Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author systematically examines one of the important issues of information — establishing the market price. He introduces the concept of «search» — where a buyer wanting to get a better price, is forced to question sellers. The article deals with various aspects of finding the necessary information.

Economics of Information

The past year has witnessed substantial advances in understanding the genetic basis of many common phenotypes of biomedical importance. These advances have been the result of systematic, well-powered, genome-wide surveys exploring the relationships between common sequence variation and disease predisposition. This approach has revealed over 50 disease-susceptibility loci and has provided insights into the allelic architecture of multifactorial traits. At the same time, much has been learned about the successful prosecution of association studies on such a scale. This Review highlights the knowledge gained, defines areas of emerging consensus, and describes the challenges that remain as researchers seek to obtain more complete descriptions of the susceptibility architecture of biomedical traits of interest and to translate the information gathered into improvements in clinical management.

http://csg.sph.umich.edu/abecasis/publications/pdf/Nat.Rev.Genet.vol.9-pp.356.pdf

Genome-wide association studies for complex traits: consensus, uncertainty and challenges

L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis and new methods of parameter estimation

https://www.jstor.org/stable/info/2345653

L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics

An account is given of a recently devised method of prediction based on sample reuse techniques. It is most useful in low structure data paradigms that involve minimal assumptions. A series of applications demonstrating the technique is presented.

The Predictive Sample Reuse Method with Applications

M OST of the measures of income inequality are derived from the Lorenz curve; indeed Morgan (1962) states that the Gini index is the best single measure of inequality. The present article reviews some of the theoretical properties of the Lorenz curve, relates them to characteristics of the frequency function underlying the income distribution and develops methods for obtaining accurate bounds on the Gini index which do not depend on curve fitting. In the process we should also like to lay to rest some myths concerning the Gini index such as: (a) its relative insensitivity (Rltet6 and Frigyes, 1968), (b) difficulty in computation (1968), and (c) problems related to the inclusion of negative incomes (Budd, 1970). The basic idea of our approach is to obtain upper and lower bounds to the Gini index from data which are grouped in intervals and the mean income in each interval is known. The usual method (Morgan, 1962) of estimating the Gini index yields a lower bound by assuming that all incomes in any interval equal the average income. We derive an upper bound to the grouping correction (Goldsmith, et al., 1954, p. 10) and hence to the Gini index by distributing the income to maximize the spread within each group. On the 1967 Internal Revenue Service tax data, the difference between our bounds is less than 0.006. As most income distributions come from a frequency function (density) which decreases in the large income range, we develop improved bounds for the Gini index based on this assumption. Fortunately, this assumption can be checked from the data so that we can use the sharper bounds only for the appropriate intervals. Using this second method the difference between our bounds is ? .002. Because Soltow (1965) detects a change in the Gini index of 0.8 of one per cent or about 0.003 or 0.004, our bound seems quite adequate. In section VI we extend our method to obtain upper and lower curves for the Lorenz curve. After reviewing the basic properties of the Lorenz curve we proceed to derive bounds on the mean difference and Gini index. In section IV we analyze an actual sample and show that the method used by the Census Bureau (1967) often leads to estimates which are outside the mathematically possible bounds we derived. Finally, in an appendix we show that the Pareto law does not give a good fit to current United States tax data.

https://www.jstor.org/stable/pdfplus/1937992.pdf

The Estimation of the Lorenz Curve and Gini Index

A General Definition of the Lorenz Curve

In many applications, the underlying scientific question concerns whether the variances of $k$ samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the $F$-test to the absolute deviations of the observations from their group means. Levene's approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances. This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or "open or closed gramophones" in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

The Impact of Levene's Test of Equality of Variances on Statistical Theory and Practice

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.

Trend tests for case-control studies of genetic markers: power, sample size and robustness.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177699058

Joseph L. Gastwirth

Papers

The Estimation of the Lorenz Curve and Gini Index

A General Definition of the Lorenz Curve

The Impact of Levene's Test of Equality of Variances on Statistical Theory and Practice

Trend tests for case-control studies of genetic markers: power, sample size and robustness.

Asymptotic Distribution of Linear Combinations of Functions of Order Statistics with Applications to Estimation