## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 1 (1973), 74-83.

### The Conditional Probability Integral Transformation and Applications to Obtain Composite Chi-Square Goodness-of-Fit Tests

Federico J. O'Reilly and C. P. Quesenberry

#### Abstract

It is shown that certain conditional distributions, obtained by conditioning on a sufficient statistic, can be used to transform a set of random variables into a smaller set of random variables that are identically and independently distributed with uniform distributions on the interval from zero to one. This result is then used to construct distribution-free tests of fit for composite goodness-of-fit problems. In particular, distribution-free chi-square goodness-of-fit tests are obtained for univariate normal, exponential, and normal linear regression model families of distributions.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 1 (1973), 74-83.

**Dates**

First available in Project Euclid: 25 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1193342383

**Digital Object Identifier**

doi:10.1214/aos/1193342383

**Mathematical Reviews number (MathSciNet)**

MR362691

**Zentralblatt MATH identifier**

0276.62025

**Keywords**

62 71 Conditional expectation minimal sufficient statistic absolute continuity MVU function estimator composite goodness-of-fit tests

#### Citation

O'Reilly, Federico J.; Quesenberry, C. P. The Conditional Probability Integral Transformation and Applications to Obtain Composite Chi-Square Goodness-of-Fit Tests. Ann. Statist. 1 (1973), no. 1, 74--83. doi:10.1214/aos/1193342383. https://projecteuclid.org/euclid.aos/1193342383