APPROXIMATION OF PEARSON TYPE IV
PROBABILITY INTEGRAL
by
WAYNE ANTHONY WOODWARD, B.A.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
May, 1971
/leu- s q s y • '
gOST
T3
197(
ACKNOWLEDGMENTS
I wish to express my appreciation to Professor Thomas L. Boullion
for his direction of this thesis and to Professor H. L. Gray for
his valuable assistance.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS
ii
LIST OF TABLES
iv
LIST OF FIGURES
v
I.
II.
INTRODUCTION
1
Background
1
Previous Estimations of the Integral
4
DERIVATION OF APPR0XII4ATING FUNCTIONS
The Use of Nonlinear Transformations in Deriving
Approximating Functions
9
The H Transformations
n
11
The L Transformation
17
The B
21
Transformation
Continued Fraction Approximations
III.
9
COMPARISON OF RESULTS
25
27
LIST OF REFERENCES
34
111
LIST OF TABLES
Table
Page
1.
Approximating Functions for j
g(x)dx
28
2.
Values of the Approximating Functions
30
iv
LIST OF FIGURES
Figure
1.
2.
3.
Page
The Effect of a Change in the Parameter k
upon the Graph of the Pearson Type IV Probability
Curve
5
The Effect of a Change in the Parameter r
upon the Graph of the Pearson Type IV Probability
Curve
6
The Effect of a Change in the Parameter a
upon the Graph of the Pearson Type IV Probability
Curve
7
CHAPTER I
INTRODUCTION
Background
Karl Pearson has derived a family of curves which over the years
has proved to be useful in representing practical distributions.
These curves are determined by solutions to the differential equation
^
=
dx
(x - a)g
,
.•
0
. ,
1^
..
2
^•'''
2^
In all, Pearson distinguished twelve types or forms of frequency
curves which arise as solutions to (1). The solution which will be
considered in this thesis is the Pearson Type IV curve.
The proba-
bility density function corresponding to this curve is given by
2
,
^
X
/ N
/ 1 . x v-m -k arctan —
^
^
g(x) = c(l + -^)
e
a -« < X < «.
r^x
(2)
a
The parameters are a, m, and k while c is the constant necessary for
the integral over the entire range to equal one.
Due to the impos-
sibility of expressing the probability integral in closed form, this
curve has been very difficult to use.
The purpose of this thesis is
to find a relatively simple approximating function for integrals of
the form
/
where g(x) is given in (2).
g(x)dx
Before undertaking this task, it seems to be beneficial to
examine further the probability density function itself. The Pearson
Type IV distribution can be determined uniquely by its first four
moments.
The moments about the mean are given by
«
ak
aVj:^
^2 ="~2
r (r - 1)
3
2
2
4a-^k(r^ + k )
Po = -
^
r^(r - l)(r - 2)
= 3a^(r^ + k^){(r + 6)(r^ + k^) - 8k^}
^
r^(r - l)(r - 2)(r - 3)
where r = 2m - 2. One should note that in order that y^ > 0 we must
2
3
Letting 3, = y^ /yj ai^d 0^ ~ ^J^o
have r > 1.
2
» ^^ obtain the
following expressions for r, k, and a in terms of the first four
moments:
6(32 - ^1 - 1>
r =
(232 " ^^1 " ^^
r(r - 2)/3T
k =
/{16(r - 1) - 3^(r - 2)"}
a = /-^ {16(r - 1) - 3, (r - 2)^} .
16
1
In order to calculate the constant c, it is necessary to evaluate
the integral
2
,
^
x
r" /i . X v-m -k arctan — ,
1
J „ (1 + -^)
e
a dx = — .
a
Making the substitutions — = tan 6 and
a
2m = r + 2 , we have
,22
1
•••
f°°
,
/I . X v-m
a
.
X^
-k arctan — ,
r^/2 , , _^ , 2 _ . - r - 2 / 2 -ke
2 , ^,
= J_.j^/2 (^ "*" t a " ^)
e
a sec 6 de
rii/2
= J
-ke
/o a e
-r
sec
„j„
ede
' -TT/Z
/•7r/2
-k
r ^ ,„
= a J /o e
cos Gd0.
Pearson has calculated the integral /
.^e
cos 0de and has pro-
vided a set of tables [8]. In his notation
•r./
1\
r"T/2
F(r,k) = /
'-nil
-k6
,o e
r^ .„
cos Ode.
1
Thus we have c = —r-f—TT • The integral F(r,k) exists in closed form
aF(r,k)
®
when r is an integer, but in other cases Pearson's tables are needed.
