Statistics of Ordinal Variation

This paper proposes an axiomatic foundation for new measures of polarization that can be applied to ordinal distributions such as self-assessed health (SAH) data. This is an improvement over the existing measures of polarization that can be used only for cardinal variables. The new measures of polarization avoid one difficulty that the related measures for evaluating health inequalities face. Indeed, inequality measures are mean based, and since only cardinal variables have a mean, SAH has to be cardinalized to compute a mean, which can then be used to calculate an inequality measure. In contrast, the new polarization measures are median based and hence do not require to impose cardinal scaling on the categories. After deriving the properties of these new polarization measures, we provide an empirical illustration using data from the British Household Panel Survey that demonstrates that SAH polarization is also a relevant question on empirical grounds, and that the polarization measures are adequate to evaluate polarization phenomena whereas inequality measures are not adequate in these cases.

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“…We first follow suggestions of Blair and Lacy (2000) to consider the observed distributions N ¼ ðN 1 ; . .…”

Section: The Families Of Polarization Measuresmentioning

confidence: 99%

“…We use a formula given by Agresti (1990) which is presented in Blair and Lacy (2000) for statistical inference. Using the multivariate delta method, we find that the sample estimator for the variance of P 2 ; which will be used to compute confidence intervals for P 2 ; is: …”

Section: Statistical Inferencementioning

confidence: 99%

Measuring health polarization with self‐assessed health data

Apouey

2007

Health Economics

106

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“…As part of a sensitivity analysis, Panel C takes into account the ordinal nature of the job satisfaction variable. That is, because satisfaction is not a cardinal variable, the variance may not be the best measure of dispersion, because it imposes a supraordinal assumption about the nature of the continuum underlying the different categories (Blair and Lacy (2000)). Therefore, in Panel C I use an index for ordinal variation based on Berry and Mielke (1992) to calculate s. It appears that the results are not much different from those in Panel A.…”

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confidence: 99%

Repeated job quits: stepping stones or learning about quality?

Gielen

2013

IZA J Labor Stud

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Increasing labor mobility is high on the political agenda because of its supposedly positive effects on labor market functioning. However, little attention has been paid to information imperfections, and to what extent they limit potential efficiency gains of labor mobility. When the quality of a new job offer is known ex ante, job quits serve as a stepping stone to better jobs. Yet, if job quality is only observed ex post, job quits may lead to worse matches. This paper argues that actual job quit behavior is characterized by a mixture of both, and investigates the relative empirical content of both extremes in quit decisions. A variance decomposition shows that for nearly 70% of job quits job quality was observed ex-ante; the remaining 30% was learned ex post. Hence, stimulating job mobility mostly improves labor market outcomes, though governments may aim to further reduce information imperfections in order to maximize the efficacy of labor policies. JEL codes: J28, J62.

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“…Numerous modifications of Leik's measure of dispersion have been published. In [24][25][26][27][28][29], the authors implicitly used φ(u) = 1/4 − (u − 1/2) 2 or equivalently φ(u) = u(1 − u). Most of the discussion was conducted in the journal "Perceptual and Motor Skills".…”

Section: Theory Of Dispersionmentioning

confidence: 99%

Cumulative Paired φ-Entropy

Klein

Mangold

Doll

2016

Entropy

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Abstract:A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general "entropy generating function" φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489-495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new "cumulative paired φ-entropy" (CPE φ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of CPE φ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider CPE φ for continuous cdfs and show that CPE φ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that CPE φ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. CPE φ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for CPE φ if the cdf is available in a closed form.

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Statistics of Ordinal Variation

Cited by 82 publications

References 9 publications

Measuring health polarization with self‐assessed health data

Measuring health polarization with self‐assessed health data

Repeated job quits: stepping stones or learning about quality?

Cumulative Paired φ-Entropy

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