Confidence Intervals For Quantiles Research Articles

Estimation of confidence intervals of quantiles for the Weibull distribution

Estimation of confidence limits and intervals for the two- and three-parameter Weibull distributions are presented based on the methods of moment (MOM), probability weighted moments (PWM), and maximum likelihood (ML). The asymptotic variances of the MOM, PWM, and ML quantile estimators are derived as a function of the sample size, return period, and parameters. Such variances can be used for estimating the confidence limits and confidence intervals of the population quantiles. Except for the two-parameter Weibull model, the formulas obtained do not have simple forms but can be evaluated numerically. Simulation experiments were performed to verify the applicability of the derived confidence intervals of quantiles. The results show that overall, the ML method for estimating the confidence limits performs better than the other two methods in terms of bias and mean square error. This is specially so for γ≥0.5 even for small sample sizes (e.g. N=10). However, the drawback of the ML method for determining the confidence limits is that it requires that the shape parameter be bigger than 2. The Weibull model based on the MOM, ML, and PWM estimation methods was applied to fit the distribution of annual 7-day low flows and 6-h maximum annual rainfall data. The results showed that the differences in the estimated quantiles based on the three methods are not large, generally are less than 10%. However, the differences between the confidence limits and confidence intervals obtained by the three estimation methods may be more significant. For instance, for the 7-day low flows the ratio between the estimated confidence interval to the estimated quantile based on ML is about 17% for T≥2 while it is about 30% for estimation based on MOM and PWM methods. In addition, the analysis of the rainfall data using the three-parameter Weibull showed that while ML parameters can be estimated, the corresponding confidence limits and intervals could not be found because the shape parameter was smaller than 2.

Confidence intervals of quantiles in hydrology computed by an analytical method

Fitting probability distributions to hydrologic data samples is widely used for quantile estimation purposes. The estimated quantile (X^T) is related to a return period (T). The confidence interval associated with each of the estimates has been calculated empirically, up until now, supposing that the quantile estimator is normally distributed. In this study, it is shown that the confidence interval follows a normal distribution only in the central part of the distribution. The real confidence limits are computed analytically, by defining and integrating the probability density function of the confidence interval. The results with an important number of hydrologic samples show that the upper confidence limits are significantly underestimated towards the tail of the distribution, when determined using the normality approximation for the quantile estimator.

Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles

Summary For a sample from a given distribution the difference of two order statistics and the Studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences. This paper gives saddlepoint approximations for densities and saddlepoint approximations of the Lugannani–Rice form for tail probabilities of these statistics. The relative errors of the approximations are n −1 uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log-concave.

Resampled quantile functions for error estimation and a relationship to density estimation

Non-parametric resampling of the data can be useful for variance estimation and constructing confidence intervals for quantiles. Parzen et al. (1994), (Biometrika, 81, 341–350) develop a technique based on resampling estimating equations that can avoid non-parametric functional estimates required to use traditional large sample variance formulas. We show that while their resampling based variance formulas are “automatic”, their performance may also be improved by adjusting the variance of the resampling mechanism. This is shown to directly relate to the usual problem of needing to adjust the span of a non-parametric density estimate used with the standard asymptotic formula to estimate standard errors for quantiles. In addition, we examine the density estimate that is obtained by the resampling method.

Direct estimation of the percentile 'p-value' for the one-sample median test

In this paper we outline and illustrate an easy-to-use inference procedure for directly calculating the approximate bootstrap percentile-type p-value for the one-sample median test, i.e. we calculate the bootstrap p -value without resampling, by using a fractional order statistics based approach. The method parallels earlier work on fractionalorder-statistics-based non-parametric bootstrap percentile-type confidence intervals for quantiles. Monte Carlo simulation studies are performed, which illustrate that the fractional-order-statistics-based approach to the one-sample median test has accurate type I error control for small samples over a wide range of distributions; is easy to calculate; and is preferable to the sign test in terms of type I error control and power. Furthermore, the fractional-order-statistics-based median test is easily generalized to testing that any quantile has some hypothesized value; for example, tests for the upper or lower quartile may be performed using the same framework.

Quantile interval estimation in finite population using a multivariate ratio estimator

A new method to derive confidence intervals for quantiles in a finite populations is presented. This method uses multi-auxiliary information through a multi-variate ratio type estimator of the population distribution function.

Approximate Confidence Intervals for Quantiles of Gamma and Generalized Gamma Distributions

In hydraulic design, one often needs to estimate flood quantiles for use as design values. It is important to assess the estimation error by constructing confidence intervals (CIs) for these quantiles. Fitting probability distributions to hydrologic data is used widely for estimating quantiles of hydrological variables. The two-parameter gamma (G2) is among the distributions commonly used, but the three-parameter generalized gamma (GG3) (also known as Kritsky-Menkel distribution) is an alternative when more shape flexibility is needed to fit the data. We use an approximate method to construct CIs for the quantiles of the G2 and GG3 distributions. This method is shown to be useful for hydrological applications where the data record is short.

