Generalized Concept of Relative Risk and Wider Applications of the Proportional Hazards Model and the Kaplan–Meier Estimator

The concepts of relative risk and hazard ratio are generalized for ordinary ordinal and continuous response variables, respectively. Under the generalized concepts, the Cox proportional hazards model with the Breslow's and Efron's methods can be regarded as generalizations of the Mantel–Haenszel estimator for dealing with broader types of covariates and responses. When ordinal responses can be regarded as discretized observations of a hypothetical continuous variable, the estimated relative risks from the Cox model reflect the associations between the responses and covariates. Examples are given to illustrate the generalized concepts and wider applications of the Cox model and the Kaplan–Meier estimator.     

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.     

Critical values of Mandel’s h and k, the Grubbs and the Cochran test statistic

According to ISO 5725-2 (1994), measurement results obtained in an interlaboratory experiment are inspected for consistency by plotting Mandel’s h and k statistics and for outliers by application of the Grubbs test and the Cochran test. Critical values of these statistics for significance levels α=5% and α=1% and for some numbers p of laboratories and n of repeated measurements in the laboratories are supplied in ISO 5725-2 without reference to methods for their calculation. In this paper, exact formulae for the critical values of Mandel’s h and k and approximate formulae for the critical values of the Single Grubbs test, the Double Grubbs test and the Cochran test are derived.     

Duality of the Multivariate Distributions of Marshall–Olkin Type and Tail Dependence

A dual class of the multivariate distributions of Marshall–Olkin type is introduced, and their copulas are presented and utilized to derive explicit expressions of the distributional tail dependencies, which describe the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. A sufficient condition under which tail dependencies of two such distributions can be compared are obtained. Some examples are also presented to illustrate our results.     

Simulated annealing for the bounds of Kendall's τ and Spearman's ρ

It has long been known that, for many joint distributions, Kendall's τ and Spearman's ρ have different values, as they measure different aspects of the dependence structure. Although the classical inequalities between Kendall's τ and Spearman's ρ for pairs of random variables are given, the joint distributions which can attain the bounds between Kendall's τ and Spearman's ρ are difficult to find. We use the simulated annealing method to find the bounds for ρ in terms of τ and its corresponding joint distribution which can attain those bounds. Furthermore, using this same method, we find the improved bounds between τ and ρ, which is different from that given by Durbin and Stuart.     

Fast computation of the exact null distribution of Spearman's ρ and Page's L statistic for samples with and without ties

We present a new algorithm for computing the exact null distribution of the Spearman rank correlation statistic ρ, which also works in the case of ties. The algorithm is based on symmetries in the representation of the probability generating function as a permanent with monomial entries. We present new critical values for sample sizes 19⩽n⩽22. Finally, we show how to derive the exact null distribution of Page's L statistic from the null distribution of ρ.     

Gender and income inequality in the UK 1968–90: the feminization of earnings or of poverty?

This paper uses data from the Family Expenditure Survey for five selected years between 1968 and 1990 to examine trends in the income distribution in the UK, highlighting the role of women's labour force participation and earnings. The increased labour force participation of married women (especially mothers of young children in the 1980s) made a greater contribution to the decline of the 'traditional' male breadwinner family than the increased number of lone parents. The lower half of the distribution of weekly earnings became increasingly dominated by women. Though women's weekly earnings remained low relative to men's, the increase in their participation meant that, over the period, an increased share of family income came from women's labour market income: in 1990 nearly a quarter of the income of families with children came from women's earnings. Women's earnings were an important factor in keeping families out of poverty. There was no trend towards increasing feminization of poverty over the sample period. Adult women were somewhat more likely to be poor than adult men were, but female-headed families were very much more likely to be in poverty, and much more dependent on state benefits, than male-headed families were. Women's increased role in the labour market affected those in male-headed families more than those in female-headed families. Alongside a broad tendency for women's earnings to reduce poverty and inequality, there is evidence that the female population has become more economically polarized.     

