Chien Pai Han

Discriminant Analysis When a Block of Observations is Missing

We consider the problem of classifying a p× 1 observation into one of two multivariate normal populations when the training samples contain a block of missing observations. A new classification procedure is proposed which is a linear combination of two discriminant functions, one based on the complete samples and the other on the incomplete samples. The new discriminant function is easy to use. We consider the estimation of error rate of the linear combination classification procedure by using the leave-one-out estimation and bootstrap estimation. A Monte Carlo study is conducted to evaluate the error rate and the estimation of it. A numerical example is given tof illustrate its use.

Conditional confidence intervals for classification error rate

An observation is to be classified into one of several multivariate normal populations with equal covariance matrix. When the parameters are unknown, independent training samples are taken from the populations. We consider the construction of confidence intervals for the conditional error rate. The cases of two populations and three populations are studied in detail. We propose the conditional jackknife confidence interval and the conditional bootstrap confidence intervals of the conditional error rate. A Monte Carlo study is conducted to compare the confidence intervals.

Interval estimation of error variance following a preliminary test in one–way random model

A confidence interval for the error variance following a preliminary test in the oneway random model is investigated. This interval is referred to as a pre-test confidence interval. The expected length of the pre-test confidence interval is shorter than that of the usual confidence interval. The coverage probability of the pre–test confidence interval is less than the nominal level. However, at some selected levels of significance, the coverage probability of the pre-test confidence interval is only slightly less than the nominal level but the maximum reduction in expected length is substantial

Truncated multiple inverse sampling in post-stratification

Multiple inverse sampling (MIS) is a sequential sampling procedure in a stratified population such that random samples are taken population-wide and continuously until a specified minimum number of observations are obtained in each stratum. The MIS procedure can be used in small sample post-stratification to solve the empty post-strata problem. However, the MIS procedure does not have control over the total sample size, in particular, when some of the stratum weights are small. A remedy for this drawback of MIS is to truncate the sampling procedure when the total sample size reaches a specified maximum number. The efficiency of the truncated MIS procedure is investigated and it is found that the truncated MIS is superior to the classical fixed sample size sampling procedure in post-stratification.

Computation of noncentral f distributions with even denominator degrees of freedom

Computational formulas for the noncentral F distributions are given when the degrees of freedom in the denominator is even and the degrees of freedom in the numerator is either odd or even.

A planning tool for maximising transit services

Transit is important for urban sustainable development, but the cost of providing transit services is high. Strategic planning for a public transit system is necessary to maximise the return of public investment. We develop a planning tool to aid decision-making about service area, using optimisation techniques for the Prize-Collecting Salesman problem and the concept of hierarchy of travel needs. We demonstrate the usefulness of the model for maximising service area under a given budget constraint and socioeconomic conditions, and suggest future research directions.

Effect of Testing Normality on Estimating Process Capability Indices

Process capability indices are used to measure whether a manufacturing process meets the specifications. The studies of these indices are usually based on the assumption that the process follows a normal distribution. These include the indices C p , C pk , C pm , and C pmk . When the investigator is uncertain whether the process follows a normal distribution, a test of normality may be used to resolve the uncertainty. If the test accepts the null hypothesis that the process follows a normal distribution, the investigator uses C p , C pk , C pm , or C pmk . If the test rejects the null hypothesis, the investigator uses indices under non-normal distributions. Therefore the test of normality is a preliminary test that determines the form of the distribution and the index to use. In this paper we study the effect of the preliminary test of normality on the estimation of the four process capability indices mentioned above.

Evaluating Estimation Techniques of Transportation Price Elasticity

Price elasticity is critical for managed lane planning. To set an effective pricing policy that can achieve a specific objective–-whether it is to maximize freeway system performance, to reduce vehicle emission, or to capitalize on revenue–-transportation planners and policy makers need to know how users would react to various pricing policies. The issue is even more important when the public has not yet fully embraced transportation pricing. Three techniques have been frequently used for estimating transportation price elasticities: shrinkage ratio, midpoint elasticity, and log arc elasticity. However, little is known about the suitability of the techniques. This paper evaluates these techniques by analyzing their statistical properties and comparing their relative efficiency. The findings suggest that the appropriateness of an estimation technique of price elasticity varies from one circumstance to another. It is important to understand the variations and apply the techniques accordingly.

Improved estimators of regression coefficients in measurement error models

In the theory of regression analysis, it is well known that the presence of measurement errors in the explanatory variables makes the ordinary least squares estimator (OLSE) biased and inconsistent, and the maximum likelihood estimator (MLE) is not obtainable without restrictive assumptions. We develop mixed-type estimators, which combine OLSE and MLE and compromise between them, utilizing a preliminary test or weighting function of a test statistic. It is shown that the weighting function estimators are superior to the classical estimators in terms of mean square errors in a large parameter space.

Adaptive plotting position and test of normality

The quantile–quantile plot is widely used to check normality. The plot depends on the plotting positions. Many commonly used plotting positions do not depend on the sample values. We propose an adaptive plotting position that depends on the relative distances of the two neighbouring sample values. The correlation coefficient obtained from the adaptive plotting position is used to test normality. The test using the adaptive plotting position is better than the Shapiro–Wilk W test for small samples and has larger power than Hazens and Bloms plotting positions for symmetric alternatives with shorter tail than normal and skewed alternatives when n is 20 or larger. The Brown–Hettmansperger T* test is designed for detecting bad tail behaviour, so it does not have power for symmetric alternatives with shorter tail than normal, but it is generally better than the other tests when β2 is greater than 3.25.