Accurate Confidence Limits for Quantiles under Random Censoring

In survival analysis, estimates of median survival times in homogeneous samples are often based on the Kaplan-Meier estimator of the survivor function. Confidence intervals for quantiles, such as median survival, are typically constructed via large sample theory or the bootstrap. The former has suspect accuracy for small sample sizes under moderate censoring and the latter is computationally intensive. In this paper, improvements on so-called test-based intervals and reflected intervals (cf., Slud, Byar, and Green, 1984, Biometrics 40, 587-600) are sought. Using the Edgeworth expansion for the distribution of the studentized Nelson-Aalen estimator derived in Strawderman and Wells (1997, Journal of the American Statistical Association 92), we propose a method for producing more accurate confidence intervals for quantiles with randomly censored data. The intervals are very simple to compute, and numerical results using simulated data show that our new test-based interval outperforms commonly used methods for computing confidence intervals for small sample sizes and/or heavy censoring, especially with regard to maintaining specified coverage.

Estimation of quantiles and confidence intervals for the log-Gumbel distribution

The log-Gumbel distribution is one of the extreme value distributions which has been widely used in flood frequency analysis. This distribution has been examined in this paper regarding quantile estimation and confidence intervals of quantiles. Specific estimation algorithms based on the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood (ML) are presented. The applicability of the estimation procedures and comparison among the methods have been illustrated based on an application example considering the flood data of the St. Mary's River.

Upper and Lower Bound Distributions That Give Simultaneous Confidence Intervals for Quantiles

Abstract We propose a new method for constructing parametric confidence intervals for quantiles of an unknown distribution. These confidence intervals are constructed so that the joint probability that all intervals simultaneously contain their respective percentiles is at least a preset value. Thus we may also use the set of either upper or lower confidence limits to define an upper (or lower) bound distribution function, which we call an upper (or lower) tolerance distribution because this distribution may also be used to construct tolerance intervals for the unknown distribution. We derive tolerance distributions with exact coverage probability for iid samples from distributions in the location-scale family. Tolerance distributions can also be used for best- and worst-case analyses, as we illustrate by considering bounds on the distribution of the time interval between the onset of infectiousness and development of detectable antibody in individuals newly infected with the human immunodeficiency virus ...

Upper and Lower Bound Distributions that Give Simultaneous Confidence Intervals for Quantiles

Upper and Lower Bound Distributions that Give Simultaneous Confidence Intervals for Quantiles

A note on coverage error of bootstrap confidence intervals for quantiles

AbstractAn interesting recent paper by Falk and Kaufmann[11] notes, with an element of surprise, that the percentile bootstrap applied to construct confidence intervals for quantiles produces two-sided intervals with coverage error of size n−½, where n denotes sample size. By way of contrast, the error would be O(n−1) for two-sided intervals in more classical problems, such as intervals for means or variances. In the present note we point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The coverage accuracy of two-sided bootstrap intervals may be improved to o(n−½) by smoothing the bootstrap. We show too that a normal approximation method, not involving the bootstrap but incorporating a density estimator as part of scale estimation, can have coverage error O(n−1+∈), for arbitrarily small ∈ > 0. Smoothed and unsmoothed versions of bootstrap percentile-t are also analysed.

Smoothed Empirical Likelihood Confidence Intervals for Quantiles

Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size $n^{-1/2}$, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order $n^{-1}$, and may be Bartlett-corrected to produce intervals with an error of order only $n^{-2}$. Necessary and sufficient conditions on the smoothing parameter, in order for these sizes of error to be attained, are derived. The effects of smoothing on the positions of endpoints of the intervals are analysed, and shown to be only of second order.

Interpolated Nonparametric Prediction Intervals and Confidence Intervals

SUMMARY In several important statistical problems, prediction intervals and confidence intervals can be constructed with coverage levels which are known precisely but cannot be rendered equal to predetermined levels such as 0.95. One solution to this difficulty is to interpolate between such intervals. We show that simple linear interpolation reduces the order of coverage error, but that higher orders of interpolation produce no further improvement. The error is reduced by a factor n −1 for prediction intervals and n −1/2 for confidence intervals, where n denotes sample size. In the case of confidence intervals for quantiles, linear interpolation provides particularly accurate intervals which err on the side of conservatism.

