Ishay Weissman

Estimation of Parameters and Large Quantiles Based on the k Largest Observations

Abstract We consider an estimation problem when only the k largest observations of a sample of size n are available. It is assumed that the underlying distribution function F belongs to the domain of attraction of a known extreme-value distribution and that k remains fixed as n → ∞. We present estimators for the location and scale parameters and for p-quantiles of F, where p is of the form 1 — c/n (c fixed). These estimators are either asymptotically maximum likelihood or minimum variance.

On blocks and runs estimators of the extremal index

Abstract Given a sample from a stationary sequence of random variables, we study the blocks and runs estimators of the extremal index. Conditions are given for consistency and asymptotic normality of these estimators. We show that moment restrictions assumed by Hsing (Stochast. Process. Appl. 37(1), 117–139; Ann. Statist. 21(4), 2043-2021) may be relaxed if a stronger mixing condition holds. The CLT for the runs estimator seems to be proven for the first time.

Confidence intervals for the threshold parameter. ii: unknown shape parameter

Confidence intervals based on the lower extreme values of a large sample are constructed for the threshold parameter (unknown minimum-life) of a life distribution. The confidence level is exact for a certain class of beta distributions and asymptotically exact for a wide class of distributions often used in life testing.

Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes

A uniform random vector over a simplex is generated. An explicit expression for the first moment of its largest spacing is derived. The result is used in a proposed diagnostic tool which examines the validity of random number generators. It is then shown that the first moment of the largest uniform spacing is related to the dependence measure of random vectors following any extreme value distribution. The main result is proved by a geometric proof as well as by a probabilistic one.

Comparison of estimation methods in extreme value theory

In this study we compare three estimators of the extreme value index: Pickands estimator, the moment estimator and a maximum likelihood estimator. The estimators are explored both theoretically and by Monte Carlo simulation. We obtain two estimators for large quantiles using Pickands and the maximum likelihood estimators. The latter and one based on the moment estimator are then compared through simulation.

A new method for stoichiometric analysis of proteins in complex mixture — reevaluation of the stoichiometry of E. coli ribosomal proteins

A novel way is presented for determination of the stoichiometry of ribosomal proteins in the ribosome. The 70S E. coli r-proteins, completely separated on a two-dimensional gel system, were used throughout our experiments. The method is based on our previous observation that the amount of Coomassie R bound to a protein molecule is directly proportional to the number of positive charges on that protein. By plotting the amount of bound Coomassie as a function of the number of positive charges of each r-protein, and relating the experimental amount of the dye bound to each r-protein to the value obtained from the linear regression line based on all (a total of some 50 proteins), one can obtain the molar concentration of every protein in the ribosome. A parallel experiment can be carried out, which relates the radioactivity contributed by 3H-labeled amino acid in each r-protein to its amino acid content in that molecule. The two sets of data, which are completely independent of each other, are well correlated. Further verification of the validity of our procedure is provided by the fact that we found the known proportions of four copies of L7/L12 and one copy of S6 per ribosome. The rationale behind the present study was our finding that recalculation of Hardys data (Hardy, S.J.S. (1975) Mol. Gen. Genet. 140, 253-274), with the accurate molecular weight value of the r-proteins provided by Giri et al. (Adv. Protein Chem. (1984) 36, 1-78), raises some doubt with regard to his experimental results, although we agree with his final conclusion that E. coli ribosome is homogeneous with respect to its proteins.

Confidence intervals for the threshold parameter

Confidence intervals for the threshold parameter (guarantee-life ) are considered. The first k failure-times from a sample of size n are observed. Under the assumption that as n →∞ the first failure-time is attracted to the Weibull distribution, confidence intervals based on the observed range are constructed. It is shown that as k(k ≥ 2) increases the expected length of the confidence interval is substantially reduced. However, when k = 10 (or 20 in some cases) the expected length is near its minimum value.

On convergence of types and processes in Euclidean space

AbstractLet ℰ be an Euclidean space; Yn, Z, U random vectors in ℰ; hn, gnaffine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that gn

Extreme value theory applied to document retrieval from large collections

g_n Y_n \xrightarrow{D}Z