## The Annals of Mathematical Statistics

### Concentration of Random Quotients

William H. Lawton

#### Abstract

The present paper proposes a definition of relative concentration of random variables about a given constant, and studies the relationship between two stochastic denominators $Z_1$ and $Z_2$ which causes the random quotient $X/Z_1$ to be more concentrated about zero than $X/Z_2$. In this paper we shall always assume that the numerator and denominator are independent. Two necessary and sufficient conditions, and several sufficient conditions for $X/Z_1$ to be more concentrated about zero than $X/Z_2$ are given in Section 4. The results of Section 4 are used in Section 5 to obtain generalizations of a theorem due to Hajek (1957) on the generalized Student's $t$-distribution. Sections 6 and 7 use these generalized theorems to produce tests and confidence intervals for several Behrens-Fisher type problems. Finally, Section 8 contains a proof of the randomization theorem stated in Section 4. In particular, Section 6 is concerned with the extension of a result in Lawton (1965) on Lord's $u$-statistic to the case of unequal sample sizes. Section 7 gives methods for constructing confidence intervals for linear combinations of means from $k$ normal populations

#### Article information

Source
Ann. Math. Statist., Volume 39, Number 2 (1968), 466-480.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177698410

Digital Object Identifier
doi:10.1214/aoms/1177698410

Mathematical Reviews number (MathSciNet)
MR225409

Zentralblatt MATH identifier
0159.48201

JSTOR

#### Citation

Lawton, William H. Concentration of Random Quotients. Ann. Math. Statist. 39 (1968), no. 2, 466--480. doi:10.1214/aoms/1177698410. https://projecteuclid.org/euclid.aoms/1177698410