Robustness of a Class of Partial Diallel Cross Designs to the Unavailability of a Complete Block of Observations

Complete and partial diallel cross designs are examined as to their construction and robustness against the loss of a block of observations. A simple generalized inverse is found for the information matrix of the line effects, which allows evaluation of expressions for the variances of the line-effect differences with and without the missing block. A-efficiencies, based on average variances of the elementary contrasts of the line-effects, suggest that these designs are fairly robust. The loss of efficiency is generally less than 10%, but it is shown that specific comparisons might suffer a loss of efficiency of as much as 40%.     

On a Dualization of Graphical Gaussian Models: A Correction Note

Abstract.  We correct two proofs concerning Markov properties for graphs representing marginal independence relations.     

COMPARISONS AND SELECTION OF TWO-PARAMETER EXPONENTIAL POPULATIONS

For two-parameter exponential populations with the same scale parameter (known or unknown) comparisons are made between the location parameters. This is done by constructing confidence intervals, which can then be used for selection procedures. Comparisons are made with a control, and with the (unknown) “best” or “worst” population. Emphasis is laid on finding approximations to the confidence so that calculations are simple and tables are not necessary. (Since we consider unequal sample sizes, tables for exact values would need to be extensive.)     

Benchmark Analysis: Shopping with Proper Confidence

We discuss the issue of using benchmark doses for quantifying (excess) risk associated with exposure to environmental hazards. The paradigm of low-dose risk estimation in dose-response modeling is used as the primary application scenario. Emphasis is placed on making simultaneous inferences on benchmark doses when data are in the form of proportions, although the concepts translate easily to other forms of outcome data.     

Premarital Cohabitation and Housework: Couples in Cross‐National Perspective

The authors examine the effect of premarital cohabitation on the division of household labor in 22 countries. First, women do more routine housework than men in all countries. Second, married couples that cohabited before marriage have a more equal division of housework. Third, national cohabitation rates have equalizing effects on couples regardless of their own cohabitation experience. However, the influence of cohabitation rates is only observed in countries with higher levels of overall gender equality. The authors conclude that the trend toward increasing cohabitation may be part of a broader social trend toward a more egalitarian division of housework.     

Bayesian locally optimal design of knockout tournaments

The elimination or knockout format is one of the most common designs for pairing competitors in tournaments and leagues. In each round of a knockout tournament, the losers are eliminated while the winners advance to the next round. Typically, the goal of such a design is to identify the overall best player. Using a common probability model for expressing relative player strengths, we develop an adaptive approach to pairing players each round in which the probability that the best player advances to the next round is maximized. We evaluate our method using simulated game outcomes under several data-generating mechanisms, and compare it to random pairings, to the standard knockout format, and to two variants of the standard format.     

Efficient Estimation of a Distribution Function under Quadrant Dependence

LetX₁,X₂, ..., be real-valued random variables forming a strictly stationary sequence, and satisfying the basic requirement of being either pairwise positively quadrant dependent or pairwise negatively quadrant dependent. LetF^ be the marginal distribution function of theX_ips, which is estimated by the empirical distribution functionF_n and also by a smooth kernel-type estimateF_n, by means of the segmentX₁, ...,X_n. These estimates are compared on the basis of their mean squared errors (MSE). The main results of this paper are the following. Under certain regularity conditions, the optimal bandwidth (in the MSE sense) is determined, and is found to be the same as that in the independent identically distributed case. It is also shown thatn MSE(F_n(t)) andnMSE (F^_n(t)) tend to the same constant, asn→∞ so that one can not discriminate be tween the two estimates on the basis of the MSE. Next, ifi(n) = min {k∈{1, 2, ...}; MSE (F_k(t)) ≤ MSE (F_n(t))}, then it is proved thati(n)/n tends to 1, asn→∞. Thus, once again, one can not choose one estimate over the other in terms of their asymptotic relative efficiency. If, however, the squared bias ofF^_n(t) tends to 0 sufficiently fast, or equivalently, the bandwidthh_n satisfies the requirement thatnh³_n→ 0, asn→∞, it is shown that, for a suitable choice of the kernel, (i(n) ?n)/(nh_n) tends to a positive number, asn→∞ It follows that the deficiency ofF_n(t) with respect toF^_n(t),i(n) ?n, is substantial, and, actually, tends to ∞, asn→∞. In terms of deficiency, the smooth estimateF^_n(t) is preferable to the empirical distribution functionF_n(t)     

An empirical Bayes' procedure for ranking players in Ryder Cup golf

We describe a model to obtain strengths and rankings of players appearing in golf's Ryder Cup. Obtaining rankings is complicated because of two reasons. First, competitors do not compete on an equal number of occasions, with some competitors appearing too infrequently for their ranking to be estimated with any degree of certainty, and second, different competitors experience different levels of volatility in results. Our approach is to assume the competitor strengths are drawn from some common distribution. For small numbers of competitors, as is the case here, we fit the model using Monte-Carlo integration. Results suggest there is very little difference between the top performing players, though Scotland's Colin Montgomerie is estimated as the strongest Ryder Cup player.     

Why Frequency is Not Well Defined for Engineering Systems with Nonexponential Failure Times

Dr. Yellman proposes to define frequency as “a time‐rate of events of a specified type over a particular time interval.” We review why no definition of frequency, including this one, can satisfy both of two conditions: (1) the definition should agree with the ordinary meaning of frequency, such as that less frequent events are less likely to occur than more frequent events, over any particular time interval for which the frequencies of both are defined; and (2) the definition should be applicable not only to exponentially distributed times between (or until) events, but also to some nonexponential (e.g., uniformly distributed) times. We make the simple point that no definition can satisfy (1) and (2) by showing that any definition that determines which of any two uniformly distributed times has the higher “frequency” (or that determines that they have the same “frequency,” if neither is higher) must assign a higher frequency number to the distribution with the lower probability of occurrence over some time intervals. Dr. Yellman's proposed phrase, “time‐rate of events … over a particular time interval” is profoundly ambiguous in such cases, as the instantaneous failure rates vary over an infinitely wide range (e.g., from one to infinity), making it unclear which value is denoted by the phrase “time‐rate of events.”