The effect which the parameters have upon the curve is also of
some interest.
Figures 1, 2, and 3 illustrate the effect upon the
curve brought about by a change in a parameter.
It can be seen that
when k = 0 the curve is symmetric about the mean which is at x = 0
in this case.
Figure 1 illustrates the fact that the curve is skew
to the right when k < 0 and skew to the left when k > 0.
Of course
a change in k also brings about a shift in the mean, since inspection
of (2) shows that a > 0.
Figure 2 indicates that if a and k remain
fixed an increase in r brings about a decrease in the variance and a
shift in the mean. One can see by Figure 3 that for r and k fixed,
an increase in a results in an increase in the variance and a shift
In the mean.
These results can also be observed by inspection of
the density function itself.
Previous Estimations of the Integral
As was mentioned before, the probability integral of the Pearson
Type IV curve cannot be found directly. This curve is the only one
of Pearson's curves for which the probability integral cannot be
reduced to known integrals such as Chi-square integrals or Incomplete
Gamma-functions.
In Pearson's Tables for Statisticians and Biometri-
cians [9] he expresses his regret for the exception.
"The series of
tables ought to include Tables of the Incomplete G-Function, i.e., the
Probability Integral of the Type IV curve, but the age of the present
Editor is likely to preclude his superintending any task, which even
exceeds in the magnitude of its calculations that of the Incomplete
B-Function."
Karl Pearson was 74 years of age when he made this statement,
and, as he said, he never did make such a set of tables. Greenwood
and Hartley [5] referring to Pearson's above statement added:
"Nor
is the project likely to commend itself to a contemporary statistician. "
Despite these pessimistic remarks, attempts have been made by
L. R. Shenton and J. A. Carpenter [10]. Through the use of Mill's
Ratio, Shenton has determined a continued fraction which converges
to r
g(x)dx when - is positive and to C g(x)dx - 1 when - is nega-
-1.0
Figure 1.
0
The Effect of a Change in the Parameter
k upon the Graph of the Pearson Type IV
Probability Curve.
1.0
Figure 2.
The Effect of a Change in the Parameter
r upon the Graph of the Pearson Type IV
Probability Curve.
1.0
Figure 3.
The Effect of a Change in the Parameter
a upon the Graph of the Pearson Type IV
Probability Curve.
0
8
tive.
This method, however, does have a few shortcomings.
value — is close to zero, convergence is very slow.
When the
This continued
a.
fraction expansion, of course, is not a simple approximating function.
In the next chapter a truncation of this continued fraction will be
made in order to have an approximating function comparable in simplicity
to the one proposed here for purposes of comparison.
Another method for evaluating the probability integral of the
Pearson Type IV curve has been given by N. L. Johnson and Eric Nixon [6].
These men along with D. E. Amos and E. S. Pearson have published a set
of tables of percentage points of Pearson curves.
Different types of
Pearson curves can be associated with different regions in a graph
J—
2 3
having /3-, and 3^ as the coordinate axes where 3, = Wo/Vo and
3^ = Vilv^'
Since the shapes of the distributions of the system change
continuously across the boundaries of the regions, a single method
was used in order to evaluate the percentage points of several types
which have adjacent regions on the graph.
Pearson Type IV curve.
Among these types is the
For this group, Johnson and Nixon used the
Runge-Kutta-Simpson method to integrate a system of differential
equations to evaluate the percentage points of the distributions.
Although this is probably an incomplete listing of results published in this area, it does appear that an extensive set of tables
does not exist for the probability integral of the Pearson Type IV
curve.
CHAPTER II
DERIVATION OF APPROXIMATING FUNCTIONS
The Use of Nonlinear Transformations in
Deriving Approximating Functions
In recent literature several nonlinear transformations have been
developed which are useful in evaluating numerically improper integrals of the first kind.
In an attempt to obtain an approximating
function for the Pearson Type IV probability integral, it seems
reasonable to investigate the use of some of these transformations.
An attempt will be made to obtain a function which approximates
S = !l
g(x)dx
where g(x) is given in (2). Throughout the remainder of this thesis,
S and g(x) will be as above and G(t) will be given by
G(t) = /^ g(x)dx.
Moreover, f will denote an arbitrary function while F(t) will be
given by
F(t) = /^ f(x)dx,
and F(«') will be given by
F(-) = !l f(x)dx.
It is desirable for the transformation which will be used to
10
possess certain properties.