Confidence Interval for Design Floods with Estimated Skew Coefficient

In 1983, Stedinger published a method, based on the noncentral t‐ distribution, for constructing approximate confidence intervals for quantiles of a Pearson type‐3 (P3) distribution when the coefficient of skewness is known. That method is extended to the case when the coefficient of skewness is estimated by either (1) The at‐site sample skew; (2) a generalized regional‐average skew; or (3) a weighted average skew. The confidence intervals perform satisfactorily in Monte Carlo simulations. Confidence intervals generated using a three‐parameter asymptotic method and the Water Resources Council method, which ignores uncertainty in the weighted skew, perform poorly. The actual confidence of confidence intervals for a design flood can be much lower than the target if uncertainty in the skewness coefficient is ignored. It is important to distinguish between the sampling variability in the sample skew coefficient at the site and the likely variation of the true skews in the region when estimating the precision of a generalized skewness coefficient for the region. Five approximations of frequency factors for the P3 distribution are also evaluated.

Subsampling quantile estimator standard errors with applications

Several smoothed quantile estimators recently have been proposed as alternatives to the conventional sample quantile, and previous studies have shown that the Harrell and Davis (1982), Kaigh and Lachenbruch (1982), and Kaigh and Cheng (1991) subsampling estimators usually improve efficiency of population quantile point estimation. Bootstrap and jackknife standard error formulas are developed here for all three sub sampling quantile estimators. Applications to one-sample quantile confidence intervals and tests for equality of two population medians are illustrated with Monte Carlo results to indicate that the proposed methods provide improvements over existing procedures.

A simulation study on the performance of bounded length confidence intervals for quantiles with censored data

Gardiner, Susarla and van Ryzin (1988) considered the problem of constructing confidence intervals of prescribed accuracy (width ≤= 2d, d > 0) and given coverage probability (l-2α, α e(0, 1/2)) for quantiles of a distribution function based on censored data. In a subsequent paper they obtained bounds on the error between this coverage probability and 1-2α as d shrinks to zero. The purpose of the presentation here is to examine the behavior of the confidence interval estimation procedure in simulations for moderate values of d, for varying amounts of censoring and different survival distributions.

A note on the accuracy of bootstrap percentile method confidence intervals for a quantile

It is shown that the bootstrap percentile method of constructing confidence intervals for quantiles is equivalent to the sign test method. As a result, and in contrast to many other problems, percentile-method intervals have coverage error of precise order n −1/2 in both one- and two-sided cases. We show that bootstrap iteration of percentile-method intervals has no role to play in quantile problems, and cannot be used to improve coverage accuracy. This is in marked distinction to more classical problems, where each iteration reduces the order of coverage error by the factor n −1/2 for one-sided intervals and n −1 for two-sided intervals. It thus emerges that standard bootstrap techniques perform poorly in constructing confidence intervals for quantiles: the percentile method is inaccurate for given levels, and produces nothing new; iteration of the percentile method fails completely; bias correction and accelerated bias correction are no more than adjustments to sign test intervals, and therefore fail; and percentile- t is hardly an efficacious alternative because of the non-availability of a suitable variance estimate.

Sequential fixed-width confidence intervals for quantiles in the presence of censoring

This paper studies the asymptotic properties of sequential fixed-width confidence intervals for quantiles of the survival distribution function in the random censorship model. The interval is formed by a pair of quantiles of the product-limit estimator. The set up requires new results on the almost sure behaviour of such empirical quantiles.

CONFIDENCE INTERVALS FOR FLOOD EVENTS UNDER A PEARSON 3 OR LOG PEARSON 3 DISTRIBUTION1

ABSTRACT: The Pearson type 3 (P3) and log Pearson type 3 (LP3) distributions are very frequently used in flood frequency analysis. Existing methods for constructing confidence intervals for quantiles (Xp) of these two distributions are very crude. Most of these methods are based on the idea of adjusting confidence intervals for quantiles Yp of the normal distribution to obtain approximate confidence inervals for quantiles Xp of the P3/LP3 distribution. Since there is no theoretical reason why this “base” distribution, Y, should be taken to be normal, we search in the present study for the best possible base distribution for producing confidence intervals for P3/LP3 quantiles. We consider a group of base distributions such as the normal, log normal, Weibull, Gumbel, and exponential. We first assume that the skew coefficient, γ of X, to be known, and develop a method for adjusting confidence intervals for Yp to produce approximate confidence intervals for Xp. We then compare this method (Method A) with another method (Method B) introduced by Stedinger. Simulation shows that the performance of each of these two methods depends on the base distribution Y that is being used, but as a whole, the normal distribution appears to be the best‐fit distribution for producing confidence intervals for P3/LP3 quantiles when γ is assumed to be known. We then extend our method (Method A) to the more important case of unknown coefficient of skewness. It is shown that by taking Y to be Weibull, fairly accurate confidence intervals for P3/LP3 quantiles can be obtained for quite a wide range of sample sizes and coefficients of skewness commonly found in hydrology. The case of the P3 distribution with negative skewness needs further research.