Journal
ELECTRONIC JOURNAL OF STATISTICS
Volume 14, Issue 1, Pages 661-701Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/20-EJS1678
Keywords
Extreme conditional quantiles; confidence interval; dimension reduction
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Funding
- French National Research Agency [ANR-19-CE40-0013-01/]
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The first part of the paper is dedicated to the construction of a gamma - nonparametric confidence interval for a conditional quantile with a level depending on the sample size. When this level tends to 0 or 1 as the sample size increases, the conditional quantile is said to be extreme and is located in the tail of the conditional distribution. The proposed confidence interval is constructed by approximating the distribution of the order statistics selected with a nearest neighbor approach by a Beta distribution. We show that its coverage probability converges to the preselected probability gamma and its accuracy is illustrated on a simulation study. When the dimension of the covariate increases, the coverage probability of the confidence interval can be very different from gamma. This is a well known consequence of the data sparsity especially in the tail of the distribution. In a second part, a dimension reduction procedure is proposed in order to select more appropriate nearest neighbors in the right tail of the distribution and in turn to obtain a better coverage probability for extreme conditional quantiles. This procedure is based on the Tail Conditional Independence assumption introduced in (Gardes, Extremes, pp. 57-95, 18(3), 2018).
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