A note on the extension of tables for the Friedman statistic

Abstract  The tables for the Friedman rank-test arc extended using a sampling method. A table is given for the traditional significance levels .01, .025, .05 and .10 for number of blocks equal to 7 and 8 and for treatments of size 6 and 7. A brief discussion of available tables is also included along with an example of an application.     

Exceptional Convergence of the Sample Variance

The case is investigated when Hoeffding's one sample U–statistic theorem for the sample variance S² is not applicable. It is shown that this occurs only when the parent distribution is the two–point distribution with jumps of equal magnitude. For this exceptional case the standardised S² is shown to converge in distribution to (1 – V )√2, where V has chi–square distribution with one degree of freedom.     

Exact finite sample properties of double k-class estimators in simultaneous equations

In our earlier paper [Srivastava, Agnihotri and Dwivedi (1980)] the dominance of double k-class over k-class with respect to exact mean squared error matrix criteria is established. It is observed that given a member of k-class, one can pick up a member of double k-class that will provide an improved estimator of the coefficients. This result prompted us to study the exact finite sample properties of the double k-class estimator. For this, we have considered a structural equation containing two endogenous variables and have investigated the properties of double k-class estimators of the coefficients of explanatory endogenous variables assuming characterizing scalars to be non-stochastic.     

The entropies: Some roots of ambiguity

Entropy concepts have developed in four general contexts—thermodynamics, communication theory, statistical information theory and social and life sciences. These concepts have five distinct mathematical forms. Ambiguity and complexity in utilization of the concept has been heightened by this multi-faceted heritage. Effective utilization may be strengthened by a sharper articulation of entropy through combinations of literal, mathematical and graphical modes of expression. Geography and planning as user disciplines have important responsibilities in enhancing cross-disciplinary communication of this important adisciplinary concept. Implications of these arguments for the social sciences include the need for caution in drawing social analogies from the ambiguous entropy concepts of statistical mechanics; and an indication of the potential benefits of social science models based on the unambiguous entropic constructs of statistical information theory.     

Robustness to the unavailability of data in block designs

We present general results for finding or boundingt _maxB , the maximum number of arbitrary whole blocks of observations which can be removed from a block design, and still leave all of the elementary treatment contrasts estimable. The block sizes may be larger than the number of treatments. The results are applied to BBDs, reinforced BIBDs and BBDs, BTIBDs, and a series of variance balanced incomplete block designs with two block sizes. Also given for most of these designs, are results fort _max, the maximum number of arbitrary, scattered observations that can become unavailable, and still leave all of the elementary treatment contrasts estimable.The work was undertaken while Dr. Whittinghill was visiting Ohio State University, and supported by a grant from the Natural Sciences Division, Colby College, Waterville, Maine.     

A probability inequality for ranges and its application to maximum range test procedures

Summary It is proved that for any fixed argument the sequence (P _k) of the distribution functions of the ranges ofk i.i.d. univariate random variables is log-concave if the random variables have a log-concave density. If the support of the distribution is an infinite interval and the density is monotonous then the theorem holds also with “log-convex” instead of “log-concave”. The resulting inequalities can be used by a quick algorithm for closed maximum range test procedures for all pairwise comparisons (Royen 1988, 1989a, 1989b). Under the above assumptions the application of this algorithm can be extended e.g. to pairwise comparisons of variances.     

Aggregation effects on price and expenditure elasticities in a quadratic almost ideal demand system

Abstract.  While it is well known that demand elasticities calculated at the macro level will in general differ from those calculated at the micro level because of aggregation effects, there remain the questions of how large the effects are and how they vary with the degree of inequality in the income distribution. We explore these questions with models based on a quadratic version of the Almost Ideal Demand System. We investigate the elasticity differences theoretically and then calibrate the models and generate numerical results, using income data for seven countries with widely different distributions. The aggregation effects are found generally to be rather small, even with highly unequal income distributions.