Tests of significance for the odds ratio in a 2×2 table based on the ratio of two independent F-variates

Following Gart (1966) a test of significance for the odds ratio in a 2×2 table is developed based on a semi-empirical method of approximating discrete distributions by their continuous analogues. The distribution of the test statistic (W), the ratio of two independent F-variates, is derived. This approximate technique is compared with the "exact" test, uncorrected X test, and a normal approximation based on lnW.     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

On the small sample estimation of the log odds ratio in 2×2 contingency tables with one set of fixed margins

For the analysis of 2 × 2 contingency tables with one set of fixed margins, a number of authors (e.g. Wolf, 1955; Cox, 1970) have proposed the use of various modified estimators based upon the empirical logistic transform. In this paper the moments of such estimators are considered and their small sample properties are investigated numerically.     

How risky is the optimal portfolio which maximizes the Sharpe ratio?

In this paper, we investigate the properties of the optimal portfolio in the sense of maximizing the Sharpe ratio (SR) and develop a procedure for the calculation of the risk of this portfolio. This is achieved by constructing an optimal portfolio which minimizes the Value-at-Risk (VaR) and at the same time coincides with the tangent (market) portfolio on the efficient frontier which is related to the SR portfolio. The resulting significance level of the minimum VaR portfolio is then used to determine the risk of both the market portfolio and the corresponding SR portfolio. However, the expression of this significance level depends on the unknown parameters which have to be estimated in practice. It leads to an estimator of the significance level whose distributional properties are investigated in detail. Based on these results, a confidence interval for the suggested risk measure of the SR portfolio is constructed and applied to real data. Both theoretical and empirical findings document that the SR portfolio is very risky since the corresponding significance level is smaller than 90 % in most of the considered cases.     

Confidence intervals and statistical power of the ‘Validation’ ratio for surrogate or intermediate endpoints

There is currently much interest in the use of surrogate endpoints in clinical trials and intermediate endpoints in epidemiology. Freedman et al. [Statist. Med. 11 (1992) 167] proposed the use of a validation ratio for judging the evidence of the validity of a surrogate endpoint. The method involves calculation of a confidence interval for the ratio. In this paper, I compare through computer simulations the performance of Fieller's method with the delta method for this calculation. In typical situations, the numerator and denominator of the ratio are highly correlated. I find that the Fieller method is superior to the delta method in coverage properties and in statistical power of the validation test. In addition, the formula for predicting statistical power seems to be much more accurate for the Fieller method than for the delta method. The simulations show that the role of validation analysis is likely to be limited in evaluating the reliability of using surrogate endpoints in clinical trials; however, it is likely to be a useful tool in epidemiology for identifying intermediate endpoints.     

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.     

Estimation of the Parameters of the Birnbaum–Saunders Distribution

Some alternative estimators to the maximum likelihood estimators of the two parameters of the Birnbaum–Saunders distribution are proposed. Most have high efficiencies as measured by root mean square error and are robust to departure from the model as well as to outliers. In addition, the proposed estimators are easy to compute. Both complete and right-censored data are discussed. Simulation studies are provided to compare the performance of the estimators.     

Computing the distribution of sequential Hölder norms of the Brownian motion

ABSTRACT

In this article, we give explicit formulas and study practical computations for the distribution function of sequential Hölder norms of a Brownian motion and of a Brownian bridge. We also discuss some statistical applications in the detection of some short “epidemic” changes in a sample.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

This paper studies the impact of jumps on volatility estimation and inference based on various realised variation measures such as realised variance, realised multipower variation and truncated realised multipower variation. We review the asymptotic theory of those realised variation measures and present a new estimator for the asymptotic ‘variance’ of the centered realised variance in the presence of jumps. Next, we compare the finite sample performance of the various estimators by means of detailed Monte Carlo studies. Here we study the impact of the jump activity, of the jump size of the jumps in the price and of the presence of additional independent or dependent jumps in the volatility. We find that the finite sample performance of realised variance and, in particular, of log-transformed realised variance is generally good, whereas the jump-robust statistics tend to struggle in the presence of a highly active jump process.     

Some further aspects of the löwner-ordering antitonicity of the moore-penrose inverse

A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz's theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.     

An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

Extended tables of the distribution of friedman’ s s-statistic in the two-way layout

Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.     

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.     

On the exact distribution of the sum of the largest n−k out of n normal random variables with differing mean values

Motivated by an application in Electrical Engineering, we derive the exact distribution of the sum of the largest n?k out of n normally distributed random variables, with differing mean values. Comparisons are made with two normal approximations to this distribution—one arising from the asymptotic negligibility of the omitted order statistics and one from the theory of L-statistics. The latter approximation is found to be in excellent agreement with the exact distribution.     

Explicit formulas for the smoother weights of the Whittaker–Henderson graduation of order 1

The Whittaker–Henderson (WH) graduation is a widely applied smoothing method. This paper contributes to the literature by providing explicit formulas for the smoother weights of the WH graduation of order 1 along with some related results, which leads to a richer understanding of the filter.     

Percentage points of the joint distribution of the extreme roots of the random matrix S1(S1+S2)−1

The numerical treatment of nonlinear model fitting problems often can be simplified by manipulating the model equations. Algebraic manipulations including nonlinear transformations of model parameters, do not change the numerical result of the adjustment and can be a powerful method to improve the performance of solution algorithms. Nonlinear transformations of the observations, on the other hand, do change the numerical results unless the normal equations are transformed accordingly. The latter transformation has been neglected by previous authors and this article provides a complete set of formulas that are needed to implement transformations of observations. The transformations are in general less useful than parameter transformations for the improvement of algorithms but may have other applications in particular situations.     

On the computation of a robusl version of the optimal c(α) test

A robust test of a parameter while in the presence of nuisance parameters was proposed by Wang (1981). The test procedure is a robust extension of the optimal C(α) tests. A numerical method for computing the solution of the orthogonality condition that is required by the test procedure is provided. An example on the testing of normal scale while in the presence of outliers is worked out to illustrate the construction of the robust test.