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Testing for unspecified periodicities in binary time series

Finn Schmidtke, Mathias Vetter

Abstract

Given independent random variables $Y_1, \ldots, Y_n$ with $Y_i \in \{0,1\}$ we test the hypothesis whether the underlying success probabilities $p_i$ are constant or whether they are periodic with an unspecified period length of $r \ge 2$. The test relies on an auxiliary integer $d$ which can be chosen arbitrarily, using which a new time series of length $d$ is constructed. For this new time series, the test statistic is derived according to the classical $g$ test by Fisher. Under the null hypothesis of a constant success probability it is shown that the test keeps the level asymptotically, while it has power for most alternatives, i.e. typically in the case of $r \ge 3$ and where $r$ and $d$ have common divisors.

Testing for unspecified periodicities in binary time series

Abstract

Given independent random variables with we test the hypothesis whether the underlying success probabilities are constant or whether they are periodic with an unspecified period length of . The test relies on an auxiliary integer which can be chosen arbitrarily, using which a new time series of length is constructed. For this new time series, the test statistic is derived according to the classical test by Fisher. Under the null hypothesis of a constant success probability it is shown that the test keeps the level asymptotically, while it has power for most alternatives, i.e. typically in the case of and where and have common divisors.

Paper Structure

This paper contains 14 sections, 8 theorems, 56 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Finite sample properties
  4. Simulation study
  5. Data analysis
  6. Proofs
  7. Proof of Theorem \ref{['thmei']}
  8. Proof of Lemma \ref{['lemei']}
  9. Proof of Lemma \ref{['lemeiA']}
  10. Proof of Theorem \ref{['thmgen']}
  11. Proof of Corollary \ref{['thmnull']}
  12. Proof of Theorem \ref{['thmaltA']}
  13. Proof of Theorem \ref{['thmalt']}
  14. Proof of Theorem \ref{['thmalt2']}

Key Result

Theorem 2.1

Let $1 \le i \le d$ be fixed and set where $r \in \mathbb{N}$ denotes the unknown period of the $p_i$. If $r \le d$, then

Theorems & Definitions (10)

  • Theorem 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Remark 2.10