Distribution‐free Approximate Methods for Constructing Confidence Intervals for Quantiles

Abstract

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.

Non-Parametric Lower Confidence Bounds for the Fifth Percentile - En 14358 in Comparison to a Fully Non-Parametric Approach

“…"lower bound" is a parameter for the truncated normal distribution that ensures that no values below the lower bound are included in the distribution. For simulation, this is implemented using accept-reject sampling (see help on rtruncnorm in [6]). Additionally, the fifth percentile of the distribution is given.…”

“…We say that these values have rank 1, 2 and , respectively. The probability with which the measurement value with rank underestimates the fifth percentile can be calculated using the binomial distribution [6]. This probability is independent of the actual values -it only depends on the ranks .…”

The use of fractional order statistics for estimating nonparametric confidence intervals for quantiles of the fatigue damage computed in service random loadings

“…A point worth to mention is that the linear interpolation in Eq. ( 11) is only valid for quantiles P in the region between 1/ n+1 and n/ n+1 , which are the lowest and highest quantiles corresponding to a sample size n. Owing to this condition, the lower and upper confidence limits cannot be computed if n'P l =0 and n'P u =n (see [19]), which is equivalent to the conditions P l <1/ n+1 and P u >n/ n+1 . For example, assume we want to construct a 99% CI for a sample of size n=50, and for two extreme quantiles P=0.1 and P=0.9.…”

Quantile Estimation Via a Combination of Conditional Monte Carlo and Randomized Quasi-Monte Carlo

“…The CI is asymptotically valid in the sense that lim n→∞ P(ξ ∈ J MC,n ) = β . There are also other approaches for constructing a CI for ξ via MC; see Nagaraja and Nagaraja (2019).…”