Note on unbiased estimation of the squared multiple correlation coefficient

Zoals bekend is het kwadraat van de correlatie-coëfficiënt bij regressie-analyse een onzuivere schatting van zijn tegenhanger in de populatie. Deze onzuiverheid wordt afgeleid tot op de orde 1/T, waarbij T het aantal waarnemingen is. Door een schatting van deze onzuiverheid van het kwadraat van de gewone correlatie-coëfficiënt af te trekken wordt een nieuwe schatting van de correlatie in de populatie verkregen, die zuiver is tot op de orde 1/T, hetgeen van de op de gebruikelgke wijzen gecorrigeerde correlatie-coëfficiënten niet gezegd kan worden.     

On interpreting the product moment correlation coefficient*

Abstract In interpreting the product moment correlation coefficient we can take advantage of the fact that functions of it exist which admit interpretation as probabilities of error in certain situations.     

Additional cautionary notes about the Pearson's correlation coefficient

In this paper we draw attention to two usually-neglected problems with the use of the Pearson's correlation coefficient for ordinal scores. We refer to the questions of substantial differences between the marginal distributions of the two variables to be correlated, and the number of ties on each one of the distributions or on both. Finally, we propose a correction factor for the Pearson's correlation coefficient when the number of ties is large. Some empirical examples are given in order to demonstrate these points.     

Analytic posteriors for Pearson's correlation coefficient

Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open‐source software package JASP.     

Comparing approximations of confidence intervals for the product-moment correlation coefficient

Five approximations in computing confidence intervals for the product-moment correlation coefficient are compared. The α-levels of the approximations are compared with three fixed nominal levels; for varying sample sizes and sample correlation values. It is recommended to use a method of R UBFN (or K RAEMER ) [section 3d (3e)] for accurate approximations and a method of S AMIUDDIN [section 3b] for somewhat less accurate but quick results.     

On testing the correlation coefficient of a bivariate normal distribution

Summary Here we propose two tests for testing in the bivariate normal population assuming that the ratio of the variances in it is known. The first test (U.M.P.U.) is derived by using the Neyman-Pearson lemma, whereas the second test is obtained through testing the scale parameter of the Cauchy distribution. The powers of the first and second tests are compared with a well-known test, based on the sample correlation coefficient for small and large samples respectively.     

Distribution of correlation coefficient under mixture normal model

The present paper obtains the nonnull distribution of the product moment correlation coefficient r when sample is drawn from a mixture of two bivariate Gaussian distributions. The moments of 1−r ² have been used to derive the nonnull density of r. Received September 2000     

Asymptotic properties of a correlation coefficient type statistic connected with the general linear model

It is proved that under certain conditions the conditional distribution of the sample multiple correlation coefficient given the ‘other’ observations in a multivariate normal distribution is asymptotically normal. Moreover a second-order approximation for the first moment is given.     

Robust feature screening for varying coefficient models via quantile partial correlation

This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

On the impact of the tests for serial correlation upon the test of significance for the regression coefficient

Monte Carlo methods are used to investigate the relationship between the power of different pretests for autocorrelation, and the Type I error and power of the significance test for a resulting two-stage estimate of the slope parameter in a simple regression. Our results suggest it may be preferable to always transform without pretesting. Moreover we find little room for improvement in the Type I errors and power of two-stage estimators using existing pretests for autocorrelation, compared with the results obtained given perfect knowledge about when to transform (i.e., given a perfect pretest). Rather, researchers should seek better estimators of the transformation parameter itself.     

Imputation methods of missing data for estimating the population mean using simple random sampling with known correlation coefficient

This paper considers three ratio estimators of the population mean using known correlation coefficient between the study and auxiliary variables in simple random sample when some sample observations are missing. The suggested estimators are compared with the estimators of Singh and Horn (Metrika 51:267–276, 2000), Singh and Deo (Stat Pap 44:555–579, 2003) and Kadilar and Cingi (Commun Stat Theory Methods 37:2226–2236, 2008). They are compared with other imputation estimators based on the mean or a ratio. It is found that the suggested estimators are approximately unbiased for the population mean. Also, it turns out that the suggested estimators perform well when compared with the other estimators considered in this study.     

