with the other three correlation coefficients. There was agreement on 8 of the 10 possible correlations, but in one case, Rg was significant when the others were not, and in yet another case, Rg was not significant when the others were. A further analysis of this data set indicated that there were three to six data points that were anomalies and had a severe effect on the other cor...

Series Expansions for Quadratic Forms in Normal Variables

Abstract A differential operator is defined and applied to the gamma density function. By formal mathematical manipulation the resulting function can be identified with linear combinations of independent gamma variates and central and noncentral definite quadratic forms in independent normal variates. Laguerre series expansions in which the choice of parameters is of a general nature are given. These Laguerre series and chi-square, power and Edgeworth series are then compared in the effectiveness in evaluating the distribution function of quadratic forms.

Some Alternative Expansions for the Distribution Function of a Noncentral Chi-Square Random Variable

A differential operator is defined and applied to the gamma density function. By formal mathematical manipulations, the resulting function is identified with the distribution function of the noncentral chi-square distribution. Several series expansions of a general nature result, and a table is presented comparing the effectiveness of seven series in evaluating this distribution function.

Markov Renewal Modelling of Poisson Traffic at Intersections having Separate Turn Lanes

Consider a two-lane road which is intersected on one side by a single-lane secondary road. A single car waiting on the secondary road may merge into either the nearest or the farthest lane. It is assumed that the traffic in each lane is independent of the other lanes and that the inter-arrival times of cars at the intersection in their respective lanes is exponential. The main purpose of this paper is to study the queue size on the secondary road when the secondary road has a special right-turn lane (or left-turn lane in some countries) which allows some cars to merge into the nearest lane of the main road even when other cars waiting to enter the far lane are present. The problem is approached by first setting up a four-state Markov Renewal process to describe the traffic on the main road. Next the merging process on the secondary road is described as a Markov Renewed process with a random environment. The event that the queue is empty is studied, and conditions are stated under which this event is recurrent or transient. Finally, the quantities which occur in the conditions for recurrence of an empty queue are derived explicitly for a one-car right turn lane.

A Polynomial Type Approximation for Bivariate Normal Variates

The probability of a general bivariate normal random variable over an angular region is expressed in terms of an angular region over a circular normal random variable. This probability over an angular region is then approximated by a polynomial in the polar coordinates related to the vertex of the angular region. The approximation is accurate to at least 5 decimal places and is more accurate over most of the plane. A table of coefficients in the required polynomial is provided for carrying out the approximation.

Location and Scale Estimation with Correlation Coefficients

This article shows how to use any correlation coefficient to produce an estimate of location and scale. It is part of a broader system, called a correlation estimation system (CES), that uses correlation coefficients as the starting point for estimations. The method is illustrated using the well-known normal distribution. This article shows that any correlation coefficient can be used to fit a simple linear regression line to bivariate data and then the slope and intercept are estimates of standard deviation and location. Because a robust correlation will produce robust estimates, this CES can be recommended as a tool for everyday data analysis. Simulations indicate that the median with this method using a robust correlation coefficient appears to be nearly as efficient as the mean with good data and much better if there are a few errant data points. Hypothesis testing and confidence intervals are discussed for the scale parameter; both normal and Cauchy distributions are covered.

A two-sample distribution-free scale test of the smirnov type

The two-sample, distribution-free statistics of Smirnov (1939) are used to define a new statistic. While the Smirnov statistics are used as a general goodness-of-fit test, a distribution-free scale test based on this new statistic is developed. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting differences in scale for some symmetric distributions with equal means/medians. The critical values of the proposed test statistic and its limiting distribution are given

The Limiting Distribution of the Rank Correlation Coefficient {R_g}

A new correlation coefficient, \({R_g}\), based on ranks and greatest deviation was defined in Gideon and Hollister (1987). In there the exact distributions were obtained by enumeration for small sample sizes, and by computer simulations for larger sample sizes. In this note, it is shown that the asymptotic distribution of n 1/2 \({R_g}\) is N(0,1) when the variables are independent and n is the sample size. This limit is derived by restating the definition of \({R_g}\) in terms of a rank measure and then using a limit theorem on set-indexed empirical processes which appears in Pyke (1985). The limiting distribution can be compared to the critical values for large samples given in Figure 2 of Gideon and Hollister (1987). Methods for deriving the limiting distribution under fixed and contiguous alternatives are also described.

Computation of the Two-Sample Smirnov Statistics

Abstract A simple method is given for evaluating the one- or two-sided Smirnov statistic for comparing two independent samples from continuous populations.

An Interpolatory Method of Approximating Distribution Functions with Applications to the Gamma and Related Variates

Abstract A method of approximating distribution functions is presented which is based on a weighted sum of exponential functions. The method is quite general but attention is confined here to approximating the distribution function of gamma and related variates. As a limiting case of X 2, the normal distribution is also considered. Tables of constants are provided for applying the approximation procedure. It can be carried out on a computer or on a desk-type calculator. Among the attractive features of the technique is the fact that the constants involved in the approximating function can be obtained by simple interpolation.