Effects of parameters choice in random sequence comparison

The aim of this thesis is to generalize the results from the article [LMT] for the score of mismatched letters. This is an introductory work and it includes complete theoretical build-up of random sequence comparison, depending on parameters. At the end we prove the large deviations inequality for the proportions of mismatches. It gives probability for proportions of mismatches being near the asymptotic proportions of mismatches. This thesis also includes a fair bit of practical examples and an introduction into algorithms, used for sequence comparison

Riemanni dzeetafunktsioon

Bakalaureusetöö eesmärk on uurida Riemanni dzeetafunktsiooni omadusi. Alustame Baseli ülesandest ning põhilistest Riemanni dzeetafunktsiooni omadustest. Töö põhitulemuseks on Riemanni dzeetafunktsiooni funktsionaalvõrrand. Töö lõpus tutvustatakse funktsiooni nullkohti ning Riemanni hüpoteesi

Cramér-von Mises tests for change points

We study two nonparametric tests of the hypothesis that a sequence of independent observations is identically distributed against the alternative that at a single change point the distribution changes. The tests are based on the Cramér–von Mises two-sample test computed at every possible change point. One test uses the largest such test statistic over all possible change points; the other averages over all possible change points. Large sample theory for the average statistic is shown to provide useful p-values much more quickly than bootstrapping, particularly in long sequences. Power is analyzed for contiguous alternatives. The average statistic is shown to have limiting power larger than its level for such alternative sequences. Evidence is presented that this is not true for the maximal statistic. Asymptotic methods and bootstrapping are used for constructing the test distribution. Performance of the tests is checked with a Monte Carlo power study for various alternative distributions

Conditional Monte Carlo revisited

Conditional Monte Carlo refers to sampling from the conditional distribution of a random vector X given the value T ( X ) = t for a function T ( X ) . Classical conditional Monte Carlo methods were designed for estimating conditional expectations of functions ϕ ( X ) by sampling from unconditional distributions obtained by certain weighting schemes. The basic ingredients were the use of importance sampling and change of variables. In the present paper we reformulate the problem by introducing an artificial parametric model in which X is a pivotal quantity, and next representing the conditional distribution of X given T ( X ) = t within this new model. The approach is illustrated by several examples, including a short simulation study and an application to goodness-of-fit testing of real data. The connection to a related approach based on sufficient statistics is briefly discussed