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.      

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.     

When does the expectation of a ratio equal the ratio of expectations?

In this paper we investigate the behaviour of a ratio of random variables and try to find conditions under which, this expectation equals the ratio of expectations.     

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.     

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.     

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.     

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.     

Segmentation of the mean of heteroscedastic data via cross-validation

This paper tackles the problem of detecting abrupt changes in the mean of a heteroscedastic signal by model selection, without knowledge on the variations of the noise. A new family of change-point detection procedures is proposed, showing that cross-validation methods can be successful in the heteroscedastic framework, whereas most existing procedures are not robust to heteroscedasticity. The robustness to heteroscedasticity of the proposed procedures is supported by an extensive simulation study, together with recent partial theoretical results. An application to Comparative Genomic Hybridization (CGH) data is provided, showing that robustness to heteroscedasticity can indeed be required for their analysis.     

A Note on “Characteristics of the Membership of the American Statistical Association, 1958”

A second course in statistics, for nonstatisticians who will use packaged statistical software in their work, is outlined. The course is directed toward the wise choice, use, and evaluation of statistical computer packages. The goal of the course is to train educated consumers of statistical programs. Particular attention is paid to computer-based data analysis, interpretation of output, comparison of competing packages, and statistical problems that arise when computers are employed to analyze large data sets.     

The Volume of Transactions and the Circulation of Money in the United States, 1950–1979

In the identity of exchange I distinguish between currency and bank payments on one side and several types of transactions and the transfer of idle money on the other. An attempt is made to measure these variables, with varying success. On the payments side I argue that currency velocity is constant (and low) and that the vast rise of bank money velocity is largely due to increased short-term investment of idle funds. The results suggest an upward shift in the level of transactions in 1968–1972, which I attribute to changes in the international role of the dollar.     

Can the Growth of Mega-Journals Affect Authors' Choice of Journal?

The growth of mega-journals has also brought about changes in authorship. This study uses collected data to identify the transitions of mega-journal publishing and changes of authorship. For authors interested in publishing in mega-journals, the publication of South Korean authors in these mega-journals is specifically sampled and checked, revealing that mega-journals are increasingly the targets for publication. Some mega-journals have an impact factor even if they have only been publishing for 5–10 years. Ten out of 12 SCIE mega-journals collected in this study show a high impact factor, which is above average. Furthermore, the number of articles published by South Korean authors in major mega-journals is also on the rise. The result of this study will be useful for mega-journal editors, who are keen to know what factors influence author choice of journals, and for those who attempt to predict what impact mega-journals will have on scholarly journal publishing in the future.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.     

On the existence and uniqueness of the maximum likelihood estimates of the parameters of Birnbaum–Saunders distribution based on Type-I,Type-II and hybrid censored samples

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution and is a common model for analysing lifetime data. In this paper, we discuss the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of BS distribution based on Type-I, Type-II and hybrid censored samples. The line of proof is based on the monotonicity property of the likelihood function. We then describe the numerical iterative procedure for determining the MLEs of the parameters, and point out briefly some recently developed simple methods of estimation in the case of Type-II censoring. Some graphical illustrations of the approach are given for three real data from the reliability literature. Finally, for illustrative purpose, we also present an example in which the MLEs do not exist.