Discussion of “On simulation and properties of the stable law” by L. Devroye and L. James

We contribute to the discussion of the paper by Devroye and James, by reviewing some of the most meaningful results that relate the unilateral stable distribution with the asymptotic behavior of the so-called Ewens-Pitman sampling model. Our focus is then on how these results have been exploited in the context of Bayesian nonparametric inference for species sampling problems.     

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.     

Discussion on the paper “On simulation and properties of the stable law” by L. Devroye and L. James

We congratulate the authors for the interesting paper. The reading has been really pleasant and instructive. We discuss briefly only some of the interesting results given in Devroye and James (Stat Methods Appl 2014) with particular attention to evolution problems. The contribution of the results collected in the paper is useful in a more wide class of applications in many areas of applied mathematics.     

Overnight and Daytime Stock-Return Dynamites on the London Stock Exchange: The Impacts of “Big Bang” and the 1987 Stock-Market Crash

We explore the time series properties of stock returns on the London Stock Exchange around the 1986 market restructuring (Big Bang) and the 1987 stock-market crash using a modified generalized autoregressive conditional heteroscedasticity model. Using this general dynamic model, which allows (a) intradaily returns to have different impacts and persistence on stock-return volatility, (b) return effects on volatility to be asymmetric, and (c) intradaily returns to follow conditional distributions with different fourth moments, we uncover important changes in return dynamics and conditional fourth moments following Big Bang and the 1987 crash not reported before.     

A Cautionary Note on the Use of the Grønnesby and Borgan Goodness-of-Fit Test for the Cox Proportional Hazards Model

Grønnesby and Borgan (1996, Lifetime Data Analysis 2, 315–328) propose an omnibus goodness-of-fit test for the Cox proportional hazards model. The test is based on grouping the subjects by their estimated risk score and comparing the number of observed and a model based estimated number of expected events within each group. We show, using extensive simulations, that even for moderate sample sizes the choice of number of groups is critical for the test to attain the specified size. In light of these results we suggest a grouping strategy under which the test attains the correct size even for small samples. The power of the test statistic seems to be acceptable when compared to other goodness-of-fit tests.     

How Long Is a Spell of Unemployment? Illusions and Biases in the Use of CPS Data

Most data used to study the durations of unemployment spells come from the Current Population Survey (CPS), which is a point-in-time survey and gives an incomplete picture of the underlying duration distribution. We introduce a new sample of completed unemployment spells obtained from panel data and apply CPS sampling and reporting techniques to replicate the type of data used by other researchers. Predicted duration distributions derived from this CPS-like data are then compared to the actual distribution. We conclude that the best inferences that can be made about unemployment durations by using CPS-like data are seriously biased.     

Characteristics and impact of faked interviews in surveys – An analysis of genuine fakes in the raw data of SOEP

Summary: Panel data offers a unique opportunity to identify data that interviewers clearly faked by comparing data waves. In the German Socio–Economic Panel (SOEP), only 0.5 percent of all records of raw data have been detected as faked. These fakes are used here to analyze the potential impact of fakes on survey results. Due to our central finding the faked records have no impact on the mean or the proportions. However, we show that there may be a serious bias in the estimation of correlations and regression coefficients. In all but one year (1998), the detected faked data have never been disseminated within the widely–used SOEP study. The fakes are removed prior to data release.* We are grateful to participants in the workshop on Item Nonresponse and Data Quality on Large Social Surveys for useful critique and comments, especially Rainer Schnell and our outstanding discussant Regina Riphahn. The usual disclaimer applies.     

Tables for the Lilliefors and Modified Cramer–von Mises Tests of Normality

We present new tabulations of the Lilliefors distribution and the modified Cramer–von Mises distribution, which are used to test for normality when the population mean and variance are unknown. Some practical remarks and an example are given.     

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.     

L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation,goodness of fit and application to earthquake data

Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum–Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962–2015, is carried out. This application identifies the extreme value Birnbaum–Saunders distribution as a better model than classic extreme value distributions for describing seismic events.     

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.