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.TABLES OF PERCENTILE POINTS OF NONCENTRAL
CHI-SQUARE DISTRIBUTIONS
by
N. L. Johnson
University of North Carolina
Institute of Statistics Mimeo Series No. 568
•
Febmary 1968
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This research was supported by the Mathematics Division
of the Air Force Office of Scientific Research Contract
No. AF-AFOSR-68-l4l5, and by National Institute of
General Medical Sciences under grant 5R01GM-12868.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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1.
Introduction:
of
The noncentral chi-square distribution is the distributim
v
I
(u. + a.)2
J.
J.
j=l
where
~,
~,
u , ••• ,
2
a , ••• , a
2
v
are independent unit normal variables and
v
are constants. The distribution depends on v - called
U
L a~
\1
the degree of freedom of the distribution - and A.
noncentrality parameter
of the distribution.
- called the
j=l
If A • 0 we have a
(central) X2 distribution.
Denoting a noncentral x2 variable, as defined in (1) by X,2(A),
v
the probability that it exceeds X can be expressed as a series in
central
x2
probabilities, as follows:
Pv[X~2(A) > X]
(2)
•
=
L
a j , APv[X~2j > X]
j=O
where a. ").
J,I\
and
= e --t>'
(lA) j / j :
~ is used to denote a (central)
(i.e.,
l
with f degrees of freedom
x.r2 (0».
The tables contain values of X such that
Q "). (X)
v,/\
for Q
= Q.
= 0.999, 0.9975, 0.995, 0.99, 0.975, 0.95, 0.9, 0.75, 0.5, 0.25,
0.1, 0.05, 0.025, 0.01, 0.005, 0.0025, 0.001.
Values of X are given to four significant figures.
2. Calculation of Tables
Equation (2) above can be written
I
ii
Qv, ~ (X) = )
a j , ~ f V+2j
j~
on
with
[t1mCD
f or.)
m
1 e -!t
= [2l-r(1m) ]-1
dt
The method of solution of the equation
was iterative, based on Newton's method.
At each stage of calculation,
a trial value of X, X say, was used, and Q ~(X) calculated.
r
'1,1\ r
The
next trial value was
where
CD
~v, ~(Xr)/OXr
=
I
a j , ~[OfV+2j(Xr)/OXr]
j=O
and
ofm(X)/OX =
- [21mr(p)]-lxim-l e-tx .
The process was continued until the inequality
was satisfied - i.e. until the correction was less than O.Ol~ of the
trial value.
of X.
The value of Xr +1 was then recorded as the required value
,
The recorded values should be correct to 1 in the fourth signifi-
cant figure.
Judging from detailed study of successive iterations in
a number of perticular cases, most of the figures should be accurate
to the fourth significant figure.
The initial value,
~,
I
-.
CD
was obtained from the value of X for the
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iii
same v and Q, but for the previous (lower) vaJ.ue of ~ by adding the
correction
- (increase in ~)
For the lowest value of ~ (0.04), Xl was taken to be equal to the
appropriate percentile of the corresponding central
x2
distribution
I wouJ.d like to thank Mrs. Carol Newman for writing the program
giving effect to these calculations.
2
Other Tables of NOncentra1 X Chi-Square Distributions.
The most entensive tables of noncentra1 chi-square distributions
are those of H8\Vtlam et. al [1].
(i)
These include
Values of Q '\ (X) to 4 decimal p1a.ces, with X equal to the upper
v, 1\
10ea;, point of the central chi-square distribution with v degrees of
freedom, for
(ii)
a
= 0.001,
~
= 0(0.1)1.0(0.2)3.0(0.5)5(1)40(2)50(5)100
v
= 1(1)30(2)50(5)100
0.005, 0.01, 0.025, 0.05, 0.1
Values of A, to 3 decimal places, satisfying
for the same values of v and X as in (i), with
Q = 0.01(0.01)0.30(0.02)0.90
(iii)
Values of v, to three decimal places, satisfying
for the same fixed values of ~, X and Q as in (i) and (ii)
iv
M:>re detailed tables (e.g [2], [.3], [4]) of Q ~(X) [or (l-Q
'V,"
for
'V
= 2 or
'V
~(X»]
'V,,,
= .3 have been caJ.culated in correction wi th 'coverage'
problems.
REFERENCES
[1]
HEl;Yllatn, G. E., Govindarajulu, Z. and Leone, F. C. (1962) Tables of the
Cumulative Non-Central
Chi-S~are Distribution. Case
StatisticaJ.
Laboratory, Report No. 104.
[2]
Guenther, W. C. (1961) On the probability of capturing a randomly
selected point in three dimensions, SIAM Rev., ~ 247-250.
[.3J Lowe, J. R. (1960) A table of the integral of the bivariate normal
distribution over an offset circle, J. R. Statist. Soc., Series B,
~ 177-187.
[4]
Owen, D. B. (1962)
Reading, Mass.
Handbook of Statistical Tables, Addison-Wesley,
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