Exploiting the quantile optimality ratio in finding confidence intervals for quantiles

Prendergast, Luke A.; Staudte, Robert G.

doi:10.1002/sta4.105

Cited by 20 publications

(17 citation statements)

References 25 publications

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“…Using standard results for the asymptotic normality and covariance structure of sample quantiles, nominal 100(1 − α)% confidence intervals for I of the form I (J) ± z 1−α/2 { Var( I (J) )} 1/2 are obtained. The required quantile function estimates are the continuous Type 8 estimates recommended by Hyndman & Fan (1996) and the quantile density estimates are those developed by Prendergast & Staudte (2016a). These large sample intervals are shown to have good coverage probabilities for nearly all of the distributions listed below in Table 1 and n =100, 200, 500, 1000 and 5000.…”

Section: Inference For Qri Component Estimatesmentioning

confidence: 98%

Decomposing the Quantile Ratio Index with Applications to Australian Income and Wealth Data

Prendergast

Staudte

2019

Eur. J. Pure Appl. Math.

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The quantile ratio index introduced by Prendergast and Staudte 2017 is a simple and effective measure of relative inequality for income data that is resistant to outliers. It measures the average relative distance of a randomly chosen income from its symmetric quantile. Another useful property of this index is investigated here: given a partition of the income distribution into a union of sets of symmetric quantiles, one can find the conditional inequality for each set as measured by the quantile ratio index and readily combine them in a weighted average to obtain the index for the entire population. When applied to data for various years, one can track how these contributions to inequality vary over time, as illustrated here for Australian income and wealth data.

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Section: Inference For Qri Component Estimatesmentioning

confidence: 98%

Decomposing the Quantile Ratio Index with Applications to Australian Income and Wealth Data

Prendergast

Staudte

2019

Eur. J. Pure Appl. Math.

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“…The ratio q(u)/q (u) is similar in shape to the density quantile f Q(u) = 1/q(u), and hence remarkably stable for F in broad classes such as all symmetric unimodal distributions, or all F with positive unimodal density on [0, +∞). Prendergast & Staudte (2016) show that by employing the optimal ratio for the Cauchy, one obtains good estimatorsq(u) of q(u) for all F in the first class, while the optimal ratio for the lognormal yields good estimators for all F in the second. In Figure 3 Figure 3: Graphs of f * (u) (solid lines) and their estimates defined in Section 2.2 (dashed lines).…”

Section: An Empirical Pdq For Smooth Distributionsmentioning

confidence: 99%

“…With regard to inference, we defined empirical pdQ s in both the discrete and continuous case. The latter are based on quantile density estimators of Prendergast & Staudte (2016), and generally require moderately large sample sizes of 500 or more. Given such a sample, we showed that one could fit a parametric shape model to it by minimizing over the shape parameter the Hellinger distance of the proposed model pdQ from the empirical pdQ .…”

Section: Summary and Further Researchmentioning

confidence: 99%

The shapes of things to come: probability density quantiles

Staudte

2017

Statistics

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For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQ s) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback-Leibler divergences. Convergent empirical estimates of these pdQ s are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQ s from the symmetric class. Further, a precise classification of shapes by tail behavior can be defined simply in terms of pdQ boundary derivatives.

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“…The problems surrounding the inference on ratios and the quantile variance estimation are well documented in the literature (Dufour, 1997;Von Luxburg and Franz, 2009;Hall et al, 1989;Hall and Sheather, 1988;Prendergast and Staudte, 2016) and motivate the proposed solutions in this chapter that eschew estimation of densities and/or are robust to ratio-induced identification issues.…”

Section: Methodsmentioning

confidence: 81%

“…= κ(. of Prendergast and Staudte (2016). For the derivation of the QOR for the GB2 distribution see the appendix below.…”

Section: Methodsmentioning

confidence: 99%

Three Essays on Statistical Inference on Inequality Measures

Zalghout¹

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Providing reliable inference on inequality measures is an enduring challenge, mainly due to the complications arising from nonlinearities in their definitions and from the complex nature of the underlying distributions which are typically characterized by extremely heavy tails. The thesis is concerned with proposing non-standard asymptotic and simulation-based inference procedures for moment-based inequality measures (general entropy family of inequality indices) and quantile-based measures (quantile ratio index). Inference on both types of measures is prone to heavy-tailed distributions complications and to the ratio-induced identification issues. In addition to that, moment-based measures are subject to the so-called Bahadur-Savage impossibility problem while quantile-based measures are not. On the other hand, the main difficulty with inference on quantile-based measures is the dependence of the quantile variance on the underlying density function which involves kernel estimation and bandwidth selection. The first chapter of my thesis introduces a Fieller-type method for the Theil Index and assess its finite-sample properties by a Monte Carlo simulation study. The fact that almost all inequality indices can be written as a ratio of functions of moments First and foremost, I would like to thank my supervisor Lynda Khalaf for her patience, trust, motivation, immense knowledge, and invaluable support; a better supervisor and mentor for my doctoral studies would be hard to find. I have benefited tremendously from Lynda's wisdom and guidance and I consider it a privilege to have worked under her supervision. I would also like to thank Jean-Marie Dufour (McGill University) and Emmanuel Flachaire (Aix-Marseille University) for their guidance with the first two chapters. Working with them has helped me sharpen my research skills and enrich my knowledge of econometrics and income inequality.

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Exploiting the quantile optimality ratio in finding confidence intervals for quantiles

Cited by 20 publications

References 25 publications

Decomposing the Quantile Ratio Index with Applications to Australian Income and Wealth Data

Decomposing the Quantile Ratio Index with Applications to Australian Income and Wealth Data

The shapes of things to come: probability density quantiles

Three Essays on Statistical Inference on Inequality Measures

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