If A represents a transformation, it is
clearly desirable that A(t) ->- S as t ->• «,
Two additional properties
which are desirable for our transformed function to possess are given
by the following definition.
Definition 1;
If A(t) and B(t) are two sequences of real numbers
such that lim A(t) = A ^ ± » and lim B(t) = B ?^ ± «, then we say A(t)
converges uniformly better than B(t) on (t ,«>) if and only if
IA - A(t)| < |B - B(t)| for every te(t ,«>). Further, if
IA -
lim \—
t-x»
A(t)I
ri)f.\ I ~ 0, then we say A(t) converges more rapidly than B(t).
Thus, if our transformed function converges uniformly better to
S than does G(t), the value of the transformation is closer to S than
is G(t) for t sufficiently large.
The following theorem shows us the
relationship between these two concepts.
Theorem 1:
If A(t) converges more rapidly than B(t), then there
exists a t such that A(t) converges uniformly better than B(t) on
o
Proof:
By the hypothesis lim lf-T"w7)l = 0-
^hus for every e > 0 there
exists a t such that if t is in the interval (t ,°°) then
e
•
^
1^ " ^^^^ I < £. Now, let e = 1. Then there is a t such that if
'B - B(t)'
*
o
tE(t ,00) then it " t/^!l < 1 or |A - A(t) | < |B - B(t)|.
11
As can be seen from the proof of Theorem 1, if the transformed
function converges more rapidly than G(t), then for a fixed e, arbitrarily small, there is a t
A(t) - sl < e |G(t) - sl.
such that if tE(t ,") then
Thus it is clear that this would be a
desirable property for our transformed function to possess.
Also,
because of the impossibility of integrating g, the approximating
function used must not involve any integration.
With these consid-
erations in mind the applicability of certain of these non-linear
transformations will be examined.
The H Transformations
n
The first type of transformation to be considered is the H
n
transformation which was given by Gray, Atchison, and McWilliams
in [3].
Definition 2:
Let f e C^ ^
^ on (b,°°).
Then
F
f
f
f
^(n-1)
^(n)
f
. . .f
(n-1)
. . .f
(n)
. . .f
(2n-l)
H [F(t)J =
n
f
•f"
. . .f
(n)
f"
•(n)
. . .f
(2n-l)
12
where F(t) = /^ f(x)dx.
Only H^[G(t)] and H [G(t)] will be considered due to the fact
that the higher derivatives involve many terms which would tend to
make the approximating function more complicated than desired.
From Definition 2 it follows that H [G(t)] is given by
G(t)
g(t)
g(t)
g'(t)
H^[G(t)] =
g'(t)
- G(t)
g,(^) .
From (2) it follows that
*.2
,
, t
, .
.- , t v-m -k arctan —
g(t) = c(l + -5") e
a
and
2
t
, ,.
/, , t .-m-1 -k arctan — , k
g' (t) = c(l + -^)
e
a [- aT
2mti
2TJ •
a
Thus
g^(t) _
g'(t)
z
a
a
and
2
t
,,
t .-m+1 -k arctan —
c(l + -2")
e
a
H^[G(t)] = G(t) +
a
.2mt k_v
^ 2 a''
a
13
Now H^[G(t)] will be examined in terms of the considerations
mentioned previously.
First, Theorem 2 shows that H [G(t)] converges
to S as t -> »,
Theorem 2: H [G(t)] -> S as t -> «,
Proof:
2. .
We see that H [G(t)] -»- S as t •> « iff ^, yi^ -> 0 as t
But, we have that
2
t
,, , t v-m+1 -k arctan —
^
c(l + —«)
e
a
g (t) _
a^
g'(t)
f 2mt
k.
^"
2 " a^
-k arctan —
-ce
a
(1 + 4)'°"' (+ ^ + !>
a
a
XT
^
^, ^
-k arctan —
-kTT
,
Now, as t -> », we see that -ce
a -*• -ce —r— and
2
2
(1 H j)
(+ —Z-) -> CO since we must have m > —. Thus lim °, ^ ^ = 0,
and we have that H [G(t)] -> S as t -> «.
Before examining for more rapid convergence, a few observations
should be made. We see that H [G(t)] is not an acceptable approximating
function since it involves G(t).
However, if we consider H.[G(t)]l
1
, ,
t=D
we have G(b) = 0, so we eliminate the necessity for integration.
