Lin-An Chen

Maximum Average-Power (MAP) Tests

The objective of this article is to propose and study frequentist tests that have maximum average power, averaging with respect to some specified weight function. First, some relationships between these tests, called maximum average-power (MAP) tests, and most powerful or uniformly most powerful tests are presented. Second, the existence of a maximum average-power test for any hypothesis testing problem is shown. Third, an MAP test for any hypothesis testing problem with a simple null hypothesis is constructed, including some interesting classical examples. Fourth, an MAP test for a hypothesis testing problem with a composite null hypothesis is discussed. From any one-parameter exponential family, a commonly used UMPU test is shown to be also an MAP test with respect to a rich class of weight functions. Finally, some remarks are given to conclude the article.

Multivariate regression splines

Abstract Multivariate regression splines of arbitrary order assuming known knots using the additional function are developed. Model description and possible parameter tests for obtaining regression splines are also stated in detail for the triplicate case. A simulation study of drawn data from the Lubricant nonlinear regression model, where “Lubricant” is referred to lubricant data that we use in application, to compare the mean squares errors for the multivariate regression spline models and the multivariate polynomial models show the need of employing the multivariate regression splines in the approximation of the nonlinear regression models.

The density control chart: a general approach for constructing a single chart for simultaneously monitoring multiple parameters

The majority of the existing literature on simultaneous control charts, i.e. control charting mechanisms that monitor multiple population parameters such as mean and variance on a single chart, assume that the process is normally distributed. In order to adjust and maintain the overall type-I error probability, these existing charts rely largely on the property that the sample mean and sample variance are independent under the normality assumption. Furthermore, the existing charting procedures cannot be readily extended to non-normal processes. In this article, we propose and study a general charting mechanism which can be used to construct simultaneous control charts for normal and non-normal processes. The proposed control chart, which we call the density control chart, is essentially based on the premise that if a sample of observations is from an in-control process, then another sample of observations is no less likely to be also from the in-control process if the likelihood of the latter is no less than the likelihood of the former. The density control chart is developed for normal and non-normal processes where the distribution of the plotting statistic of the density control chart can be explicitly derived. Real examples are given and discussed in these cases. We also discuss how the density control chart can be constructed in cases when the distribution of the plotting statistic cannot be determined. A discussion of potential future research is also given.

Parametric coverage interval

Parametric estimation of coverage interval is useful since the parametric intervals are generally narrower than the non-parametric ones; however, it has been considered only for the measurement variable with normal distribution. Here we propose a general technique for constructing parametric coverage intervals that may deal with all distributions, both symmetric and asymmetric, in measurement science.

Extending the discussion on coverage intervals and statistical coverage intervals

Willink (2004 Metrologia 41 L5–6) is concerned that, in the society of metrology, there is potential for confusion between coverage interval and statistical coverage interval and he makes a precise interpretaion of these two terms. We further clarify that the confidence of a coverage interval is actually a statistical coverage interval.

Bivariate regression splines

Abstract Towards the construction of multivariate spline functions, we introduce a way to set linear restrictions in the generation of bivariate regression splines. The hyperplanes in R 2 are used in the role of “knot” to slice the domain of explanatory variables; hence, we have the flexibility in domain partition which includes rectangle, parallelogram, trapezoid and trapezium.

Symmetric quantiles and their applications

Abstract To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) and Chen & Chiang (1996), implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.

A non-parametric coverage interval

Classically the non-parametric coverage interval is estimated by empirical quantiles. We introduce an alternative way for estimating the coverage interval by symmetric quantiles given by Chen and Chiang (1996 J. Nonparametric Stat. 7 171–85). We further show that this alternative has a better precision in the sense that its asymptotic variances are smaller than the classical one.

Mode type quasi-range and its applications

Building from the consideration of closeness, we propose the mode quasi-range as an alternative scale parameter. Application of this scale parameter to formulate the population standard deviation is investigated leading to an efficient sample estimator of standard deviation from the point of asymptotic variance. Monte Carlo studies, in terms of finite sample efficiency and robustness of breakdown point, have been performed for the sample mode quasi-range. This study reveals that this closeness consideration-based mode, quasi-range, is satisfactory because these statistical procedures based on it are efficient and are less misleading for drawing conclusion from the sample results.

A multivariate parallelogram and its application to multivariate trimmed means

Summary This paper introduces a multivariate parallelogram that can play the role of the univariate quantile in the location model, and uses it to define a multivariate trimmed mean. It assesses the asymptotic efficiency of the proposed multivariate trimmed mean by its asymptotic variance and by Monte Carlo simulation.