The Annals of Mathematical Statistics

Note on Multivariate Goodness-of-fit Tests

Judah Rosenblatt

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Abstract

Let $X_1, X_2, \cdots, X_n$ be $m$-dimensional statistically independent random vectors with common distribution function $F$. It is frequently desirable to test the hypothesis that $F$ is a member of some class of distribution functions $\mathscr{H}_0$. For the scalar case, $(m = 1)$, much research has been done; see for example [1], [2], [3]. For $m > 1$ comparatively little has been accomplished, and a useful extension of the techniques used for $m = 1$ awaits the solution of certain problems in stochastic processes with a vector parameter; see, for example, [4]. In this paper consistent tests are developed for any given class $\mathscr{H}_0$. These tests can be constructed to have size $\alpha$ and prescribed power $1 - \beta$ against alternatives whose probability assignment to at least one of a certain given class of sets $\{B(v)\}$ differs from that of each member of $\mathscr{H}_0$ by at least a prescribed value $K$. The range of such alternatives is seen in Section 3 to be rather wide, so that at least in theory, the suggested tests would seem to be rather useful. The tests are constructed by mapping the set of all $m$-dimensional distribution functions in a one to one measurable manner into a subset of the set of one-dimensional distribution functions. Such mappings are, of course, reasonably well known; see Halmos [7], p. 153. The purpose of this note is to show how such mappings offer sufficient flexibility for the construction of a class of tests which are useful for most ordinary purposes. Simpson [6] suggested tests based on mapping bivariate distributions into univariate distributions. However no mention was made there of consistency, or of power in terms of the type of alternative here mentioned.

Article information

Source
Ann. Math. Statist., Volume 33, Number 2 (1962), 807-810.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177704601

Digital Object Identifier
doi:10.1214/aoms/1177704601

Mathematical Reviews number (MathSciNet)
MR137235

Zentralblatt MATH identifier
0124.10105

JSTOR
links.jstor.org

Citation

Rosenblatt, Judah. Note on Multivariate Goodness-of-fit Tests. Ann. Math. Statist. 33 (1962), no. 2, 807--810. doi:10.1214/aoms/1177704601. https://projecteuclid.org/euclid.aoms/1177704601


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