arXiv subjects

Andrey Novikov

Publications and source records attributed to Andrey Novikov.

11 recordsLinked to original sources

arXiv2024

A class of sequential multi-hypothesis tests

In this paper, we deal with sequential testing of multiple hypotheses. In the general scheme of construction of optimal tests based on the backward induction, we propose a modification which provides a simplified (generally speaking, suboptimal) version of the optimal test, for any particular criterion of optimization. We call this DBC version (the one with Dropped Backward Control) of the optimal test. In particular, for the case of two simple hypotheses, dropping backward control in the Bayesian test produces the classical sequential probability ratio test (SPRT). Similarly, dropping backward control in the modified Kiefer-Weiss solutions produces Lorden's 2-SPRTs . In the case of more than two hypotheses, we obtain in this way new classes of sequential multi-hypothesis tests, and investigate their properties. The efficiency of the DBC-tests is evaluated with respect to the optimal Bayesian multi-hypothesis test and with respect to the matrix sequential probability ratio test (MSPRT) by Armitage. In a multihypothesis variant of the Kiefer-Weiss problem for binomial proportions the performance of the DBC-test is numerically compared with that of the exact solution. In a model of normal observations with a linear trend, the performance of of the DBC-test is numerically compared with that of the MSPRT. Some other numerical examples are presented. In all the cases the proposed tests exhibit a very high efficiency with respect to the optimal tests (more than 99.3\% when sampling from Bernoulli populations) and/or with respect to the MSPRT (even outperforming the latter in some scenarios).

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arXiv2022

A Numerical Approach to Sequential Multi-Hypothesis Testing for Bernoulli Model

In this paper we deal with the problem of sequential testing of multiple hypotheses. The main goal is minimizing the expected sample size (ESS) under restrictions on the error probabilities. We take, as a criterion of minimization, a weighted sum of the ESS's evaluated at some points of interest in the parameter space aiming at its minimization under restrictions on the error probabilities. We use a variant of the method of Lagrange multipliers which is based on the minimization of an auxiliary objective function (called Lagrangian) combining the objective function with the restrictions, taken with some constants called multipliers. Subsequently, the multipliers are used to make the solution comply with the restrictions. We develop a computer-oriented method of minimization of the Lagrangian function, that provides, depending on the specific choice of the parameter points, optimal tests in different concrete settings, like in Bayesian, Kiefer-Weiss and other settings. To exemplify the proposed methods for the particular case of sampling from a Bernoulli population we develop a set of computer algorithms for designing sequential tests that minimize the Lagrangian function and for the numerical evaluation of test characteristics like the error probabilities and the ESS, and other related. We implement the algorithms in the R programming language. The program code is available in a public GitHub repository. For the Bernoulli model, we made a series of computer evaluations related to the optimality of sequential multi-hypothesis tests, in a particular case of three hypotheses. A numerical comparison with the matrix sequential probability ratio test is carried out. A method of solution of the multi-hypothesis Kiefer-Weiss is proposed, and is applied for a particular case of three hypotheses in the Bernoulli model.

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arXiv2022

Group sequential hypothesis tests with variable group sizes: optimal design and performance evaluation

In this paper, we propose a computer-oriented method of construction of optimal group sequential hypothesis tests with variable group sizes. In particular, for independent and identically distributed observations we obtain the form of optimal group sequential tests which turn to be a particular case of sequentially planned probability ratio tests (SPPRTs, Schmitz, 1993) . Formulas are given for computing the numerical characteristics of general SPPRTs, like error probabilities, average sampling cost, etc. A numerical method of designing the optimal tests and evaluation of the performance characteristics is proposed, and computer algorithms of its implementation are developed. For a particular case of sampling from a Bernoulli population, the proposed method is implemented in R programming language, the code is available in a public GitHub repository. The proposed method is compared numerically with other known sampling plans.

