Gilbert W. Bassett

Strong Consistency of Regression Quantiles and Related Empirical Processes

The strong consistency of regression quantile statistics (Koenker and Bassett [4]) in linear models with iid errors is established. Mild regularity conditions on the regression design sequence and the error distribution are required. Strong consistency of the associated empirical quantile process (introduced in Bassett and Koenker [1]) is also established under analogous conditions. However, for the proposed estimate of the conditional distribution function of Y, no regularity conditions on the error distribution are required for uniform strong convergence, thus establishing a Glivenko-Cantelli-type theorem for this estimator.

Breaking recent global temperature records

Global surface temperature was a record in 1988. What is the probability that this record will be surpassed in the next few years? Answers are provided given a variety of simple statistical models for temperature. The answers illustrate how record breaking is influenced by alternative model specifications. Estimates for the probability of a record are shown to range widely. If annual temperature is independent and identically distributed then a new record is unlikely. But probabilities increase rapidly if there is a trend or autocorrelation. Estimates of the probability of a record using data on global temperature suggest that another record in the next few years would not be a rare event.

Robustness of the p-Subset Algorithm for Regression with High Breakdown Point

Regression techniques with high breakdown point can withstand a substantial amount of outliers in the data. One such method is the least trimmed squares estimator. Unfortunately, its exact computation is quite difficult because the objective function may have a large number of local minima. Therefore, we have been using an approximate algorithm based on p-subsets. In this paper we prove that the algorithm shares the equivariance and good breakdown properties of the exact estimator. The same result is also valid for other high-breakdown-point estimators. Finally, the special case of one-dimensional location is discussed separately because of unexpected results concerning half samples.

The St. Petersburg Paradox and Bounded Utility

I. Bernoulli and Crarner In the 1738 Papers of the Imperial Academy of Sciences in Petersburg, Daniel Bernoulli originates the distinction between the mathematical and moral expectation of a lottery. The value of a lottery differs between persons not because one is thought luckier, expects his desires to be more closely fulfilled, but because the utility of an additional dollar differs for a poor man and a rich man. The value of a lottery is determined by its expected utility, utility depends on wealth, utility is a concave function of wealth, and, most specifically, the utility of additional wealth varies inversely with current wealth or u(x) = lo&). Bernoulli’s paper, which had been presented to the Academy in 173 1, contains a letter written by Gabriel Cramer (1728) which shows that Cramer independently arrived at the expected utility idea. Bernoulli and Cramer’s development of expected utility came about because both were seeking a solution to what is now called the St. Petersburg paradox. The paradox arises in a game where a fair coin is flipped until at the n-th flip heads first comes up and one then wins 2” dollars. The realistic value of the game in actual life does not seem large even though the mathematical expectation is unbounded. Why? Bernoulli says it is because the value of the game is determined by expected utility, and he shows that the expected (log) utility value of the game is bounded and accords roughly with what persons would pay to play. This much of the early history of expected utility and the St. Petersburg game is well-known. Also, well-known today is that Bernoulli’s resolution of the paradox is unsatisfactory. As long as the utility function is unbounded, as lo&) is, the game can be always modified (by making prizes grow sufficiently fast) so that expected utility becomes unbounded while willingness to pay remains finite. (Let y,, the prize when heads first comes up, grow so that ~ ( y , ) = 2 ~ . ) So, the paradox returns. To really avoid all paradox it is necessary to invoke both the expected utility principle and bounded utility (or else attribute the paradox to some other feature of the game). When was bounded utility first noticed to be essential to bound the value

The Effects of Alternative HOME-AWAY Sequences in a Best-of-Seven Playoff Series

Abstract In the NBA and NHL, the usual playoff format is a best-of-seven series where the stronger team (based on regular season performance) is given the benefit of four games scheduled in its home building. Typical HOME-AWAY schedules are HHAAAHH (the 2–3 format) for the NBA and HHAAHAH (the 2–2 format) for the NHL. Assuming that games are independent Bernoulli trials, we show that each teams probability of winning the series is unaffected by HOME-AWAY sequencing but that the average length of a series is affected by HOME-AWAY sequencing. For instance, if one team is stronger than the other in both buildings, the 2–3 format has a higher expected number of games than does the 2–2 format. The results follow from simple probability calculations. The sporting context makes this an interesting exercise for students of statistics.

Four (Pathological) Examples in Asymptotic Statistics

Abstract Four simple examples illustrating varieties of pathological asymptotic behavior are presented. The examples are based on some recent work on l 1 asymptotics. The examples have some pedagogical value in clarifying the role of certain standard regularity conditions.

A p -subset property of L 1 and regression quantile estimates

Abstract The L 1 estimate for the p -variable linear model has the well known property of fitting p observations exactly. A less well known property is that certain subsets of p observations will not be fit by the L 1 estimate for any realization of the dependent variables. This property is shown to generalize to other regression quantiles and to the set of all regression quantiles. This identifies subsets of the data which seem to be unimportant. The analog of the property for the location submodel is a situation where one observation, say the first, would not be any quantile for any sample. The implications of the property for other estimates which are based on p -observation subsets are discussed, but the property is considered mainly because it seems strange.

Robust Strategies for Quantitative Investment Management

We show how quantile estimation combined with robust methods can be used in quantitative investment management. A portfolio manager uses a quantitative model to select securities. The objective is to outperform a benchmark portfolio, subject to risk constraints. Traditional stock selection models express expected returns as a function of factors where all parts of the return distribution are affected similarly. This is subsumed by the quantile approach in which a stock’s entire return distribution is a conditional function of factors. Robust methods insure that our estimates do not depend on a small subset of the data. Regression quantile estimates then detect the potentially different impact of factors at the center and tails of the return distribution. This is illustrated in assessing a model’s forecasting accuracy while controlling for return differences between economic sectors. We are thereby able to detect forecasting properties that would have been missed by a conventional analysis of the data.

A property of the observations fit by the extreme regression quantiles

Abstract The extreme regression quantile estimates have recently been proposed as a computationally fast method for detecting discrepant points and constructing high breakdown estimates for the linear model. Further support for this proposal comes from the following property. The convex hull of the observations which exactly fit an extreme regression quantile estimate must contain the mean values of the design variables. Hence, the observations identified by an extreme regression quantile cannot all be clustered on one side of the overall mean values of the design variables.

Conditional quantile regression models of melanoma tumor growth curves for assessing treatment effect in small sample studies.

Tumor growth curves provide a simple way to understand how tumors change over time. The traditional approach to fitting such curves to empirical data has been to estimate conditional mean regression functions, which describe the average effect of covariates on growth. However, this method ignores the possibility that tumor growth dynamics are different for different quantiles of the possible distribution of growth patterns. Furthermore, typical individual preclinical cancer drug study designs have very small sample sizes and can have lower power to detect a statistically significant difference in tumor volume between treatment groups. In our work, we begin to address these issues by combining several independent small sample studies of an experimental cancer treatment with differing study designs to construct quantile tumor growth curves. For modeling, we use a Penalized Fixed Effects Quantile Regression with added study effects to control for study differences. We demonstrate this approach using data from a series of small sample studies that investigated the effect of a naturally derived biological peptide, P28, on tumor volumes in mice grafted with human melanoma cells. We find a statistically significant quantile treatment effect on tumor volume trajectories and baseline values. In particular, the experimental treatment and a corresponding conventional chemotherapy had different effects on tumor growth by quantile. The conventional treatment, Dacarbazine (DTIC), tended to inhibit growth for smaller quantiles, while the experimental treatment P28 produced slower rates of growth in the upper quantiles, especially in the 95th quantile.