Ghettos of the mind: the empirical behaviour of indices of segregation and diversity

Summary.  The effect of immigration on social cohesion is a political issue, expressed as a fear that racially skewed residential patterns represent ghettos which prevent integration. Residential patterns have been measured by indices of segregation. The range of indices is reviewed in the paper and measured empirically for England and Wales by using census data for 1991 and 2001, including a new index of migration dispersal. There has been an increase in residential mixing as a result of growing minority populations and their more even spread across localities. These two trends are identified by two commonly used indices of segregation which are moving in opposite directions for the most recent immigrant groups. The sensitivity of each index to modifiable area boundaries makes them unsuitable for evaluation of cities' relative performance. The residential patterns of cities after immigration are more clearly understood by using demographic measures of migration and age structure.     

On Measuring Segregation in Samples With Small Units

Standard indexes of segregation measure a sample's distance from evenness, which occurs when each sample unit (e.g., an occupation) has the population share of both the minority and majority groups. We show that random allocation of individuals to units generates substantial unevenness among small units and hence that standard segregation indexes reflect random allocation as well as systematic group segregation. We then modify two popular indexes so that they measure deviations from random allocation rather than deviations from evenness. An empirical example suggests that these modified indexes provide improved measures of the systematic component of group segregation.     

Three New Multivariate Process Capability Indices

Different multivariate process capability indices are developed by researchers to evaluate process capability when vectors of quality characteristics are considered in a study. This article presents three indices referred to as NCpM, MCpM, and NMC _PM in order to evaluate process capability in multivariate environment. The performance of the proposed indices is investigated numerically. Simulation results indicate that the proposed indices have descended estimation error and improved performance compared to the existing ones. These results can be important to researchers and practitioners who are interested in evaluating process capability in multivariate domain.     

Simple Estimators for Invertible Index Models

This article considers estimation of the unknown linear index coefficients of a model in which a number of nonparametrically identified reduced form parameters are assumed to be smooth and invertible function of one or more linear indices. The results extend the previous literature by allowing the number of reduced form parameters to exceed the number of indices (i.e., the indices are “overdetermined” by the reduced form parameters. The estimator of the unknown index coefficients (up to scale) is the eigenvector of a matrix (defined in terms of a first-step nonparametric estimator of the reduced form parameters) corresponding to its smallest (in magnitude) eigenvalue. Under suitable conditions, the proposed estimator is shown to be root-n-consistent and asymptotically normal, and under additional restrictions an efficient choice of a “weight matrix” is derived in the overdetermined case.     

Theoretical and practical issues in purchasing power parities illustrated with reference to the 1993 Organization for Economic Co-operation and Development data

This paper deals first with some theoretical issues in purchasing power parities, including the definition of a generalized class of additive indices, which includes the standard Geary–Khamis and Iklé indices as special cases, and the definition of indicators of price and quantity structure which are independent of the particular units in which quantities are originally expressed. The paper then provides a comparative analysis of the results of applying these techniques to the 1993 Organization for Economic Co-operation and Development data and relates the results to underlying features of the price and expenditure data.     

Reliable error estimation for Sobol’ indices

In the field of sensitivity analysis, Sobol’ indices are sensitivity measures widely used to assess the importance of inputs of a model to its output. The estimation of these indices is often performed through Monte Carlo or quasi-Monte Carlo methods. A notable method is the replication procedure that estimates first-order indices at a reduced cost in terms of number of model evaluations. An inherent practical problem of this estimation is how to quantify the number of model evaluations needed to ensure that estimates satisfy a desired error tolerance. This article addresses this challenge by proposing a reliable error bound for first-order and total effect Sobol’ indices. Starting from the integral formula of the indices, the error bound is defined in terms of the discrete Walsh coefficients of the different integrands. We propose a sequential estimation procedure of Sobol’ indices using the error bound as a stopping criterion. The sequential procedure combines Sobol’ sequences with either Saltelli’s strategy to estimate both first-order and total effect indices, or the replication procedure to estimate only first-order indices.     