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arXiv2021

A computational approach to the Kiefer-Weiss problem for sampling from a Bernoulli population

We present a computational approach to solution of the Kiefer-Weiss problem. Algorithms for construction of the optimal sampling plans and evaluation of their performance are proposed. In the particular case of Bernoulli observations, the proposed algorithms are implemented in the form of R program code. Using the developed computer program, we numerically compare the optimal tests with the respective sequential probability ratio test (SPRT) and the fixed sample size test, for a wide range of hypothesized values and type I and type II errors. The results are compared with those of D.~Freeman and L.~Weiss (Journal of the American Statistical Association, 59(1964)). The R source code for the algorithms of construction of optimal sampling plans and evaluation of their characteristics is available at https://github.com/tosinabase/Kiefer-Weiss.

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arXiv2021

Optimal Group-Sequential Tests with Groups of Random Size

We consider sequential hypothesis testing based on observations which are received in groups of random size. The observations are assumed to be independent both within and between the groups. We assume that the group sizes are independent and their distributions are known, and that the groups are formed independently of the observations. We are concerned with a problem of testing a simple hypothesis against a simple alternative. For any (group-) sequential test, we take into account the following three characteristics: its type I and type II error probabilities and the average cost of observations. Under mild conditions, we characterize the structure of sequential tests minimizing the average cost of observations among all sequential tests whose type I and type II error probabilities do not exceed some prescribed levels.

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arXiv2010

Locally most powerful sequential tests of a simple hypothesis vs. One-sided alternatives for independent observations

Let $X_1,X_2,..., X_n,...$ be a stochastic process with independent values whose distribution $P_θ$ depends on an unknown parameter $θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. The problem of testing $H_0:$ $θ=θ_0$ vs. a composite alternative $H_1:$ $θ>θ_0$ is considered, where $θ_0\inΘ$ is a fixed value of the parameter. The main objective of this work is the characterization of the structure of the locally most powerful (in the sense of Berk) sequential tests in this problem.

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arXiv2009

Optimal sequential procedures with Bayes decision rules

In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. The problem is to minimize an average sample number given that Bayesian risk due to incorrect decision does not exceed some given bound. We characterize the form of optimal sequential stopping rules in this problem. In particular, we have a characterization of the form of optimal sequential decision procedures when the Bayesian risk includes both the loss due to incorrect decision and the cost of observations.

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arXiv2009

Locally most powerful sequential tests of a simple hypothesis vs one-sided alternatives

Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $θ=θ_0$ versus a composite alternative $H_1:$ $θ>θ_0$, where $θ_0\inΘ$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(ψ,ϕ)$ with a (randomized) stopping rule $ψ$ and a (randomized) decision rule $ϕ$ let $α(ψ,ϕ)$ be the type I error probability, $\dot β_0(ψ,ϕ)$ the derivative, at $θ=θ_0$, of the power function, and $\mathscr N(ψ)$ an average sample number of the test $(ψ,ϕ)$. Then we are concerned with the problem of maximizing $\dot β_0(ψ,ϕ)$ in the class of all sequential tests such that $$ α(ψ,ϕ)\leq α\quad{and}\quad \mathscr N(ψ)\leq \mathscr N, $$ where $α\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(ψ)$ is calculated under some fixed (not necessarily coinciding with one of $P_θ$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.

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arXiv2008

Sequential multiple hypothesis testing in presence of control variables

Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $θ$, and we consider the problem of testing multiple hypotheses $H_1: θ=θ_1$, $H_2: θ=θ_2, ...$, $H_k: θ=θ_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another value $X_2$ for the control variable is chosen, and $Y_2$ as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations $Y_1, Y_2,..., Y_n$ are independent, given controls $X_1,X_2,..., X_n$, $n=1,2,...$.

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arXiv2008

Optimal sequential testing of two simple hypotheses in presence of control variables

Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ can be used. The distribution of $Y$ depends on some unknown parameter $θ$, and we consider the classical problem of testing a simple hypothesis $H_0: θ=θ_0$ against a simple alternative $H_1: θ=θ_1$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, we choose another value $X_2$ for the control variable, and observe $Y_2$ as a response, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favour of $H_0$ or $H_1$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential procedures, based on this type of data, for testing a simple hypothesis against a simple alternative.

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arXiv2008

Optimal sequential multiple hypothesis tests

This work deals with a general problem of testing multiple hypotheses about the distribution of a discrete-time stochastic process. Both the Bayesian and the conditional settings are considered. The structure of optimal sequential tests is characterized.

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