scholarly journals Statistical sequential hypotheses testing on para meters of probability distributions of random binary data

2021 ◽  
pp. 60-66
Author(s):  
Alexey Yu. Kharin
Keyword(s):  
Binary Data ◽  
Random Number ◽  
Probability Distributions ◽  
Mathematical Problem ◽  
Decision Rules ◽  
Performance Characteristics ◽  
Sequential Testing ◽  
Computer Data ◽  
Independent Observations ◽  
Error Probabilities

An important mathematical problem of computer data analysis – the problem of statistical sequential testing of simple hypotheses on parameters of probability distributions of observed binary data – is considered in the paper. This problem is being solved for two models of observation: for independent observations and for homogeneous Markov chains. Explicit expressions of the sequential tests statistics are derived, transparent for interpretation and convenient for computer realisation. An approach is developed to calculate the performance characteristics – error probabilities and mathematical expectations of the random number of observations required to guarantee the requested accuracy for decision rules. Asymptotic expansions for the mentioned performance characteristics are constructed under «contamination» of the probability distributions of observed data.

scholarly journals Robustness Analysis in Sequential Statistical Decisions

2020 ◽  
Vol 49 (4) ◽  
pp. 69-75
Author(s):  
Alexey Kharin
Keyword(s):  
Decision Making ◽  
Decision Rules ◽  
Robustness Analysis ◽  
Test Construction ◽  
Performance Characteristics ◽  
Statistical Decision ◽  
Statistical Decision Making ◽  
Composite Hypotheses ◽  
Simple And Composite Hypotheses ◽  
Error Probabilities

The sequential statistical decision making is considered. Performance characteristics (error probabilities and expected sample sizes) for the sequential statistical decision rules (tests) are analysed. Both cases of simple and composite hypotheses are considered. Asymptotic expansions under distortions are constructed for the performance characteristics enabling robust sequential test construction.

scholarly journals Invariant Bipartite Random Graphs on Rd

2014 ◽  
Vol 51 (3) ◽  
pp. 769-779
Author(s):  
Fabio Lopes
Keyword(s):  
Point Processes ◽  
Random Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Stable Marriage ◽  
Simple Point ◽  
Stable Marriage Problem ◽  
Blue Point ◽  
Translation Invariant ◽  
Positive Integers

Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.

Rate of convergence for computing expectations of stopping functionals of an α-mixing process

1998 ◽  
Vol 30 (2) ◽  
pp. 425-448
Author(s):  
Mohamed Ben Alaya ◽  
Gilles Pagès
Keyword(s):  
Monte Carlo ◽  
Probability Space ◽  
Random Number ◽  
Shift Operator ◽  
Probability Distributions ◽  
Random Number Generator ◽  
Stopping Time ◽  
Mixing Process ◽  
Canonical Process ◽  
Image Position

The shift method consists in computing the expectation of an integrable functional F defined on the probability space ((ℝd)N, B(ℝd)⊗N, μ⊗N) (μ is a probability measure on ℝd) using Birkhoff's Pointwise Ergodic Theorem, i.e. as n → ∞, where θ denotes the canonical shift operator. When F lies in L2(FT, μ⊗N) for some integrable enough stopping time T, several weak (CLT) or strong (Gàl-Koksma Theorem or LIL) converging rates hold. The method successfully competes with Monte Carlo. The aim of this paper is to extend these results to more general probability distributions P on ((ℝd)N, B(ℝd)⊗N), namely when the canonical process (Xn)n∊N is P-stationary, α-mixing and fulfils Ibragimov's assumption for some δ > 0. One application is the computation of the expectation of functionals of an α-mixing Markov Chain, under its stationary distribution Pν. It may both provide a better accuracy and save the random number generator compared to the usual Monte Carlo or shift methods on independent innovations.

scholarly journals Performance and Robustness Analysis of Sequential Hypotheses Testing for Time Series with Trend

2017 ◽  
Vol 46 (3-4) ◽  
pp. 23-36 ◽  
Author(s):  
Alexey Kharin ◽  
Ton That Tu
Keyword(s):  
Time Series ◽  
Asymptotic Expansions ◽  
Robustness Analysis ◽  
Numerical Results ◽  
Sequential Testing ◽  
Hypotheses Testing ◽  
Analytic Expressions ◽  
Error Probabilities

The problem of sequential testing of simple hypotheses for time series with a trend is considered. Analytic expressions and asymptotic expansions for error probabilities and expected numbers of observations are obtained. Robustness analysis is performed. Numerical results are given.

