Nonparametric tests for multivariate locations based on data depth

The present paper deals with the problem of testing equality of locations of two multivariate distributions using a notion of data depth. A notion of data depth has been used to measure centrality/outlyingness of a given point in a given data cloud. The paper proposes two nonparametric tests for testing equality of locations of two multivariate populations which are developed by observing the behavior of the depth versus depth plot. Simulation study reveals that the proposed tests are superior to the existing tests based on the data depth with regard to power. Illustrations with real data are provided.     

On the estimation of homogeneous population size from a complex dual-record system

ABSTRACT

A dual-record system (DRS) (equivalently two sample capture–recapture experiments) model, with time and behavioural response variation, has attracted much attention specifically in the domain of official statistics and epidemiology, as the assumption of list independence often fails. The relevant model suffers from parameter identifiability problem, and suitable Bayesian methodologies could be helpful. In this article, we formulate population size estimation in DRS as a missing data problem and two empirical Bayes approaches are proposed along with the discussion of an existing Bayes treatment. Some features and associated posterior convergence for these methods are mentioned. Investigation through an extensive simulation study finds that our proposed approaches compare favourably with the existing Bayes approach for this complex model depending upon the availability of directional nature of underlying behavioural response effect. A real-data example is given to illustrate these methods.     

Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.     

Robust logistic regression modelling via the elastic net-type regularization and tuning parameter selection

The penalized logistic regression is a useful tool for classifying samples and feature selection. Although the methodology has been widely used in various fields of research, their performance takes a sudden turn for the worst in the presence of outlier, since the logistic regression is based on the maximum log-likelihood method which is sensitive to outliers. It implies that we cannot accurately classify samples and find important factors having crucial information for classification. To overcome the problem, we propose a robust penalized logistic regression based on a weighted likelihood methodology. We also derive an information criterion for choosing the tuning parameters, which is a vital matter in robust penalized logistic regression modelling in line with generalized information criteria. We demonstrate through Monte Carlo simulations and real-world example that the proposed robust modelling strategies perform well for sparse logistic regression modelling even in the presence of outliers.     

Economic statistical design of a multivariate control chart under a Burr XII shock model with multiple assignable causes

Control charts show the distinction between the random and assignable causes of variation in a process. The real process may be affected by many characteristics and several assignable causes. Therefore, the economic statistical design of multiple control chart under Burr XII shock model with multiple assignable causes can be an appropriate candidate model. In this paper, we develop a cost model based on the optimization of the average cost per unit of time. Indeed, the cost model under the influence of a single match case assignable cause and multiple assignable causes under a same cost and time parameters were compared. Besides, a sensitivity analysis was also presented in which the changeability of loss-cost and design parameters were evaluated based on the changes in cost, time and Burr XII distribution parameters.     

Sensitivity analysis approaches to high-dimensional screening problems at low sample size

Sensitivity analysis is an essential tool in the development of robust models for engineering, physical sciences, economics and policy-making, but typically requires running the model a large number of times in order to estimate sensitivity measures. While statistical emulators allow sensitivity analysis even on complex models, they only perform well with a moderately low number of model inputs: in higher dimensional problems they tend to require a restrictively high number of model runs unless the model is relatively linear. Therefore, an open question is how to tackle sensitivity problems in higher dimensionalities, at very low sample sizes. This article examines the relative performance of four sampling-based measures which can be used in such high-dimensional nonlinear problems. The measures tested are the Sobol' total sensitivity indices, the absolute mean of elementary effects, a derivative-based global sensitivity measure, and a modified derivative-based measure. Performance is assessed in a ‘screening’ context, by assessing the ability of each measure to identify influential and non-influential inputs on a wide variety of test functions at different dimensionalities. The results show that the best-performing measure in the screening context is dependent on the model or function, but derivative-based measures have a significant potential at low sample sizes that is currently not widely recognised.     

The Design of and R Control Charts for Skew Normal Distributed Data

The Shewhart-type control chart is traditionally developed under the normality assumption. In practice, however, this assumption may not hold. Because the skew normal distribution represents a broad distribution class and is more flexible than is the normal distribution, we propose two new control charts to monitor process mean and spread for skew normal distributed data. Moreover, to facilitate practical implementation, tables of charting constants are provided. We conducted simulation studies to compare the false alarm rates, and the results show that new proposed charts perform better than others as skewness increases. Finally, an illustrative example is provided.     

An economic statistical design of the Poisson EWMA control charts for monitoring nonconformities

An economic statistical model of the exponentially weighted moving average (EWMA) control chart for the average number of nonconformities in the sample is proposed. The statistical and economic performance of proposed design are evaluated using the average run length (ARL) and the hourly expected cost, respectively. A Markov chain approach is applied to derive expressions for ARL. The cost model is established based on the general cost function given in Lorenzen and Vance [The economic design of control charts: a unified approach. Technometrics. 1986;28:3–11]. An example is provided to illustrate the application of the proposed model. A sensitivity analysis is also carried out to investigate the effects of model parameters on the solution of the economic statistical design by using the design of experiments (DOE) technique.     

Power of depth-based nonparametric tests for multivariate locations

Li and Liu [New nonparametric tests of multivariate locations and scales. Statist Sci. 2004;19(4):686–696] introduced two tests for a difference in locations of two multivariate distributions based on the concept of data depth. Using the simplicial depth [Liu RY. On a notion of data depth based on random simplices. Ann Stat. 1990;18(1):405–414], they studied the performance of these tests for symmetric distributions, namely, the normal and the Cauchy, in a simulation study. However, to the best of our knowledge, the performance of these tests for skewed distributions has not been studied in the current literature. This paper is a contribution in that direction and examines the performance of these depth-based tests in an extensive simulation study involving ten distributions belonging to five well-known families of multivariate skewed distributions. The study includes a comparison of the performance of these tests for four popular affine-invariant depth functions. Conclusions and recommendations are offered.