JOURNAL OF THE IRANIAN STATISTICAL SOCIETY (JIRSS)
Year:2003 | Volume:2 | Issue:1|Papers Count:5

Bioinformatics is an emerging field of science emphasizing the application of mathematics, statistics, and informatics to study and analysis of very large molecular biological (mostly, genetic and genomic) systems (data sets). In a comparatively broader setup of large biological systems without necessarily having a predominant genetic undercurrent, and having genesis in biometry to biostatistics, biostochastics has evolved as the primary vehicle for the much needed statistical reasoning. It is intended to point out the genuine need for statistical reasoning in this evolving interdisciplinary field, and in that way, to appraise the limitations of current (mostly, algorithm based) statistical resolutions.

We consider the second-order asymptotic properties of the bootstrap of L1 regression estimators by looking at the deference between the L1 estimator and its first-order approximation, where the latter is the minimizer of a quadratic approximation to the L1 objective function. It is shown that the bootstrap distribution of the normed deference does not converge (either in probability or with probability 1) to the "correct" limiting distribution but rather converges in distribution to a random distribution. A characterization of this random distribution is given. Some applications and extensions are given.

Given an i.i.d. sequence of n letters from a finite alphabet, we consider the length of the longest run of any letter. In the equiprobable case, results for this run turn out to be closely related to the well-known results for the longest run of a given letter. For coin-tossing, tail probabilities are compared for both kinds of runs via Poisson approximation

This paper reviews Polya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • Polya-Eggenberger’s urn• Bernard Friedman’s urn• Generalized Polya urns• Extended urn schemes• Invertible urn schemesConnections to random trees are surveyed. Numerous applications to trees common in computer science are discussed, including:• Binary search trees• Fringe-balanced trees• m-ary search trees• 2–3 trees• Paged binary trees• Bucket quad trees• Bucket k–d treesThe applications also include various types of recursive trees:• Standard recursive trees• Pyramids• Plane-oriented recursive trees• Phylogenetic trees• Bucket recursive trees• SproutsLimit distributions, and phase changes therein are presented within the unifying theme of Polya urn models.

This paper establishes the first four moment expansions to the order o(a-1) of Sta/Ö ta, where S´n =åni=1 Vi i=1 Yi is a simple random walk with E(Yi) = 0, and tα is a stopping time given by ta = inf {n ³1 : n + Sn + zn > a} where Sn = ån i= Xi is another simple random walk with E(Xi) = 0, and { zn, n³1} is a sequence of random variables satisfying certain assumptions. These moment expansions complement the classical central limit theorem for a random number of i.i.d. random variables when the random number has the form ta, which arises from many sequential statistical procedures. They can be used to correct higher order bias and/or skewness in S´ta/Öta to make asymptotic approximation more accurate for small and moderate sample sizes.