Nicholas R. Farnum

USING JOHNSON CURVES TO DESCRIBE NON-NORMAL ROCESS DATA

Process capability indices were originally defined under the assumption that process data can be adequately described by a normal distribution. Applied to non-normal processes, however, these indices can give misleading results. In this article, we outl..

A generalized binomial distribution

A new generalization of the binomial distribution is introduced that allows dependence between trials, nonconstant probabilities of success from trial to trial, and which contains the usual binomial distribution as a special case. Along with the number of trials and an initial probability of ‘success’, an additional parameter that controls the degree of correlation between trials is introduced. The resulting class of distributions includes the binomial, unirnodal distributions, and bimodal distributions. Formulas for the moments, mean, and variance of this distribution are given along with a method for fitting the distribution to sample data.

Control Charts for Short Runs: Nonconstant Process and Measurement Error

When the usual assumption of constant variances does not hold, the recommended substitute for a deviations from nominal (DNOM) chart is to chart standardized deviations. This has the added difficulty of estimating different standard deviations for each ..

Using Bilinear Transformations to Induce Probability Distributions

Abstract We use bilinear transformations to map points z = cos(α) + isin(α) on the unit circle in the complex plane into points x on the real line. Given any density function g(α) on the interval (−π, π), we show how a corresponding density function f(x) on (−∞, ∞) is induced. When α is uniformly distributed on (−π, π), we show that x has a Cauchy distribution in (−∞, ∞). When g(α) = Kn (1 + cos(α)) n , we show that x has a t-distribution in (−∞, ∞).

Uniqueness of maximum likelihood estimators of the 2-parameter Weibull distribution

We present a simple statistic, calculated from either complete failure data or from right-censored data of type-I or -II. It is useful for understanding the behavior of the parameter maximum likelihood estimates (MLE) of a 2-parameter Weibull distribution. The statistic is based on the logarithms of the failure data and can be interpreted as a measure of variation in the data. This statistic provides: (a) simple lower bounds on the parameter MLE, and (b) a quick approximation for parameter estimates that can serve as starting points for iterative MLE routines; it can be used to show that the MLE for the 2-parameter Weibull distribution are unique.

Using Counts to Monitor a Process Mean

A procedure for controlling a process mean to a target midway between two specified values is presented. It is assumed that the process variable has a normal distribution with unknown mean and variance, and that the data consists solely of counts of the..

A New Fixed Point Iteration to Find Percentage Points for Distributions on the Positive Axis

In this article, we apply a fixed point approach to derive an iterative method that converges quadratically to any percentage point for a wide variety of commonly used distributions on the positive axis, often with no restriction on the starting point.

Improving the Relative Error of Estimation

Abstract A condition often imposed on estimated models is that they produce estimates or predictions that have a small average (absolute) relative error. Usually, however, the relative error criterion is not used anywhere in the estimation process, but only to evaluate a model estimated by other means. A procedure is given for reducing the relative error of a model fit by any method. The procedure adjusts the original model by a multiplicative factor that is easily calculated from the raw data and the models estimates of that data. I show that the relative error for the adjusted model never exceeds that of the original model.

A correlated poisson distribution for correlated events

In this paper we introduce a correlated Poisson distribution (CPD) that incorporates possible nonzero correlation between successive events. The CPD is a two-parameter distribution that reduces to the usual Poission distribution in the case of zero correlation between successive events. Computational experimentation with various data show the usefulness of the CPD in modelling such correlated data.

Closed-Form Approximation for the AOQL of Attributes Acceptance Sample Plans

We derive upper and lower bounds at the point at which the average outgoing quality limit (AOQL) of an attributes acceptance sampling plan is achieved. Using a simple average of these bounds to approximate the ordinate of the AOQL, we develop an accurate, closed-form approximation to the AOQL. The bounds and approximation show how the parameters of a sampling plan affect the AOQL and can be used to study the behavior of the AOQL and other measures of the plans performance.