'.
ON BOUNDS FOR THE
NORV~L
INTEGRAL
by
John T. Chu
Institute of Statistics
Unive~s1ty of North Carolina
....
.".;..
Special report to the Office of Naval
Research of work at Chapel Hill under
Contract NR 042 031, Project N7-onr284(02), for research in probability
and statistics.
~~.
,.:
,.~
Institute of Statistics
Mimeograph Series No. 119
September, 1954,
e··,·'
•
ON
BUUl\JDS
FOR 1j,l H1
NOl~~.L IN'l'.J.;;GRAL 1,2
John T. Chu, Institute of-Statistics, University of North Carolina
1.
Summary.
Bounds are der.ived for the normal integral.
Some comparisons
are made with several known results.
2.
Let
x
( l)
=
v
(
)
1
(23t)-2 e- t
2/ 2
dt
x
> G.
o
G. Polya and J. D. Williams proved independently that
(2)
Two simple questions follow naturally.
2/n
1.
Is it possible to replace the constant
in (2) by a smaller quantity without breaking the inequality?
exist a lower bound, in a similar form, for v?
If for all x
~
2.
Does there
We find the following answer.
0, the integral v given by (l) satisfies
(3)
then it is necessary and sufficient that 0 ~ a ~ ~ and b ~
2/n.
1. Work sponsored by the Office of Naval Research under Contract NR 042 031
at Chapel Hill.
2. Revised and enlarged from "A note on the normal integral", Institute of
Statistics Mimeograph Series No. 114.
2
'_
The proof of this statement is simple.
x
Lim
_>
2
bx )
= (2nb)-I.
0 v /(l - e2
Hence if (}) is true, b ~ 2/n.
all real x.
First,
On the other hand, x
2
/1=
Since the limit of this ratio, as x
--->
00,
x
2n
x
log(I-4v
2
)_7 ~ l/a for
is 2, we have a ~ 1/2.
Finally
4v2
=
x
f
f
-x
-x
2
(21<)-1 e-(i+t )/2 dB dt
>
f
f
o
o
(21<)-1 e-
r2 2
/ r dr dQ •
Therefore
1
v ~ ~(l
(4)
2
-x /2
- e ,
l..
2
).
Polya showed that as x varies from 0 to
00,
the ratio of the LHS (Left
hand side) of (2) to the RHS decreases steadily from 1 to a minimum value and then
increases steadily.
Williams' calculations indicate that, approximately, the
minimum value .99}0 is taken at x
= 1.6.
j
Using a similar method to that of Polya,
it can be shown that the ratio of the LHS of (4) to the RHS is a steadily decreasing function of x for all x
~
OJ
for the derivative of this ratio has the same
sign as
x
2(ex2 /2 _ 1) _ x eX 2/2
f
e- t 2/2 dt,
o
which is non-positive since
x
e- t
o
2/
2 dt
=
00
1:
n=l
x 2n-1
l.} ••• (2n-1)
As a c-onsequence, we obtain that this ratio (of the LHS of (4) to the BHS) has an
1
upper bound 2/1t2 •
3.
A
different lower bound for v can be obtained easily from a result
proved by Chu and Hotelling. There we showed that for all x
~
0,
222
x (1 - 4v )/4v < 1t/2.
(6)
Hence it follows that
For easy reference, we will give here a proof of (6).
(8)
gO(x)
then x
Lim
_>
0
BO(X) = 1t/2.
Let
22
= x2
(1-4v )/(4v ),
\ve will show that gO(x) is decreasing.
differentiation with respect to x.
Let" I " denote
Then,
where
I
(10)
(11)
xv •
f
gl (x)
2
= (12/1t) e- x B (X), where
3
= (1t/6)xe x2_ eX 2/2
4
From (5), we have g;(x) = ~o
./
n=O
[~7
on.
2
.,1 + } x n+l
1. 3••• ( 2n+ 1)
•
_
It can be shown" by a
similar argument used by Pblya for a similar purpose, that
... ) ,
(12)
where a
O
= 1,
< 0 and a i >0, i
S3(x) ~ 0 if 0 ~ x ~
X
2, ••• •
o and g3(x)
Hence there exists an X > 0 such that
o
~ 0 if x ~ xO.
