Howard G. Tucker

Abnormalities in hair trace elements as indicators of aberrant behavior

There are long-standing viewpoints that impulsive and violent behavior may stem from brain dysfunction or damage secondary to head injury, disease, or toxic chemical substances. This research has aimed to examine the relationship between potentially toxic metals and aberrant behavior, especially violent activity, through the nonintrusive technique of hair analysis for trace elements. In an initial study, phase I, it was not possible to replicate findings of others who reported high levels of lead, cadmium, and copper in violent offenders. However, high levels of manganese were found in prison versus control groups. In phase II, the possibility of artifactual results arising from prison cooking utensils was controlled for by sampling early after incarceration. Phase III was included to substantiate the initial post hoc findings in an additional jail population. In both latter phases, significantly elevated manganese levels were found in the hair of violent versus nonviolent subjects (P less than .0001). A review of the effects of manganese at deficient and toxic levels does not provide a simple answer as to why manganese levels are elevated in the hair of individuals who have been incarcerated for violent behavior. Our study does not implicate the prison environment or soaps and shampoos used in California prisons. Other factors, such as alcohol, dietary, or psychosocial factors, might influence manganese levels in hair, or any of these factors might function in combination with mild manganese toxicity to contribute to aberrant behavior.

Confidence intervals for the number of unseen types

This paper finds the unique maximum likelihood estimator of, and conservative confidence intervals for, the unknown number of different coupons in the coupon collectors problem. This problem is also known as the problem of estimating the abundance of wildlife. The techniques developed here can be easily implemented, are valid without regard to sample size, and validate previous methods based on large sample theory when those methods apply.

Pareto Tail Index Estimation Revisited

Abstract An estimator of the tail index of a Pareto distribution is given that is based on the use of the probability integral transform. This new estimator provides performance that is comparable to the best robust estimators, while retaining conceptual and computational simplicity. A tuning parameter in the new estimator can be adjusted to control the tradeoff between robustness and efficiency. The method used to compute the estimator also can be used to find a confidence interval for the tail index that is guaranteed to have the nominal confidence level for any given sample size. Guidelines for the use of the new estimator are provided.

Convergence in law of random sums with non-random centering

We extend results obtained in Kruglov,(7) and Finkelstein and Tucker(3) to obtain necessary and sufficient conditions for convergence in law of random sums of non-identically distributed independent random variables under non-random centering. Thei.i.d. case is also considered for random variables attracted to a stable law. Necessary and sufficient conditions for convergence in law of these random variables under non-random centering, and in some cases, under non-random norming, are also obtained. The distribution functions for the limit laws are determined as well, generalizing results of Robbins.(10)

Limit distributions of sums of m-dependent Bernoulli random variables

SummaryThe set of limit distributions of row sums of a triangular array of Bernoulli random variables which is strictly stationary and m-dependent in each row is characterized. Necessary and sufficient conditions for the convergence of the row sums to a given limit distribution are found. The case of convergence to a Poisson distribution is given special attention.

Conservative confidence intervals for a single parameter

The method of constructing confidence intervals from hypothesis tests is studied in the case in which there is a single unknown parameter and is proved to provide confidence intervals with coverage probability that is at least the nominal level. The confidence intervals obtained by the method in several different contexts are seen to compare favorably with confidence intervals obtained by traditional methods. The traditional intervals are seen to have coverage probability less than the nominal level in several instances, This method can be applied to all confidence interval problems and reduces to the traditional method when an exact pivotal statistic is known.

Mathematical methods in sample surveys

This book is about both the mathematics of sample surveys and about sample surveys. The mathematics is both elementary and rigorous. It is suitable for a one year junior-senior level course for mathematics and statistics majors as well as for students in the social sciences who are not handicapped by a fear of proofs in mathematics. It requires no previous knowledge of statistics, and it could actually serve as an introduction to statistics. A sizeable part of the book covers the discrete probability needed for the sampling methods covered. Topics then covered are: simple random sampling, sampling with unequal probabilities, linear relationships, stratified sampling, cluster sampling and two-stage sampling.

The limit distribution of the number of rare mutants.

We study the number of mutants in a mutation process in which reverse mutation is allowed and in which both normal and mutant organisms reproduce at the same rate. Under a mild side condition on the rate of forward mutation we find necessary and sufficient conditions for the number of mutants to converge in distribution. We find the probability generating function of the limit distribution, when it exists. We present an example which shows that the mild side condition cannot be relaxed.

EXTINGUISHING THE DISTINGUISHED LOGARITHM PROBLEMS

The existence, uniqueness, and convergence properties of distinguished logarithms of multivariate characteristic functions are proved.

The distribution of the likelihood ratio for additive processes

Let {X(t), 0 ≤ t ≤ T} and {Y(t), 0 ≤ t ≤ T} be two additive processes over the interval [0, T] which, as measures over D[0, T], are absolutely continuous with respect to each other. Let μX and μY be the measures over D[0, T] determined by the two processes. The characteristic function of ln(dμXdμY) with respect to μY is obtained in terms of the determining parameters of the two processes.