J. Jpn. Soc. Comp. Statist., xx(200x), 1–3

COMPUTATION OF THE TEST STATISTIC AND THE NULL
DISTRIBUTION IN THE MULTIVARIATE ANALOGUE OF
THE ONE-SIDED TEST
Yoshiro Yamamoto∗ , Akio Kudô† and Katsumi Ujiie‡

ABSTRACTThe multivariate analogue of the one sided test derived in Kudô (1963) is considered. A handy method of computing the test statistic and its significance probability
is given. The method is based on applying sweep out operations on a certain matrix ina systematic manner and applying the Fortran subroutine of Sun (1988).

1. Introduction
The problem of this paper is as follows. Given a p-variate normal distribution: Np (θ, Λ),we consider the testing problem: H0 : θ = 0 versus H1 − H0 : θ ≥ 0, where Λ is assumed
to be known and θ = (θ1 , · · · , θp )0 ≥ 0 means Max θi > 0, Min θi ≥ 0 , thus the space of
the alternative hypothesis is a cone. The MLE of θ in the space of H0 ∪ H1 is often called“the projection into the cone” because of the very nature of the alternative hypothesis. Inthis paper we also set the hypothesis H2 : θ 6= 0, and consider a testing problem : H1 versus
H2 − H1 .
More than 3 decades have passed since the paper “Multivariate Analogue of the Onesided Test” was published in Biometrika where the above problem was treated and thelikelihood ratio test was derived. This paper has been criticized in two aspects. Difficulties
in computing (i) the test statistic and (ii) the significance probability. (See for instanceTang et al. (1989). See also Barlow et al. (1972) and Robertson et al. (1988).)
Our purpose is to demonstrate methods to compute these two. In Section 2 we quickly
review the outline of the problem, and Section 3 is devoted to the method for computing
the MLE, which will find its application in computing an optimum linear test given in Shi
et al. (1987), and Section 4 to a numerical example. In Section 5 an application of Sun’s
subroutine is demonstrated and its accuracy examined, and the final Section is for concluding
discussions.
The first of two distribution functions given in Section 2 was given by Kudô (1963)
and the second can be derived from Theorem 2.3.1. of Robertson et al. (1988). The second
distribution was treated in Robertson and Wegman (1978) and we documented the same in
the present paper’s form.
∗ Graduate

School of Natural Science and Technology, Okayama University, 3-1-1, Tsushima-naka,Okayama 700-0082, Japan
E-mail: yamamoto@xxx.okayama-u.ac.jp† Faculty of Economics and Information Science, Hyogo University, 2301, Shinzaike, Hiraoka-Cho, Kakogawa 675-0101, Japan‡ Department of Mathematics, School of Science, Tokai University, 1117, Kitakaname, Hiratsuka 2591292, Japan
Key words: Level probability; Order restricted statistical inference; Orthant probability

J. Jpn. Soc. Comput. Statist., Vol. xx, 200x1.1. How to use the .cls or .sty files
At first, place appropreate class/style file in the same directory with your TEX document(s).
• jscs sub.cls: LATEX 2ε , for submission• jscs.cls: LATEX 2ε , for the final image• jscs sub.sty: LATEX2.09, for submission• jscs.sty: LATEX2.09, for the final image
If you use LATEX 2ε , you should begin your TEX document with\documentclass{jscs_sub}(for submission), or\documentclass{jscs}(for the final image). Or if you use LATEX2.09, you should begin with\documentstyle[jscs_sub]{article}(for submission), or\documentstyle[jscs]{article}(for the final image).
We already defined some environment for theorems, propositions, etc. See this TEX
source file and the following examples.
Lemma 1 This is an example of the lemma environment.
n
X

k=k=1

n(n + 1).
2(1)Theorem 1 This is an example of the theorem environment.
nX
k=1

k2 =n(n + 1)(2n + 1).
6(2)In theorem environment, the reference to an equation(s) should not be italicized. Use \rmcommand (LATEX2.09) or \textup{} command (LATEX 2ε ) like “From the equation (1), ...”,“According to the conditions (i)–(iv), ...”, etc.
Proof. In our .cls and .sty files, \proof command and \qed command are defined. You
can use \proof for opening the proof and \qed for closing the proof.
2–2–

Compiled on June 5, 2003

Table 1:
No.
1
2
34..
.
2p−2+1..
.

1···110 · · · 0..
.

111 · · · 10000 · · · 01..
.

2p−1

11 · · · 1

111 · · · 11

p−2

2The first 2p − 1 binary numbers
Binary No. Sweep out position
11000 · · · 0100100 · · · 0111100 · · · 0
1000010 · · · 0..
..
.
.

2. A brief review
As the variance matrix Λ is assumed to be known, we can assume there is only one
observation x. Let P be the set of integers P = {1, · · · , p} and M = {i1 , · · · , im } be a subsetof P . It was shown that there exists uniquely a set satisfying the condition stated below.
(see Beale (1959))
Without loss of generality, we state the condition in the case when the set is the firstm natural numbers: M = {1, · · · , m} where m ≤ p.
Let the observation vector x0 = (x1 , · · · , xp ) be partitioned into m and p−m components
and the mean vector and the variance matrix be partitioned accordingly:
µ¶
µ¶
µ
¶
x(1)
θ (1)
Λ11 Λ12x=, θ=, Λ=.
x(2)
θ (2)
Λ21 Λ22
There exists uniquely the subset M satisfying the condition.
½
Λ−1< 011 x(1)x(2) − Λ21 Λ−1x
≥ 011 (1)

(3)

When m = 0, M = φ, the empty set, and when m = p, the second condition in (3) isvacuous.
REFERENCES
Akaike, M. (1973). Information theory and extension of the maximum likelihood principle. In Second
International Symposium of Information Theory, N. B. Petrov and F. Czaki (eds.), 267–281,
Budapest: Akademiai Kiado.
Beale, E. M. L. (1959). On quadratic programming. Naval Research Logistic Quarterly 6, 227–243.
Miyahara, H. and Tango, T. (eds.) (1995). Handbook of Medical Statistics. Tokyo: Asakura Shoten.
(in Japanese)
Swayne, D. F., Cook, D. and Buja, A. (1998). Xgobi: Interactive dynamic data visualization inthe X Window System. Journal of Computational and Graphical Statistics 7, 113–130. See also
http://www.research.att.com/areas/stat/xgobi/.
Tarumi, T. and Kudô, A. (1974). An algorithm related to all possible regression and discriminate
analysis. Journal of the Japan Statistical Society 4, 47–50.

–3–Table 1 is inserted here.