Fractional order statistic approximation for nonparametric conditional quantile inference

Goldman, Matt; Kaplan, David

doi:10.1016/j.jeconom.2016.09.015

Cited by 9 publications

(44 citation statements)

References 42 publications

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“…The accuracy of each new unconditional CI is precisely characterized in terms of coverage error, defined as the difference between the true and nominal coverage probabilities. The foundation of our methods' high‐order accuracy is the use of the probability integral transform and a Dirichlet (instead of Gaussian) approximation of the distribution of a linear combination of ‘fractional'‐order statistics (linearly interpolated between observed order statistics), formally studied in Goldman and Kaplan (, hereafter GK). For our confidence set for a vector of quantiles, the Dirichlet approximation is the only source of error, so coverage error is

O (n^{- 1})

.…”

Section: Introductionmentioning

confidence: 99%

“…The acronyms used include those for [conditional] interquantile range ([C]IQR), [conditional] quantile difference ([C]QD), [conditional] quantile treatment effect ([C]QTE), confidence interval (CI), confidence set (CS), coverage probability (CP), coverage probability error (CPE), cumulative distribution function (CDF), mean squared error (MSE), mean value theorem (MVT), and probability density function (PDF), and GK denotes Goldman and Kaplan (). Notationally,

Beta (a, b)

is a beta distribution with parameters a and b , or such a random variable if clear from context, and

Dir (k_{1}, k_{2}, \dots)

is a Dirichlet distribution (or random variable);

Φ (\cdot)

and

φ (\cdot)

are the standard normal CDF and PDF, respectively; ≐ should be read as ‘is equal to, up to smaller‐order terms’, and ≈ as ‘has exact (asymptotic) rate/order of’; random and non‐random (column) vectors are typeset as

Z = {(Z_{1}, Z_{2}, \dots)}^{'}

and

z = {(z_{1}, z_{2}, \dots)}^{'}

, respectively, with random and non‐random matrices

\underset{̲}{Z}

and

\underset{̲}{z}

, and scalar random variables and values Z and z .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics

Goldman

Kaplan

2018

The Econometrics Journal

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We provide novel, high-order accurate methods for non-parametric inference on quantile differences between two populations in both unconditional and conditional settings. These quantile differences correspond to (conditional) quantile treatment effects under (conditional) independence of a binary treatment and potential outcomes. Our methods use the probability integral transform and a Dirichlet (rather than Gaussian) reference distribution to pick appropriate L-statistics as confidence interval endpoints, achieving highorder accuracy. Using a similar approach, we also propose confidence intervals/sets for vectors of quantiles, interquantile ranges and differences of linear combinations of quantiles. In the conditional setting, when smoothing over continuous covariates, optimal bandwidth and coverage probability rates are derived for all methods. Simulations show that the new confidence intervals have a favourable combination of robust accuracy and short length compared with existing approaches. Detailed steps for confidence interval construction are provided in online Appendix E as supporting information, and code for all methods, simulations and empirical examples is provided.

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O (n^{- 1})

.…”

Section: Introductionmentioning

confidence: 99%

Beta (a, b)

is a beta distribution with parameters a and b , or such a random variable if clear from context, and

Dir (k_{1}, k_{2}, \dots)

is a Dirichlet distribution (or random variable);

Φ (\cdot)

and

φ (\cdot)

Z = {(Z_{1}, Z_{2}, \dots)}^{'}

and

z = {(z_{1}, z_{2}, \dots)}^{'}

, respectively, with random and non‐random matrices

\underset{̲}{Z}

and

\underset{̲}{z}

, and scalar random variables and values Z and z .…”

Section: Introductionmentioning

confidence: 99%

Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics

Goldman

Kaplan

2018

The Econometrics Journal

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show abstract

“…With a simulation study, they showed that Hutson's method performed at least as well as Beran and Hall's, in terms of confidence interval coverage and short interval widths. Their method also performed similarly to Beran and Hall's and better than the bootstrap method (see Goldman & Kaplan, , p. 337–338). Chapter 5 in Meeker et al () also contains some discussion of interpolation methods.…”

Section: Quantile Confidence Interval Construction Methodsmentioning

confidence: 74%

“…Goldman and Kaplan () applied a calibration method similar to Ho and Lee's () to Hutson's () procedure to improve the asymptotic performance of the confidence intervals. As with Hutson (), the method cannot be applied for extreme order statistics in some cases.…”

Section: Quantile Confidence Interval Construction Methodsmentioning

confidence: 99%

Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles

Nagaraja

2019

Int Statistical Rev

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Summary Quantile estimation is important for a wide range of applications. While point estimates based on one or two order statistics are common, constructing confidence intervals around them, however, is a more difficult problem. This paper has two goals. First, it surveys the numerous distribution‐free methods for constructing approximate confidence intervals for quantiles. These techniques can be divided roughly into four categories: using a pivotal quantity, resampling, interpolation, and empirical likelihood methods. Second, a method based on the pivotal quantity that has received limited attention in the past is extended. Comprehensive simulation studies are used to compare performance across methods. The proposed method is simple and performs similarly to linear interpolation methods and a smoothed empirical likelihood method. While the proposed method has slightly wider interval widths, it can be calculated for more extreme quantiles even when there are few observations.

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“…This CDF can be computed immediately by any modern statistical software. The only difficulty is if a specific α is desired for a specific τ , in which case one cannot find an exact, non-randomized test (but for solutions using interpolation, see Beran and Hall, 1993;Goldman and Kaplan, 2017a).…”

Section: Basic Method Fwer and Computationmentioning

confidence: 99%

Comparing distributions by multiple testing across quantiles or CDF values

Goldman

Kaplan

2018

Journal of Econometrics

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When comparing two distributions, it is often helpful to learn at which quantiles or values there is a statistically significant difference. This provides more information than the binary "reject" or "do not reject" decision of a global goodness-of-fit test. Framing our question as multiple testing across the continuum of quantiles τ ∈ (0, 1) or values r ∈ R, we show that the Kolmogorov-Smirnov test (interpreted as a multiple testing procedure) achieves strong control of the familywise error rate. However, its well-known flaw of low sensitivity in the tails remains. We provide an alternative method that retains such strong control of familywise error rate while also having even sensitivity, i.e., equal pointwise type I error rates at each of n → ∞ order statistics across the distribution. Our one-sample method computes instantly, using our new formula that also instantly computes goodness-of-fit p-values and uniform confidence bands. To improve power, we also propose stepdown and pre-test procedures that maintain control of the asymptotic familywise error rate. One-sample and two-sample cases are considered, as well as extensions to regression discontinuity designs and conditional distributions. Simulations, empirical examples, and code are provided.JEL classification: C12, C14, C21

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Fractional order statistic approximation for nonparametric conditional quantile inference

Cited by 9 publications

References 42 publications

Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics

Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics

Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles

Comparing distributions by multiple testing across quantiles or CDF values

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