The Annals of Statistics

Tests for Change of Parameter at Unknown Times and Distributions of Some Related Functionals on Brownian Motion

Ian B. MacNeill

Full-text: Open access

Abstract

Statistics are derived for testing a sequence of observations from an exponential-type distribution for no change in parameter against possible two-sided alternatives involving parameter changes at unknown points. The test statistic can be chosen to have high power against certain of a variety of alternatives. Conditions on functionals on $C\lbrack 0,1\rbrack$ are given under which one can assert that the large sample distribution of the test statistic under the null-hypothesis or an alternative from a range of interesting hypotheses is that of a functional on Brownian Motion. We compute and tabulate distributions for functionals defined by nonnegative weight functions of the form $\psi(s) = as^k, k > -2$. The functionals for $-1 \geqq k > -2$ are not continuous in the uniform topology on $C\lbrack 0, 1\rbrack$.

Article information

Source
Ann. Statist., Volume 2, Number 5 (1974), 950-962.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342816

Digital Object Identifier
doi:10.1214/aos/1176342816

Mathematical Reviews number (MathSciNet)
MR426253

Zentralblatt MATH identifier
0289.62059

JSTOR
links.jstor.org

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60E05: Distributions: general theory 60G50: Sums of independent random variables; random walks 60H05: Stochastic integrals 60J65: Brownian motion [See also 58J65] 62E15: Exact distribution theory 62E20: Asymptotic distribution theory 62F05: Asymptotic properties of tests

Keywords
Parameter changes at unknown times functionals on Brownian motions weak convergence weight functions exponential family Krhunen-Loeve expansions stochastic integrals Bessel functions

Citation

MacNeill, Ian B. Tests for Change of Parameter at Unknown Times and Distributions of Some Related Functionals on Brownian Motion. Ann. Statist. 2 (1974), no. 5, 950--962. doi:10.1214/aos/1176342816. https://projecteuclid.org/euclid.aos/1176342816


Export citation