Haimeng Zhang

Efficiency of the maximum partial likelihood estimator for nested case control sampling

This invention relates to a method for forming an aqueous polymeric dispersion incorporating a wax. To form the aqueous dispersion, a polymer is formed by polymerizing at least one ethylenically unsaturated monomer under conditions for forming a polymer suspended in an aqueous medium in the presence of said wax. The polymerization of the monomer in the presence of a wax in finely divided form permits the formation of a stable dispersion having excellent properties. The surfactant system is a multiple system consisting of a nonionic and/or anionic surfactant and/or an alkali metal salt of a fatty acid ester.

Information and asymptotic efficiency of the case-cohort sampling design in Cox's regression model

Efficiencies of the maximum pseudolikelihood estimator and a number of related estimators for the case-cohort sampling design in the proportional hazards regression model are studied. The asymptotic information and lower bound for estimating the parametric regression parameter are calculated based on the effective score, which is obtained by determining the component of the parametric score orthogonal to the space generated by the infinite-dimensional nuisance parameter. The asymptotic distributions of the maximum pseudolikelihood and related estimators in an i.i.d, setting show that these estimators do not achieve the computed asymptotic lower bound. Simple guidelines are provided to determine in which instances such estimators are close enough to efficient for practical purposes.

Asymptotic efficiency of estimation in the partial Koziol-Green model

In this paper, the convolution theorem and the minimax theorem for estimating the survival function in the partial Koziol–Green model (PKG) are presented. The result indicates that the partial Abdushukurov–Cheng–Lin (ACL) estimator in the PKG model is asymptotically efficient in the sense of being the least dispersed regular estimator. Consequently, the calculation shows that the ACL estimator in the KG model is also asymptotically efficient.

A Berry-Esseen bound for the lightbulb process

In the so-called lightbulb process, on days r = 1,…,n, out of n lightbulbs, all initially off, exactly r bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With X the number of bulbs on at the terminal time n, an even integer, and μ = n/2, σ2 = var(X), we have sup z ∈ R | P((X - μ)/σ ≤ z) - P(Z ≤ z) | ≤ nΔ̅0/2σ2 + 1.64n/σ3 + 2/σ, where Z is a standard normal random variable and Δ̅0 = 1/2√n + 1/2n + e−n/2/3 for n ≥ 6, yielding a bound of order O(n −1/2) as n → ∞. A similar, though slightly larger bound, holds for odd n. The results are shown using a version of Steins method for bounded, monotone size bias couplings. The argument for even n depends on the construction of a variable X s on the same space as X that has the X-size bias distribution, that is, which satisfies E[X g(X)] = μE[g(X s )] for all bounded continuous g, and for which there exists a B ≥ 0, in this case B = 2, such that X ≤ X s ≤ X + B almost surely. The argument for odd n is similar to that for even n, but one first couples X closely to V, a symmetrized version of X, for which a size bias coupling of V to V s can proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

A discrete probability problem in card shuffling

Abstract There are n cards serially numbered from 1 to n. The cards are shuffled and placed in a line one after the other on top of a table with faces up. The numbers on the faces are read from left to right. If there are consecutive numbers in increasing order of magnitude the corresponding cards are merged into one. After the merger, the cards are numbered serially from one to whatever the number of cards we now have. The cards are shuffled and placed in a line one after another on top of the table with faces up. The process continues until we have only one card left. In this paper, we develop a probabilistic recurrence relation approach to obtain the mean, variance, and distribution of the number of shuffles needed. A Markov chain formulation and its properties are discussed in the paper as well.

On Nelson-Aalen type estimation in the partial Koziol-Green model

In this paper, a Nelson–Aalen (NA) type estimator is derived and its sample properties are compared with the partial Abdushukurov–Cheng–Lin (PACL), generalized maximum likelihood (GMLE), and Kaplan–Meier (KM) estimators under the partial Koziol–Green model. These comparisons are made through Monto Carlo simulations under various sample sizes. The results indicate that the NA estimator always performs better than the KM estimator and is competitive with other estimators. Moreover, the PACL, GMLE, and NA estimators are shown to be asymptotically equivalent.

On weak consistency in linear models with equi-correlated random errors

A new estimator in linear models with equi-correlated random errors is postulated. Consistency properties of the proposed estimator and the ordinary least squares estimator are studied. It is shown that the new estimator has smaller variance than the usual least squares estimator under some mild conditions. In addition, it is observed that the new estimator tends to be weakly consistent in many cases where the usual least squares estimator is not.