However, the question remains as to whether or not H [G(t)]|
good approximation.
is a
If we know that H [G(t)] converges more rapidly
14
to S than does G(t), given an arbitrarily small e > 0, there is a t
such that |H^[G(t)] - S| < e|G(t) - S| for te(t ,»). If be(t ,») we
would know that |H-,^[G(t) ] | ^^^ - S| < £|G(b) - s| = e|s|. Thus we see
that if e is small and b is larger than t , then we should obtain good
results with H^[G(t)]|^^^,
If H^[G(t)] does not converge more rapidly
than G(t) but does converge uniformly better on an interval (t , " ) ,
then we only know that H.[G(t)]|
i
, will be a better estimate for S
t=D
than G(b) = 0 for b larger than t . Even though this does not give
us sufficient reason to believe that H [G(t)]|
will be a good
estimate in this case we will use it as an approximating function and
compare the results we obtain with those for which we do have more
rapid convergence.
In both cases we expect to obtain the best results
when b is "large", but we do not know how large it must be.
In order
to assure ourselves that b is "large", in this thesis the approximations will only be applied to integrals such that
r g(t) ^ .1.
We should first note that H [G(t)l does not converge more rapidly
than G(t). To accomplish this we need only show that
S - H^[G(t)]
^^^
S - G(t)
We know that
^ °-
2
S-H,[G(t)]
1-
S - G(t)
^
= J^"
'^^'^"^g'(t)
^ - "^'^
. l.nm
^-^
^^„g'(t)[G(t) -
S]
15
Thus we need to show that
lim
J. 1
g^(^>
J ^ g'(t)[G(t) - S] '^ 1Using L'Hospital's rule, we have
2
Z(ti
lim g'(t)
t-x» G(t)
= li^ g'(t)2g(t)g'(t) - g'(t)g"(t)
S
^^^
[g'(t)]^g(t)
=lim^tg'(t)]^-g(t)g"(t)
t^
[g'(t)]^
2m - 1
2m
i 1.
Even though H [G(t)] does not converge more rapidly than G(t),
H^[G(t)] does possess the less restrictive property that it converges
uniformly better on an interval (t^,~) for some t^. Gray and
Schucany [4] give a necessary and sufficient condition for the exist2
ence of such a t^ to be that 0 < lim ^^^^ _ p(^) < 2. From the preceding paragraph we see that
,, g^t)/g'(t) _ 2m - 1
^"-^ G(t) - S
2m '
t->a>
and we know that 0 < "^^^-^ < 1-
Later in this thesis we will give
some transformations which possess the property that the transformed
G(t) converges more rapidly than G(t) to S.
Next, we will consider H2[G(t)].
H2[G(t)l possesses the same
properties as does H^[G(t)] in that it converges to S but docs not
converge more rapidly than G(t).
The criteria for this can be found
16
in [3], For an approximating function we will use H-[G(t)]I
which
2
t=b
Is given by
G
g
g
g
g'
g
II
g'
g"
g
III
H2(G(t)]|^^^ =
g'
g"
g"
g"'
b
_ 2gg'g" - g V ' - (gM^
u^2
g'g"' - (g")
We have
g"(x) = c(i+4""""^ ^"^ ^'''^'^ ^ [(1+4^f^)+4
+ l)kx +^ 4(m + l)mx],
,
+. 2(2m 3
a
a
and
2
X
111 / \
/ 1 . x v-m-3 - k a r c t a n — rk /^
, 2 , „v
6x
g"' (x) = c ( l + —2)
e
a [-j(6m - k + 2) + ~
9
6kx
• (m + l)(2m - k ) - - ^ ^ (2m + 1) (m + 1)
4mx' (2m + l)(m + 1)].
Thus
»2[^<'>]|t=b=
t
2
-k arctan — ,. , t v-m+1 .^.^
ce
a (1 + -j)
[2AB - C
a
[AC - B^]
A^l
17
where A = - (— + —TT-)
a
z
a
Tj _ k^ - 2m ^ 2kt(2m + 1) ^ 2mt^(2m + 1)
B
2
+
3
+
4
a
a
a
r - k(6m - k^ + 2) ^ 6t(m + 1)(2m - k^)
a
a
6kt^(2m + l)(m + 1) _ 4mt^(2m + 1)(m + 1)
5
6
a
a
Some results have been recorded in Table 2. As can be seen, these
two approximations give quite good results even though the corresponding transformed functions do not converge more rapidly to S than
does G(t).
The L Transformation
Although H^[G(t)]|^^^ and H2[G(t)]| ^^^ give satisfactory results,
it would seem desirable to use a transformation which does possess the
property that the transformed function converges more rapidly than
G(t).
It is possible to find such a transformation, and we shall
follow the line of development of such a transformation below.