Inference on stochastic time-varying coefficient models

Recently, there has been considerable work on stochastic time-varying coefficient models as vehicles for modelling structural change in the macroeconomy with a focus on the estimation of the unobserved paths of random coefficient processes. The dominant estimation methods, in this context, are based on various filters, such as the Kalman filter, that are applicable when the models are cast in state space representations. This paper introduces a new class of autoregressive bounded processes that decompose a time series into a persistent random attractor, a time varying autoregressive component, and martingale difference errors. The paper examines, rigorously, alternative kernel based, nonparametric estimation approaches for such models and derives their basic properties. These estimators have long been studied in the context of deterministic structural change, but their use in the presence of stochastic time variation is novel. The proposed inference methods have desirable properties such as consistency and asymptotic normality and allow a tractable studentization. In extensive Monte Carlo and empirical studies, we find that the methods exhibit very good small sample properties and can shed light on important empirical issues such as the evolution of inflation persistence and the purchasing power parity (PPP) hypothesis.     

时空动态半参数变系数随机效应面板模型的估计

研究目标：克服半参数变系数面板模型中扰动项和因变量存在时空动态性问题。研究方法：提出一类更加一般化的时空动态半参数变系数随机效应面板模型,并构建截面似然估计量。研究发现：估计量具有良好的小样本性质,估计误差随着样本总量的提高而减小,在Case空间矩阵下,空间滞后和时空滞后系数的估计精度随空间复杂度的增大而降低,用该方法分析我国外商直接投资、知识产权保护与经济增长关系,进一步证实了模型的适用性。研究创新：证明了估计量满足一致性和渐近正态性,数值模拟考察了估计量的小样本性质。研究价值：拓展了现有半参数变系数空间面板模型的形式,增强了模型的适用性和解释力,有益于经济问题实证研究的开展。     

Images of entrepreneurship: towards a new categorization of entrepreneurship

Based on a systematic literature review of contemporary entrepreneurship process research, the article provides a supplementary way of categorizing research on the entrepreneurial process which takes philosophy of science into account to a higher extent than done in previous categorizations. The motivation for doing so is to minimize problems of communication and comparison of concepts, results and theoretical contributions often experienced in entrepreneurship research. Applying Burrell and Morgan’s (1979) paradigmatic framework to analyse the selected literature, it is suggested that the essence of past and present entrepreneurship process theory can be captured in five images of entrepreneurship: 1) The Image of Machines; 2) The Image of Evolution; 3) The Image of Contingencies; 4) The Image of Mind; and 5) The Image of Social Becoming. The five images are considered to be concise and easily recalled tools, which will be valuable in the continuous development and organization of future ideas in entrepreneurship research.     

An analysis of the bounds for the Gini coefficient

The bounds on the Gini coefficient obtained by Gastwirth for the case of grouped data are considered. While the population bounds will always include the value of the population Gini coefficient and the estimated bounds will always include a suitably chosen estimate of the Gini coefficient, estimated bounds need not include the value of the population Gini coefficient.The distributions of the estimators of the bounds are considered and it is shown that a failure to take account of sampling variation can lead to very misleading results. In fact, increasing the number of income groups used tends to decrease the difference between the bounds, but the relative frequency with which the estimated bounds includes the population Gini coefficient decreases. The relationship between sample size, the nature of income groups and estimator precision is considered.     

Inequalities for the Gini coefficient of composite populations

Unlike other popular measures of income inequality, the Gini coefficient is not decomposable, i.e., the coefficient G($\text{X}$) of a composite population $\text{X}\text{=}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}$ cannot be computed in terms of the sizes, mean incomes and Gini coefficients of the components $\text{X}$_i. In this paper upper and lower bounds (best possible for r=2) for G($\text{X}$) in terms of these data are given. For example, $\text{G(}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}\text{)≧Σα}{}_{\mathrm{i}}\text{G(}\text{X}{}_{\mathrm{i}}\text{)}$, where α_i is the proportion of the pop ulation in $\text{X}$_i. One of the tools used, which may be of interest for other applications, is a lower bound for ∫^∞₀f(x)g(x)dx (converse to Cauchy's inequality) for monotone decreasing functions f and g.