Jonathan D. Cryer

The Estimation of Sigma for an X Chart: /d2 or S/c4?

The standard approach to estimating sigma for an individual observations control chart is to use moving ranges. We show that, for independent, normally distributed observations, this approach is very inefficient compared to using the sample standard dev..

Mirror-Image and Invariant Distributions in ARMA Models

The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome.

An Iterative Procedure for Estimating the Mode

Abstract An estimate of the mode of a distribution is proposed. The estimate is obtained from a set of nested intervals each of which is a shortest interval containing a certain number of observations. The convergence properties of the estimator are presented along with some results of simulation studies.

A time-series analysis of public transit ridership in Portland, Oregon, 1971–1982

This paper examines the factors affecting changes in transit ridership in Portland, Oregon, during the period 1971 through 1982. A time-series methodology is used to investigate the effects of service level, travel costs, and market size at the system, sector, and route levels. Transfer function and multiple time-series models are compared. Intervention analysis is used to determine the impact of 81 service-level changes and 5 fare changes. A comparison is made of the elasticities estimated for these changes with elasticities developed from other studies.

Forecast Error Symmetry in ARIMA Models

Abstract Forecasts and residuals from estimated autoregressive integrated moving average (ARIMA) models are investigated with respect to the symmetry of their distributions. For models estimated with and without intercept terms and, more generally, for regression models with correlated errors, it is shown that both the set of residuals and the set of forecast errors at leads 1, 2, …, l have joint distributions that are symmetric about 0 under mild symmetry conditions on the true process generating the observations. We consider most common estimation methods including maximum likelihood, conditional least squares, and unconditional least squares. We also show that the forecast error t statistics are symmetrically distributed about 0.

Isotonic Estimation of the Probability of Extinction of a Branching Process

Abstract Suppose X 1, X 2, ···, Xn is a random sample from a distribution which assigns all of its mass to the nonnegative integers. Let p j = P[X 1 = j]; j = 0, 1, ··· and let f(·) be the generating function of the sequence {p j }∞ j=0. An estimate of the smallest nonnegative root of f(s) = s which is based on isotonized estimates of p 0, p 1, ··· is considered. Results of a simulation study, almost sure convergence rates and the asymptotic distribution theory are discussed.