We noticed that H [G(t)] did not converge more rapidly since
2
H m g (^)/g ^^^ J: 1^ Thus it would seem reasonable to examine a
G(t) - S
t-x»
transformation of the form L[F(t)] = F(t) - a ^,^^^ where a is given
by a = lim ^(^^ " ^(°°)^ and a is assumed to exist.
It would seem
t-Ko f^(t)/f'(t)
that the values of the integrals F(t) and F(o.) are necessary in order
to evaluate a.
However, this is not always true as the following
18
If lim -pcn
considerations show.
rule.
^ ^» ^^^" ^^ ^^^ ^^^ L'Hospital'
Thus we obtain
'M
a = lim ^ ^ ) - ^(°°> = lim
2
- xxm
t ^ f^(t)/f'(t)
t-^ 2[f'(t)]^f(t) - f^(t)f"(t)
~2
[f'(t)]
1-
[f'(t)]^
= lim
*^
^
t-^oo 2[f'(t)]^ - f(t)f"(t)
One should notice that we may use this alternative form for a since
E^(t)
we know that lim ^,; : = 0. Theorems 3 and 4 show that L[F(t)]
possesses the properties which we desire.
Theorem 3: L[F(t)] -> F(«) as t -> «.
Proof:
2
We see that L[F(t)] -> F(«) as t -> «iff a jT^r
We consider two cases:
(1) a = 0: The result is obvious.
(2) a Tf* 0:
,.
In this case we have
af^t) _ ,.
l^^-^Tj^-^^^
t-x»
F(t) - F(cx>)
-2—
t-x»
I (t)
f'(t)
= lim [F(t) - F(oo)]
= 0.
f^t)
f'(t)
-• 0 as t -»- «.
19
Theorem 4;
t
If a ?4 0, then L[F(t)] -^ F(») more rapidly than F(t) as
-»• 00.
Proof:
iVe n e e d t o show t h a t l i m
^^'
F(<») - LfFTt')!
p L x - W T " °*
F(«>) - L [ F ( t ) ] _ F(oo)
FC ) - F
F (Ut ); +
+ aa ^^ ,-^ ^^^ ^ a^ f ' ^( t ) ^
+ 1
F(«') - F ( t )
F(co) - F ( t )
F(«) - F ( t )
Tlius
, . F(c») - L [ F ( t ) ]
^^"^
F(oo) - F ( t )
t-x»
a ^ ^ ^
_ . . ., . ° f ' ( t )
,
- l^"^ f^ ^ F(oo) - F ( t ) ^
t-^^"
1 + a ( - •^)
a
= 0.
It should be noted that more rapid convergence is not achieved if a = 0
since in this case L[F(t)] = F(t). Also, one should note that if
a = 1, then L[F(t)] = H [F(t)].
The following theorem gives a further
criterion for L[F(t)] to reduce to H^[F(t)].
Theorem 5:
If lim ^^^^^"^^^ = 1, then L[F(t)] = H [F(t)l.
t-x" [f'(t)]^
20
Proof:
a = 1,, F(t) - F(oo) ^ ^.^
t-K»
= lim
t^
f^(t)
f (t)
f(tl_^
t-x" f'(t)2f(t)f'(t) - f'^(t)f"(t)
[f (t)]
[f'(t)]
= lim
2[f'(t)]" - f(t)f"(t)
t-K»
2 -
f(t)f"(t)
[f (t)]
u
T
f(t)f"(t) _ ,
= 1
1 when
lim
-z- = 1.
t ^ [f'(t)]^
Now we will consider the use of L[G(t)] as an approximating
function.
As before, we must use L[G(t)l|^^^ to avoid having to
perform any integration.
a = lim -^^-^
Z(ti
7 . V
^ = T^ \
To evaluate a we notice that
as previously noted.
zm - ±
g'(t)
Thus for the approximating function L[G(t)]|^^^, we have
,rp,,^l
2m
g"(t)
2m - 1 g'(t) t=b
t2^-m+l -k arctan —
e
a
c(l + -^)
L[G(t)]^^^ -
2m
2m - 1
(2mt^^)
z
a
a
t=b •
The fact that L[G(t)] converges more rapidly than G(t) leads us to
expect that beyond a certain point we will get better results with this
approximation than with G(t), H^[G(t)], and H2[G(t)].
However, Table 2
shows us that these results are generally not better in the ranges of
b under consideration.