So as x increases from 0 to
g2(x) decreases steadily from 0 to a minimum and then increases steadily to
Consequently gl(x) first decreases steadily and then increases steadily.
Lim
x
> O.
x
-:>
00
gl(x)
= 0,
00,
00.
As
it becomes clear that gl(X) ~ 0 for all
Therefore gO(x) is a decreasing function of x.
Hence we have
(6).
Comparison can be made easily of the two lower bounds for v given by (4)
and
(7). For simplicity, they will be denoted by a(x) and b(x) respectively. Now
a{x)
I
C
:>
<
b(x)
according as c(x)
= eX 2/2
2
- 2x I~
-
1
>
< o.
As x varies from 0 to
(x), the derivative of c(x), changes sign from negative to positive.
c(x).
If x
= Xo is
o=1
the solution of c(x) = 0, then X
value is slightly smaller).
that in (4) if 0
~
x
~
00,
So does
approximately (the exact
Therefore, the lower bound in (7) is closer to v than
1 (approximately) and less close if x
> 1.
Further,the following statement is of similar nature to the one made in
If for all x
(13)
~
0,
v ~
r 2
'2 J.. ax / (l
1
1
+ ax
2"2
then it is necessary and sufficient that 0 ~ a ~ 2/~.
)_7
,
On the other hand, for no
i 2:
finite a can the RHS of (13) 'be, for all x
> 0,
an upper bound for v.
The above statement can be shown easily by considering the limit, as x ---> 0,
of the ratio of v to the RES of (13); and the limit, as x
--->
00,
of
2
2
(1-4v )(l+ax ).
4. Several authors have derived inequalities for Mills' ratio. Their
results can be written in the form of bounds for the normal integral.
For example,
in our notation, Gordon's inequalities are equivalent to
,
for x
> O.
Birnbaum improved Gordon's upper bound in (14) and obtained
(15)
for x
> O.
,More recently, Tate showed what amounts to
2
/2
2
e
<
v < -l( 1-e -x) for x
- - 2
'
X(21t)2
_X
(16)
----~.
We will now compare briefly (2) and (4) with (111-), (15), and (16).
in (16) is obviously not so good as that in (2).
negative for all real x.
2
x2 _ (B/n)(l _ e- x /2)
o to
~
It is
O.
>
Z
As x varies from 0 to
imately (the exact value is slightly smaller).
The upper bound
The lo#er bound in (16) is non-
the RHS of (4) according as h(x)
a minimum and then increases steadily to
> O.
00;
00,
=
h(x) decreases steadily from
and vanishes at x
= 1.01
approx-
Therefore the lower bound in (16)
is closer to v than that in (4) if and only if x
~
1.01 approximately.
6
The lower bound in (14) is an increasing function of x for all x
non-negative when x ~ .65 (approximately); and in this case it is
= 2(2/n)21
according as g(x)
x - x 2 _ (2/n) e-x
2/
2 ~ O.
5< the
> O.
It is
RHS of (4)
As x varies from 0 to
00,
g(x) increases steadily from - 2/n to a maximum and then decreases steadily to
The two roots of g(x) = 0 are approximately x
= .5
-00.
and x = 1.45. Hence the lower
bound in (14) is closer to v than that of (4) if and only if x
~
1.45 approximately.
Finally we point out that, for values of x close to 0, the upper bound in
(2) is better than those in (14) and (15); while for large values of x, the latter
two are better.
No detailed comparison is attempted.
The author wishes to thank the referee for calling his attention to R. F.
Tate's work and suggesting adding to the original note some comparison of the new
and known results.
Thanks are also due to Professor Harold Hotelling for his
critical reading of the manuscript.
Rl!FERmCBS
Birnbaum, Z. W. (1942).
Ana. Math. Stat. 13, 245-246.
Chu, J. T. and Hotelling, H. (1954).
To be pUblished.
"The moments of the sample median".
Gordon, R. D. (1941). Ann. Math. Stat. 12, 364-366.
Polya, G. (1949). Proceedings of the Berkeley Symposium on Mathematical
Statistics and Probability, University of California Press, Berkeley, 63-78.
Tate, R. F. (1953). Ann. Math. Stat. 24, 132-134.
Williams, J. D. (1946). Ann. Math. Stat. 17, 363-365.