Estimating the distribution function using k-tuple ranked set samples

The basic assumption underlying the concept of ranked set sampling is that actual measurement of units is expensive, whereas ranking is cheap. This may not be true in reality in certain cases where ranking may be moderately expensive. In such situations, based on total cost considerations, k-tuple ranked set sampling is known to be a viable alternative, where one selects k units (instead of one) from each ranked set. In this article, we consider estimation of the distribution function based on k-tuple ranked set samples when the cost of selecting and ranking units is not ignorable. We investigate estimation both in the balanced and unbalanced data case. Properties of the estimation procedure in the presence of ranking error are also investigated. Results of simulation studies as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

A multivariate non-parametric kernel estimator for global sensitivity analysis

The fact of estimating how a model output is influenced by the variations of inputs has become an important problematic in reliability and sensitivity analysis. This article is interested in estimating sensitivity indices useful to quantify the contribution of inputs to the variance of model output. A multivariate mixed kernel estimator is investigated since, until now, discrete and continuous inputs have been separately considered in kernel estimation of sensitivity indices. To illustrate the differences between the influence of mixed, discrete, and continuous inputs, analytical expressions of Sobol sensitivity indices are expressed in these three cases for the Ishigami test function. Besides, the performance of the mixed kernel estimator is illustrated through simulations in which the Bayesian procedure is applied for bandwidth parameter choice. An application is also realized on a real example. Finally, the use of an appropriate kernel estimator according to the type of inputs is found to be influential on the accuracy of sensitivity indices estimates.     

A two-stage method for the estimation of global sensitivity indices of non-parametric models

This article aims to propose a method to effectively estimate global sensitivity indices under non-parametric models. The new method involves two stages. First, all the non-influential sensitivity indices are filtered out by an adjustive W-statistic test process with low cost, and then the remaining significant sensitivity indices are precisely estimated by an orthogonal array (OA) with large number of levels and low strength. The method avoids complicated prototype building and shows a much lower experimental cost. The performance of this method as well as comparisons with polynomial regression method, Gaussian Process (GP) method, and component selection and smoothing operator (COSSO) method are tested on three numerical models that are widely used in engineering and statistical areas. Finally, a real data example is analyzed.     

Effects of some design factors on the distribution of similarity indices in cluster analysis

This article investigates the effects of number of clusters, cluster size, and correction for chance agreement on the distribution of two similarity indices, namely, Jaccard and Rand indices. Skewness and kurtosis are calculated for the two indices and their corrected forms then compared with those of the normal distribution. Three clustering algorithms are implemented: complete linkage, Ward, and K-means. Data were randomly generated from bivariate normal distributions with specified means and variance covariance matrices. Three-way ANOVA is performed to assess the significance of the design factors using skewness and kurtosis of the indices as responses. Test statistics for testing skewness and kurtosis and observed power are calculated. Simulation results showed that independent of the clustering algorithms or the similarity indices used, the interaction effect cluster size x number of clusters and the main effects of cluster size and number of clusters were found always significant for skewness and kurtosis. The three way interaction of cluster size x correction x number of clusters was significant for skewness of Rand and Jaccard indices using all clustering algorithms, but was not significant using Ward's method for both Rand and Jaccard indices, while significant for Jaccard only using complete linkage and K-means algorithms. The correction for chance agreement was significant for skewness and kurtosis using Rand and Jaccard indices when complete linkage method is used. Hence, such design factors must be taken into consideration when studying distribution of such indices.     

A Study on Reliability for A Two-Item Cold Standby Markov Repairable System with Neglected Failures

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-channel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article.     

Comparisons of normalization methods for relative quantitative in real-time polymerase chain reaction

The real-time polymerase chain reaction (rtPCR) provides sensitive and accurate quantitative results and becomes a widespread technique in analyzing gene expressions. House-keeping genes are required as references to normalize data of target genes, which may be unstable. This normalization process is similar to the normalization in analyzing high-density oligonucleotide arrays. This article evaluates the feasibility of normalizations for high-density oligonucleotide arrays to normalize data collected in rtPCR experiments. Since data features are different, simulations are used to evaluate the performance of these normalizations to rtPCR data based on five indices. Their feasibilities are illustrated by a rtPCR data.     

Measuring Social Tension from Income Class Segregation

We develop an index that effectively measures the level of social tension generated by income class segregation. We adopt the basic concepts of between-group difference (or alienation) and within-group similarity (or identification) from the income [bi]polarization literature; but we allow for asymmetric degrees of between-group antagonism in the alienation function, and construct a more effective identification function using both the relative degree of within-group clustering and the group size. To facilitate statistical inference, we derive the asymptotic distribution of the proposed measure using results from U-statistics. As the new measure is general enough to include existing income polarization indices as well as the Gini index as special cases, the asymptotic result can be readily applied to these popular indices. Evidence from the Panel Study of Income Dynamics data suggests that, while the level of social tension shows an upward trend over the sample period of 1981 to 2005, government’s taxes and transfers have been effective in reducing the level of social tension significantly.     