21
The B
Transformation
Another transformation which seems to have possibilities for our
use is the B^ transformation given by Gray and Schucany in [4],
Definition 3;
If f is differentiable on (b,«>) and F(t) = /^ f(x)dx,
'' b
then
2
B^[F(t)J = F(t) - tf'(t)^+^(t) ^^^^ ^f'(*^> + ^M
^ 0.
This transformation has the properties which we would like our transformation to have as the following theorems show.
Theorem 6:
lim B [G(t)] = S
t-x»
Proof:
We need to show that
2
, .g^
— T — T -• 0 as t ->- «.
tg^t)
tg'(t) + g(t)
2
2,, , t v-2m -2k arctan
tc (1 + —y)
e
2
ta
.,
t v-m-1 -k arctan — , k
2mtx , /-,
tc(l + —z)
e
^ ^~ ^
2^ + c(l
a
a
t
-k arctan —
-ce
a
2
(1 + \ ) ' " - ' (| + 2H|)
(^ + 4 "
f—
t
—
a
2
t
, t .-m -k arctan —
+ —)
e
a
a
22
Now as t -> 00, the numerator approaches -ce - ~
, and as t -> «», the
denominator approaches «. Thus as t ^ oo^
isict)—.0.
tg*(t) + g(t)
Now B^[G(t)] also has the desirable property that B [G(t)] converges to S more rapidly than G(t).
In order to prove this we must
first prove the following lemma.
Lemma 1:
If lim tf(t) = 0 and lim ^^,/5^1 ^ . .
t-Ko
t-Xx> ^^
^^^ +
exists and is finite
^(t)
but not zero, then B-[F(t)] converges more rapidly than F(t) to F(«').
Proof:
We need to show
2
~ ^ ' ^ ^^ f(^)
. . F(oo) - Bi[F(t)] _ - . F(«')
-V ^- F(t)
-vw +• ^.fi(^)
^^
F(«) - F(t) - [^
F(») - F(t)
^ , _ ,.
tf^(t)
.
^ ^ [tf'(t) + f(t)][F(t) - F(<»)] - "•
2
Thus we need to show that lim [tf.(t) + f(t)][F(t) - F(«)] " ^•
.,
tf^(t)
^ ,.
f(t)
t^"^ [tf'(t) + f(t)][F(t) - F(«>)] - ^ ^ [tf(t)]'
tf(t)
F(t) - F(c.)-
23
By hypothesis ^ ^ . ^ ^ ^ ^ j ,
exists and is not
zero.
Now tf (t) and F(t) - F(oo) approach zero as t ^ oo, so by L'Hospital's
rule,
Hn,
tf(t)
_
.
[tf(t)]'
t!!^(t)-F(»)-j--T(Fr-2
Thus lim
tf (t)
_ .
f (t)
tf (t)
\Z
[tf'(t) +f(t)][F(t) -F(»)] - ^ ^ [tf(t)]' • F(t) - F(co) = 1
It should be noted that Lemma 1 is a special case of a result published by Gray and Schucany in [4].
Theorem 7:
B [G(t)] converges to S more rapidly than G(t).
Proof:
We will show that our function g(t) satisfies the hypothesis
of Lemma 1.
t
2
T7J .. 1 • .. /..\
1•
~k arctan ~" /, , t v-m
^
.
First, lim tg(t) = lim ce
a (1 + — )
• t = 0 since
t->«>
t-too
a
m > — (actually m > — ) .
Also
g(t)
tg'(t)
+ g(t)
2
,
t
,. . t v-m -k arctan —
c(l + —2)
e
a
t^,-m-l , ka 2mt, ^ ,, ^ t^N-iHi -k arctan [tc(l + -2)
(- ~
2^
^^ + -2") ]e
a
a
a
a
-1
a
a
24
Thus lim ^ ,M,—Tzr
= lim
=
t-xx> tS*(t) + g(t)
2_
^
^
t(l + -^) ^ (- + ^ )
z
a
z
a
a
-1
2m - 1
- 1
^ 0,
We see that -—^^
exists since m ^ —.
2m — 1
z
Thus, since g(t) satisfies
the conditions of Lemma 1, we have that B^[G(t)] converges more
rapidly than G(t) to S.
As was done with the preceding transformations, B^[G(t)]|^^^
will be used as the corresponding approximating function. Explicitly
this function is given by
B^[G(t)]|^^^
2
,
t
,, . t v-m -k arctan —
tc(l + "2
-y)
e
a
t=b
a
a
a
It was originally thought that this type of transformation would
be our best since it did have the properties desired.
However, in
the ranges desired, the value of b in the integration was not sufficiently large to give good answers.
