## The Annals of Statistics

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

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

#### 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