On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

Some theoretical results concerning non parametric estimation by using a judgment poststratification sample

ABSTRACT

In this paper, some of the properties of non parametric estimation of the expectation of g(X) (any function of X), by using a judgment poststratification sample (JPS), have been discussed. A class of estimators (including the standard JPS estimator and a JPS estimator proposed by Frey and Feeman (2012 Frey, J., Feeman, T.G. (2012). An improved mean estimator for judgment post-stratification. Comput. Stat. Data Anal. 56(2):418–426.[Crossref], [Web of Science ®] , [Google Scholar], Comput. Stat. Data An.) is considered. The paper provides mean and variance of the members of this class, and examines their consistency and asymptotic distribution. Specifically, the results are for the estimation of population mean, population variance, and cumulative distribution function. We show that any estimators of the class may be less efficient than simple random sampling (SRS) estimator for small sample sizes. We prove that the relative efficiency of some estimators in the class with respect to balanced ranked set sampling (BRSS) estimator tends to 1 as the sample size goes to infinity. Furthermore, the standard JPS mean estimator, and Frey–Feeman JPS mean estimator are specifically studied and we show that two estimators have the same asymptotic distribution. For the standard JPS mean estimator, in perfect ranking situations, optimum values of H (the ranking class size), for different sample sizes, are determined non parametrically for populations that are not heavily skewed or thick tailed.     

Disability in the UK: measuring equality

In this article we identify the key survey data for examining the issue of equality in the lives of disabled people in the UK. Such data is essential for assessing change in quality of life over time and for the evaluation of the impact of policy initiatives. For each data source we consider definitions, data collection, issue coverage, sample size and data access. It is evident that there is only limited survey data on the lives of disabled people in the UK. A number of national surveys include questions on disability, but many offer only limited coverage. There is a tendency to focus on the medical aspects of disability and a failure to clearly distinguish disability from other health issues. Moreover, disability definitions vary and sample sizes are often too small to compare differences by age, gender, ethnicity or locality. For effective policy development there is a need to identify and measure what leads to change in disabled peoples' lives.     

A generalization of the Grizzle model to the estimation of treatment effects in crossover trials with non-compliance

Compliance with one specified dosing strategy of assigned treatments is a common problem in randomized drug clinical trials. Recently, there has been much interest in methods used for analysing treatment effects in randomized clinical trials that are subject to non-compliance. In this paper, we estimate and compare treatment effects based on the Grizzle model (GM) (ignorable non-compliance) as the custom model and the generalized Grizzle model (GGM) (non-ignorable non-compliance) as the new model. A real data set based on the treatment of knee osteoarthritis is used to compare these models. The results based on the likelihood ratio statistics and simulation study show the advantage of the proposed model (GGM) over the custom model (GGM).     

Bootstrap power of the generalized correlation coefficient

We present a bootstrap Monte Carlo algorithm for computing the power function of the generalized correlation coefficient. The proposed method makes no assumptions about the form of the underlying probability distribution and may be used with observed data to approximate the power function and pilot data for sample size determination. In particular, the bootstrap power functions of the Pearson product moment correlation and the Spearman rank correlation are examined. Monte Carlo experiments indicate that the proposed algorithm is reliable and compares well with the asymptotic values. An example which demonstrates how this method can be used for sample size determination and power calculations is provided.     

Class Position,Class Ideology and Class Voting: Mobilization of Support for the New Democratic Party in the Canadian Election of 1984*

La recherche canadienne sur le vote des classes a surtout fait ressortir l'importance des forces structurelles qui contribuent à faire naître l'appui au Nouveau Parti démocratique (NPD) en insistant moins sur l'idéologie des classes. Le présent article contient un modèle LISREL faisant état des relations entre les positions des classes, leur idéologic et le vote en faveur du NPD selon la base de données provenant de l'étude des élections nationales du Canada de 1984. Les observations soulignent l'importance que revêtent, en tant que sources des suffrages des classes, trois dimensions de l'idéologie des classes qui repré-sentent en partie le résultat de la position de ces dernières. L'idéologie égalitaire et l'appui au maintien des syndicate forts ont des effete directs et prononcés sur le vote en faveur du NPD alors que l'identifi-cation à une classe exerce un effet indirect par le biais de deux autres variables. Même si la position objective des classes agit indépendam-ment sur les suffrages accordés au NPD, ses répercussions semblent limitées comparativement à celles que produisent les trois variables liées à l'étude. En modélisant les effete de la position et de l'idéologie des classes, les auteurs ont employé trois variables non encore utili-sées dans les analyses précédentes des suffrages des classes, soit l'appui des père et mère au NPD, la présence d'un syndiqué (autre que le répondant) dans le ménage et l'appartenance à une «classe nouvelle». Chacune de ces variables a un effet majeur sur le soutien donné au NPD. Quant aux répercussions attribuables à la région, elles semblent assez limitées une fois que l'on a tenu compte du vote des père et mère en faveur du NPD et de l'appui accordéà ce dernier à l'occasion des élections de 1980. Canadian research on class voting has primarily emphasized structural forces that produce support for the NDP, with less attention to class ideology. This paper presents a LISREL model of relationships among class position, class ideology, and voting for the NDP, based on data from the Canadian National Election Study of 1984. The findings point to the importance of three dimensions of class ideology, produced in part by class position, as sources of class voting. Egalitarian ideology and support for union strength have strong direct effects on NDP voting, and class identification exerts an indirect effect through the other two variables. Although objective class position has an independent effect on NDP voting, its impact appears to be modest in relation to that of the three attitudinal variables. In modelling the effects of class position and ideology, three variables not used in previous Canadian analyses of class voting have been employed: parental support for the NDP, presence of a union member (other than the respondent) in the household, and membership in a “New Class.” Each has a solid effect on NDP support. The effects of region appear as quite modest, after controlling for parental support for the NDP and for NDP support in the 1980 election.     

What Level of Statistical Model Should We Use in Small Area Estimation?

If unit‐level data are available, small area estimation (SAE) is usually based on models formulated at the unit level, but they are ultimately used to produce estimates at the area level and thus involve area‐level inferences. This paper investigates the circumstances under which using an area‐level model may be more effective. Linear mixed models (LMMs) fitted using different levels of data are applied in SAE to calculate synthetic estimators and empirical best linear unbiased predictors (EBLUPs). The performance of area‐level models is compared with unit‐level models when both individual and aggregate data are available. A key factor is whether there are substantial contextual effects. Ignoring these effects in unit‐level working models can cause biased estimates of regression parameters. The contextual effects can be automatically accounted for in the area‐level models. Using synthetic and EBLUP techniques, small area estimates based on different levels of LMMs are investigated in this paper by means of a simulation study.