In fact, as can be seen from
Table 2, some of the answers are very poor.
In order to alleviate some of this trouble and still be able to
use this transformation, a translation was considered.
One should
note that /;f(x)dx = /;^, f(x - h)dx = j;^^ f,(x)dx where
25
f^(x) - f(x - h ) .
Now, using our approximating function, we have that
(b + h)f^(b + h)
- (b + h)f]^(b + h) + f^(b + h)
/^ f(x)dx = /^^^ f^(x)dx.
^^ ^"^ approximation for
Since f^(b + h) = f(b), we have that
R \v(t>i^ =
(b + h)f^(b)
i5j^Li^i.t;j
- (b + h)f'(b) + f(b)
should be a good approximating function for F(o°).
The problem next
arises as to which values of h give the best results.
In order to
have one approximating function it was hoped that there would be one
value of h which would give good results for all values of the parameters in g(x) and all five of the ranges involved.
Empirical results
show that such an h does not exist, and even an h which gives good
results for all values of the parameters for one range does not exist.
Thus it seems that this approximation will not be the one we desire
to use.
Continued Fraction Approximations
In order to obtain a relatively simple approximating function
with which to compare the results given by those we have derived, we
will truncate the continued fraction expansion mentioned in Chapter I
Truncating the expansion after the second convergent gives
^-k arctan f ^^^2m-2 ^^^^^^^ 1^
'2^'^ =
H^)
2m
(2m - 1) [k + ^ ] *
26
Algebraic and trigonometric manipulations reveal that C«(t) is actually
identical to L[G(t)].
Thus we see that since convergence of the con-
tinued fraction is slow when the value - is close to zero, the value
a
of C2(t) and consequently of L[G(t)] would not be expected to be
very accurate in this case. Also, the results should be very poor
when — is negative since the continued fraction does not converge to
the desired value in this case.
For further comparison we will consider the third convergent of
the continued fraction. We obtain the expression
-k arctan —
2m-2 ,
tv
i
a cos
(arctan —)
C,(t) =
3^^
F(r,k)
4m(m - 1)A
AB + C
where A = 2(2m + 1) [k(m - 1) + 2m(m + 1)-],
a
B = 2(2m - l)(m - 1) [k + - ^ ] ,
Si
C = 4(m - l)(m + l)(2m - l)[k^ + 4m^].
We see that this expression is comparable in complexity with H-[G(t)].
Table 2 gives a comparison of results given by C^(t) with those of
the other approximations.
CHAPTER III
COMPARISON OF RESULTS
In Chapter II several functions were proposed as candidates for
use as approximations for the area under the right tail of a Pearson
Type IV distribution.
In the present chapter we will present a
complete listing of the approximating functions in Table 1 along with
a comparison of the accuracy of these functions in Table 2.
Finally,
based upon the results obtained, recommendations concerning the use
of the approximations will be made.
Table 2 gives a comparison of the results given by the approximating functions with a "true value" which is taken to be the convergent of the continued fraction given by Shenton and Carpenter which
differs from its predecessor by at most 10
.
Since the convergents
eventually straddle the actual true value, we see that our "true
value" should be within 10~
of the actual true value.
One should note that the value of the parameter a is always
taken to be 1 in Table 2.
To illustrate the reasoning for this, we
will let X be a random variable with the corresponding density function
2
X
/, . X v-mo -ko arctan ^0
1
0
0
0
a
0
which is the Pearson Type IV function with parameters a^, k^, and m^
If we make the change of variable Y = |, we obtain a random variable
Y which has the density function
27
28
TABLE 1
APPROXIMATING FUNCTIONS FOR /* g(x)dx
TYPE
H,
APPROXIMATION
^/i . b v-m+1 -k arctan —
c(l + —j)
e
a
a
^2mb k_v
'^ 2 a''
a
^
- k a r c t a n — , , . b v-m+1 , „
3,
ce
a (1 + - ^ )
[2AB - C - A"^]
H,
[AC - B^]
2
b
/ , . b v-m+1 - k a r c t a n —
(1 + —j)
e
a
/
2m
V
c(Tr:
a
r)
'2m -
1'
,2mb
Icv
^
a''
a
2
£'
B.
u2
,
,
b
, / , , b v-m - k a r c t a n —
c b ( l + —)
e
a
a
b(i+4)"^ (^+^)
I
a
a
-1
i.
a
4c*
-k arctan—
2m-2 ,
^
bv,, /
ix-,!
e
a cos
(arctan —)[4m(m - 1)D]
a
F(r,k)[DE + F]
*Footnotes are on the following page.