A Superstructure of Process Capability Indices for Circular Specification Region

Circular specification regions can be seen in processes like hitting a target (in ballistics), drilling a hole (in manufacturing industries) and so on. However, only a few process capability indices are available in the literature to address the problem. Most of these indices, in turn, make some assumptions like equality of variance and independence of the two axes of the circular tolerance region. Since, in most of the cases, these assumptions are not practically viable, in the present article, we have proposed a few of the process capability indices which do not need the above assumptions to be valid. Also, we propose a superstructure which unifies all the proposed indices. Some properties of these indices have been studied including the threshold value and the relationship of the proportion of non-conformance with the member indices of the superstructure. These strengthen the practical utility of the superstructure. Distributional properties like expectations and variances of the member indices of the superstructure are also studied to have a better insight about the indices. A real life example has been discussed to carry out a comparative study of the performance of the existing as well as the newly developed indices.     

Inference for a Linear Functional of Cumulative Hazard Function Via Empirical Likelihood

This article discusses sampling plans, that is, the allocation of sampling units, for computing tolerance limits in a balanced one--way random-effects model. The expected width of the tolerance interval is derived and used as the basis for comparing different sampling plans. A well-known cost function and examples are used to facilitate the discussion.     

基于信息可替代性的评价指标筛选研究

提出一种指标筛选方法,旨在筛选出对评价结果影响显著、反映信息重叠程度低的评价指标。以相对离散系数作为指标的信息含量,依据提出的累计信息贡献率标准删除信息含量明显偏小的指标进行指标的初筛,克服现有研究仅利用相对离散系数进行指标筛选的不足。借鉴聚类分析的思想,取初筛后保留下来的一个指标与其余各指标构成Person相关系数平方的均值,反映该指标的信息可被其余全部指标替代的程度。通过信息可替代性标准剔除信息可替代性较大的指标,保证最终被保留的指标间反映的信息重叠程度低,克服现有研究仅通过两个指标间的相关性筛选指标难以有效降低评价指标集信息重叠的不足。此外,针对剔除信息重叠的指标与剔除对评价结果影响不显著的指标何者优先为宜的问题,提出了显著再相关的指标筛选标准。最后,通过一个实例说明指标筛选方法的可行性。     

A general function of axiomatic index numbers

Summary This paper describes an axiomatic approach to index number theory. A general bilaeral formula which generates a set of indices of prices and quantities is proposed. This formula is written as a geometric mean, weighted with logarithmic means of relative values, and includes all of those indices and related cofactors which satisfy the following axiomatic properties: strong identity, commensurability, linear homogeneity and associativity (or monotonicity). Moreover, two subsets of indices are identified: the first subset includes those which can be expressed either as geometric means or as expenditure ratios; the corresponding bounds are given by the Laspeyres' and Paasche's indices. The second subset also satisfies the desired properties of base and factor reversibility; in this case the bounds are given by Sato Vartia's and Fisher's indices. In addition it is shown that the intersection between the two subsets identifies a new bilateral ideal index which satisfies the axiomatic and reversibility properties (base and factor) and may also be written as an expenditure ratio. All other formulas, which do not belong to the family of the proposed indices, violate some of the axiomatic properties. For multilateral comparisons, a mixed system of direct and indirect indices which satisfies the transitivity condition is proposed.     

Toward the establishment of a family of distributions that may fit any dataset

This article presents a review and comparison of the most important expanded families of distributions. We set the essential requirements by which an expanding family can fit any dataset successfully. A new method is proposed to construct families, which fulfill these essential requirements. Consequently, two families are suggested, which are more tractable than many other known families and possess very wide range of the indices of skewness and kurtosis. The article is motivated by two applications to real dataset.     

Response

This article focuses on the roles consulting units can play in statistics departments, universities, and the statistics profession. An opinion on what constitutes legitimate academic activity is given. The roles endorsed are consistent with the view that statistics is both a science and an integral part of scientific methodology and hence that statistics departments need to be concerned with both the development of new methods and the proper application of those methods in the pure and applied sciences. Consulting units serve as an interface between the developers and users of statistical methodology. Communication between developers and users can be carried on effectively through consultants who are actively involved in the education of users and in the education of statistics students, either formally or informally. Such communication is also enhanced when consultants are publishing in both statistics and nonstatistics journals. Routine data analysis is not endorsed as a justifiable academic activity for consulting faculty. Supervision of graduate students on such analyses, however, can be justified. The success a consulting unit can have in playing its roles depends on appropriate funding, staffing, and administration of the unit.