THE PARTICLE SWARM AS COLLABORATIVE SAMPLING OF THE SEARCH SPACE

2007 ◽  
Vol 10 (supp01) ◽  
pp. 191-213 ◽  
Author(s):  
JAMES KENNEDY
Keyword(s):  
Social Psychology ◽  
Random Number ◽  
Probability Distributions ◽  
Particle Swarm ◽  
Search Space ◽  
Particle Swarm Algorithm ◽  
Test Functions ◽  
Excellent Performance ◽  
Random Number Generators ◽  
Swarm Algorithm

The particle swarm algorithm uses principles derived from social psychology to find optimal points in a search space. The present paper decomposes and reinterprets the particle swarm in order to discover new ways of implementing the algorithm. Some essential characteristics of the method are illuminated, and some inessential features are discarded. Various new forms are tested and found to perform well on a suite of test functions. In particular, it is shown that the traditional trajectory formulas can be replaced with random number generators sampling from various symmetrical probability distributions. The excellent performance of these new versions demonstrates that the strength of the algorithm is in the interactions of the particles, rather than in their behavior as individuals.

scholarly journals Clustering Objects Described by Juxtaposition of Binary Data Tables

2008 ◽  
Vol 2008 ◽  
pp. 1-13
Author(s):  
Amar Rebbouh
Keyword(s):  
Survey Data ◽  
Binary Data ◽  
Probability Distributions ◽  
Hierarchical Classification ◽  
Classification Method ◽  
Physical Systems ◽  
Numerical Example ◽  
Data Tables ◽  
Ascending Hierarchical Classification ◽  
Data Matrices

This paper seeks to develop an allocation of 0/1 data matrices to physical systems upon a Kullback-Leibler distance between probability distributions. The distributions are estimated from the contents of the data matrices. We discuss an ascending hierarchical classification method, a numerical example and mention an application with survey data concerning the level of development of the departments of a given territory of a country.

scholarly journals Regenerative Partition Structures

10.37236/1869 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Alexander Gnedin ◽  
Jim Pitman
Keyword(s):  
Random Number ◽  
Probability Distributions ◽  
Random Allocation ◽  
Random Set ◽  
Random Partition ◽  
Independent Increments ◽  
Increasing Process ◽  
Composition Structure ◽  
Partition Structure ◽  
Two Parameters

A partition structure is a sequence of probability distributions for $\pi_n$, a random partition of $n$, such that if $\pi_n$ is regarded as a random allocation of $n$ unlabeled balls into some random number of unlabeled boxes, and given $\pi_n$ some $x$ of the $n$ balls are removed by uniform random deletion without replacement, the remaining random partition of $n-x$ is distributed like $\pi_{n-x}$, for all $1 \le x \le n$. We call a partition structure regenerative if for each $n$ it is possible to delete a single box of balls from $\pi_n$ in such a way that for each $1 \le x \le n$, given the deleted box contains $x$ balls, the remaining partition of $n-x$ balls is distributed like $\pi_{n-x}$. Examples are provided by the Ewens partition structures, which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator (that is an increasing process with stationary independent increments). The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator, for which exponent an integral representation is provided by the Lévy-Khintchine formula. The extended Ewens family of partition structures, previously studied by Pitman and Yor, with two parameters $(\alpha,\theta)$, is characterised for $0 \le \alpha < 1$ and $\theta >0$ by regeneration with respect to deletion of each distinct part of size $x$ with probability proportional to $(n-x)\tau+x(1-\tau)$, where $\tau = \alpha/(\alpha+\theta)$.

scholarly journals Robustness Evaluation in Sequential Testing of Composite Hypotheses

2016 ◽  
Vol 37 (1) ◽  
Author(s):  
Alexey Kharin
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Likelihood Ratio ◽  
Asymptotic Expansions ◽  
Likelihood Ratio Statistic ◽  
Sequential Testing ◽  
New Approach ◽  
Generalized Likelihood Ratio ◽  
Composite Hypotheses ◽  
Error Probabilities

The problem of sequential testing of composite hypotheses is considered. Asymptotic expansions are constructed for the conditional error probabilities and expected sample sizes under “contamination” of the probability distribution of observations. To obtain these results a new approach based on approximation of the generalized likelihood ratio statistic by a specially constructed Markov chain is proposed. The approach is illustrated numerically.

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