Large-sample confidence intervals for risk measures of location–scale families

For a loss distribution belonging to a location–scale family, _Fμ,σ$F_{\mu ,\sigma }$, the risk measures, Value-at-Risk and Expected Shortfall are linear functions of the parameters: μ+τσ$\mu +\tau \sigma$ where τ$\tau$ is the corresponding risk measure of the mean-zero and unit-variance member of the family. For each risk measure, we consider a natural estimator by replacing the unknown parameters μ$\mu$ and σ$\sigma$ by the sample mean and (bias corrected) sample standard deviation, respectively. The large-sample parametric confidence intervals for the risk measures are derived, relying on the asymptotic joint distribution of the sample mean and sample standard deviation. Simulation studies with the Normal, Laplace and Gumbel families illustrate that the derived asymptotic confidence intervals for Value-at-Risk and Expected Shortfall outperform those of Bahadur (1966) and Brazauskas et al. (2008), respectively. The method can also be effectively applied to Log-location-scale families whose supports are positive reals; an illustrative example is given in the area of financial credit risk.     

Adjusting Cornish–Fisher expansions and confidence intervals for the effect of roundoff

Suppose we want to estimate some smooth function of two types of parameters. The first can be estimated by sample means, while the second is known exactly up to the number of decimal places recorded, that is they are subject to roundoff. We obtain the Cornish–Fisher expansions and associated nonparametric confidence intervals for such functions. These results are illustrated by a simulation study.     

A comparison of confidence intervals generated by the scheffé and bonferroni methods

Simultaneous confidence intervals for the p means of a multivariate normal random variable with known variances may be generated by the projection method of Scheffé and by the use of Bonferroni's inequality. It has been conjectured that the Bonferroni intervals are shorter than the Scheffé intervals, at least for the usual confidence levels. This conjecture is proved for all p≥2 and all confidence levels above 50%. It is shown, incidentally, that for all p≥2 Scheffé's intervals are shorter for sufficiently small confidence levels. The results are also applicable to the Bonferroni and Scheffé intervals generated for multinomial proportions.     

Non parametric confidence intervals for the extinction probability of a Galton–Watson branching process

It is demonstrated that the confidence intervals (CIs) for the probability of eventual extinction and other parameters of a Galton–Watson branching process based upon the maximum likelihood estimators can often have substantially lower coverage when compared to the desired nominal confidence coefficient, especially in small, more realistic sample sizes. The same conclusion holds for the traditional bootstrap CIs. We propose several adjustments to these CIs, which greatly improves coverage in most cases. We also make a correction in an asymptotic variance formula given in Stigler (1971 Stigler, S.M. (1971). The estimation of the probability of extinction and other parameters associated with branching processes. Biometrika 58(3):499–508.[Crossref], [Web of Science ®] , [Google Scholar]). The focus here is on implementation of the CIs which have good coverage, in a wide variety of cases. We also consider expected CI lengths. Some recommendations are made.     

A note on scheffé's method of deriving confidence sets for directions and ratios of normals

Scheffé (1970) introduced a method for deriving confidence sets for directions and ratios of normals. The procedure requires use of an approximation and Scheffé provided evidence that the method performs well for cases in which the variances of the random deviates are known. This paper extends Scheffé's numerical integrations to the case of unknown variances. Our results indicate that Scheffé's method works well when variances are unknown     

Bayesian confidence interval for the risk ratio in a correlated 2 × 2 table with structural zero

This paper studies the construction of a Bayesian confidence interval for the risk ratio (RR) in a 2 × 2 table with structural zero. Under a Dirichlet prior distribution, the exact posterior distribution of the RR is derived, and tail-based interval is suggested for constructing Bayesian confidence interval. The frequentist performance of this confidence interval is investigated by simulation and compared with the score-based interval in terms of the mean coverage probability and mean expected width of the interval. An advantage of the Bayesian confidence interval is that it is well defined for all data structure and has shorter expected width. Our simulation shows that the Bayesian tail-based interval under Jeffreys’ prior performs as well as or better than the score-based confidence interval.     

Inferences for the ratio: Fieller’s interval,log ratio,and large sample based confidence intervals

In sample surveys and many other areas of application, the ratio of variables is often of great importance. This often occurs when one variable is available at the population level while another variable of interest is available for sample data only. In this case, using the sample ratio, we can often gather valuable information on the variable of interest for the unsampled observations. In many other studies, the ratio itself is of interest, for example when estimating proportions from a random number of observations. In this note we compare three confidence intervals for the population ratio: A large sample interval, a log based version of the large sample interval, and Fieller’s interval. This is done through data analysis and through a small simulation experiment. The Fieller method has often been proposed as a superior interval for small sample sizes. We show through a data example and simulation experiments that Fieller’s method often gives nonsensical and uninformative intervals when the observations are noisy relative to the mean of the data. The large sample interval does not similarly suffer and thus can be a more reliable method for small and large samples.     

Approximate confidence intervals for the weighted kappa coefficient of a binary diagnostic test subject to a case–control design

Case–control design to assess the accuracy of a binary diagnostic test (BDT) is very frequent in clinical practice. This design consists of applying the diagnostic test to all of the individuals in a sample of those who have the disease and in another sample of those who do not have the disease. The sensitivity of the diagnostic test is estimated from the case sample and the specificity is estimated from the control sample. Another parameter which is used to assess the performance of a BDT is the weighted kappa coefficient. The weighted kappa coefficient depends on the sensitivity and specificity of the diagnostic test, on the disease prevalence and on the weighting index. In this article, confidence intervals are studied for the weighted kappa coefficient subject to a case–control design and a method is proposed to calculate the sample sizes to estimate this parameter. The results obtained were applied to a real example.