**Footnotes are on the following page,
29
a
2
a
B = k^ - 2m ^ 2kb(2m + 1) ^ 2mb^(2m + 1)
r - k(6m - k^ + 2) . 6b(m + 1)(2m - k^)
a
6kb^(2m + 1) (m + 1)
a
4mb^(2m + 1) (m + 1)
** D = 2(2m + l ) [ ( m - l ) k + 2(m + 1)—]
E = 2(2m - l)(m - 1) [k +
^ ]
3.
F = 4(m - l)(2m - 1)(m + 1)[k^ + 4m^]
30
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33
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32
0 *• o' o'
O
0
O
which is the Pearson Type IV density function with parameters 1, k ,
and m^ for which the table is applicable.
As one can see from Table 2, all of the approximations give the
best results at the .01, .005, and .001 levels, which we would expect
The approximation B [G(t)]|
seems to be the poorest approximation
even though B [G(t)] possesses the property of more rapid convergence
As can be seen, the results with this approximation do improve for
the larger values of b.
Before considering the use of the remaining four approximations,
one should check the value of —.
If this value is negative, then
3.
Go(t)I _, and L[G(t)]| _, will most likely give very poor results
since the continued fraction does not converge to the value required.
Actually in this case the values of H^[G(t)]|^^^ and H2[G(t)]|^^^
are also poor, but as can be seen from Table 2 they are better than
C3(t)|^^^ and L[G(t)]|^^^.
For small positive values of —, less than .2, the results of
^
a
the approximations are also relatively poor with H [G(t)]|
and
H^[G(t)]| ,_ again giving the better results, with H [G(t)]|
2
t=b
^
*- ^
giving the better results of the two.
Considering values of — greater than .2, we will investigate
a
the accuracy of the approximations. At the .1 level H2[G(t)]|^^^
and C^(t)| ^ give the best results, varying by less than .007 from
3
t=b
33
the true value while H^[G(t)]|^^^ varies by less than .017 and
L[G(t)]|^^^ by less than .026. At the .05 level H [G(t)]|
C^LGCt)]!
and
give the best results, each varying from the true value
by no more than .005. At the .01 level C^(t)|
gives the best
results, varying from the true value by less than .0007 while
H^[G(t)]|^^^ and L[G(t)]|^^^ vary by less than .0015. C3(t)|^^^
also gives the best results at the .005 and .001 levels varying at
the .005 level by less than .0002 and at the .001 level by less than
.00003 from the true value. Of the two simpler functions, H [G(t)]| ,
gives best results at the .005 level, varying by less than ,0008 from
the true value; but at the .001 level L[G(t)]|
results, varying by less than .0001.
accuracy is not required H [G(t)]|
, gives the best
Thus it seems that if great
, should be used for levels other
than .001, and at this level and below L[G(t)]|^^^ will probably
give better results. However, if greater accuracy is required, one
should probably use H2[G(t)]|^^^ or C2(t)|^^^ at the .1 level and
C-^(t)| ^, for the lower levels.
LIST OF REFERENCES
1.
Elderton, William Palin. Frequency Curves and Correlation.
4th ed. Washington, D. C.: Harren Press, 1953.
2.
Gray, H. L., and Atchison, T. A. "A Note on the G-Transformation." Journal of Research of the National Bureau of
Standards; B, Mathematical Sciences, Vol. 72B. No. 1
(January - March, 1968), 29-31.
3.
Gray, H. L. ; Atchison, T. A.; and McWilliams, G. V. "Higher
Order G-Transformations." SIAM Journal Numerical Analysis,
(March, 1971).
4.
Gray, H. L., and Schucany, W. R. "Some Limiting Cases of the
G-Transformations." Mathematics of Computation, Vol. 23,
No. 108 (October, 1969), 849-61.
5.
Greenwood, J. Arthur, and Hartley, H. 0. Guide to Tables in
Mathematical Statistics. Princeton: Princeton University
Press, 1962.
6.
Johnson, N. L.; Nixon, Eric; Amos, D. E.; and Pearson, E. S.
"Table of Percentage Points of Pearson Curves, for given
/B, and 6p, expressed in Standard Measure." Biometrika,
Vol. 50 (1963), 459-98).
7.
Kendall, M. G., and Stuart, A. The Advanced Theory of Statistics , Vol. I: Distribution Theory. 2nd ed. New York:
Hafner, 1963.
8.
Pearson, Karl, ed. Tables for Statisticians and Biometricians.
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