Alternative Approximations to Value-At-Risk: A Comparison

This article compares three value-at-risk (VaR) approximation methods suggested in the literature: Cornish and Fisher (1937 Cornish, E.A., Fisher, R.A. (1937). Moments and cumulants in the specification of distributions. Revue de l’Institut International de Statistique 5:307–320.[Crossref] , [Google Scholar]), Sillitto (1969 Sillitto, G.P. (1969). Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample. Biometrika 56:641–650.[Crossref], [Web of Science ®] , [Google Scholar]), and Liu (2010 Liu, W.-H. (2010). Estimation and testing of portfolio value-at-risk based on L-comoment matrices. Journal of Futures Markets 30:897–908.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results are obtained for three families of distributions: student-t, skewed-normal, and skewed-t. We recommend the Sillitto approximation as the best method to evaluate the VaR when the financial return has an unknown, skewed, and heavy-tailed distribution.     

Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT

In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.     

A Poisson Limit Theorem for Reliability Models Based on Markov Chains

Abstract

In this article, we deal with a class of discrete-time reliability models. The failures are assumed to be generated by an underlying time inhomogeneous Markov chain. The multivariate point process of failures is proved to converge to a Poisson-type process when the failures are rare. As a result, we obtain a Compound Poisson approximation of the cumulative number of failures. A rate of convergence is provided.     

Edgeworth Expansion and Bootstrap Approximation for Studentized Product-Limit Estimator with Truncated and Censored Data

The Yule-Walker estimators of the AR coefficients of a causal multidimensional AR model are obtained by replacing the autocovariances with their estimators in the Yule-Walker equations. It is shown that only unbiased-type estimators of the autocovariances yield consistency of the Yule-Walker estimators. Also, the asymptotic joint distribution of the Yule-Walker estimators is presented.     

Testing the Difference of Two Binomial Proportions: Comparison of Continuity Corrections for Saddlepoint Approximation

We carried out a simulation study based on the methodology of Newcombe (1998 Newcombe , R. G. ( 1998 ). Interval estimation for the difference between independent proportions: comparison of eleven methods . Statist. Med. 17 : 873 – 890 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to compare tests for the difference of two binomial proportions by applying different continuity corrections on saddlepoint approximation to tail probabilities. In this article, we proposed a new continuity correction based on the least common multiple of two sample sizes. We evaluated that the best test should have the actual Type I error rates that are, on the whole, closest to α, but not exceeding α, where α is nominal level of significance.     

Saddlepoint Approximation for Sample Quantiles with Some Applications

In this article, we use the integral form of the binomial distribution to derive saddlepoint approximations for sample quantiles. As an application, we present the calculation of the tail probability of the empirical log-likelihood ratio statistic for quantiles. Simulation results are also given to show that our approximations are extremely accurate.     

Stirling's Formula and Its Extensions: Heuristic Approaches

Walsh (1995 Walsh , D. P. ( 1995 ). Equating Poisson and normal probability functions to derive Stirling's formula . Amer. Statist. 49 : 270 – 271 .[Taylor & Francis Online] , [Google Scholar]) introduced a heuristic approach to motivate Stirling's formula by equating a Poisson probability to an analogous value from a normal density function. We explore similar heuristics to derive approximations for various binomial, negative binomial, and multinomial coefficients. Also, using heuristics markedly different from those of Walsh, we develop an approximation of (nk)! for positive integers n (large) and k. These heuristics are then used to validate Stirling's formula for Γ(nα) where α is a positive real number. To derive each of our approximations we use a different probability distribution, and hence each section may serve as pedagogical module.     

Application of Anbar's Approach to Hypothesis Testing to Detect the Difference between Two Proportions

In this paper, Anbar's (1983) approach for estimating a difference between two binomial proportions is discussed with respect to a hypothesis testing problem. Such an approach results in two possible testing strategies. While the results of the tests are expected to agree for a large sample size when two proportions are equal, the tests are shown to perform quite differently in terms of their probabilities of a Type I error for selected sample sizes. Moreover, the tests can lead to different conclusions, which is illustrated via a simple example; and the probability of such cases can be relatively large. In an attempt to improve the tests while preserving their relative simplicity feature, a modified test is proposed. The performance of this test and a conventional test based on normal approximation is assessed. It is shown that the modified Anbar's test better controls the probability of a Type I error for moderate sample sizes.     

A Generalized Chain Binomial Model with Aggregated Data

In large cohort studies it can be impractical to report individual data that only summary or aggregated data are available. Using aggregated data from Bernoulli trials is expected to result in overdispersion so that a quasi-binomial approach would seem feasible. We show that when applied to aggregated data arising from cohorts of individuals according to a chain binomial model, the quasi-binomial model results in biased estimates. We propose an alternate calibration estimator and demonstrate its improved performance by simulations. The calibration method is then applied to model the probability of leaving a personal emergency link service in Hong Kong.     

The sampling distribution of the W estimator of the number of valid signatures on a petition

Abstract

A new non linear estimator, W, for the number of valid, unique signatures on a petition has been shown better, for the cases enumerated and with certain restrictions, than a popular Goodman-type statistic, G. This article extends those results with relaxed conditions by developing the exact probability mass function and mean of W and a close approximation of the variance (Var(W)). If the proportion of valid signatures among unique and duplicated signatures is the same, then Var(W) is approximately a function of the means and variances of the two sample statistics. Using the delta method, we estimate Var(W), with the resulting approximation shown to be good, even when the condition of equal proportions does not hold. We compare W to G and establish which estimator is preferred for different intervals of the design parameters. Data from a Washington State petition illustrate the findings.