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SOME DISTRIBUTIONS ON THE POSITIVE
REAL LINE WHICH HAVE NO MOMENTS
By
Peter A. Lachenbruch and Donna R. Brogan
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 700
July 1970
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SOME DISTRIBUTIONS ON THE POSITIVE REAL LINE WHICH HAVE NO MOMENTS
By
Peter A. Lachenbruch and Donna R. Brogan
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Department of Biostatistics
University of North Carolina at Chapel Hill
A recent letter to the editor (Gross, 1969) prompts us to report on a distribution, defined on the positive real line, which has no moments.
of this distribution are also considered.
X>o.
00
Clearly,
J
o
x
(X+a)2
C'X d"
1verges, so no moments ex i st.
'.
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has moments up to order r-l.
I
This distribution is defined as
(1)
that the distribution
,.
Some generalizations
defined as
C
f(X) = - - - (X+a)r+l
(2)
More generally, we may show
where
C=ro.
r
The distribution (2) has the interesting property that its percentiles are
very simple to obtain.
For
r
r
ro.
d X = 1 __.::::.0._ _
(X+a)r+l
(t +a)r
(3)
p
and hence
(4)
t
=
P
I-V
aV
V
where
Note that for distribution (1), this gives t
p
=
(l_p)l/r
for the pth percentile.
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The distribution (1) was not created arbitrarily"
It arose when we were con-
sidering the distribution of the ratio of two independent exponential variables,
say Yl and Y2"
Let
1
f (Y ) = - e
l
Sl
(5)
1
f(Y2) = - e
62
-yl/S l
-Y/6 2
and
(6)
Y =XZ
1
y =Z "
2
Then
(7)
(8)
z
X
o
1
z.
=
So
f(X,Z) =
(9)
1_ Z e
SlS2
-xz/6
1
- z/S
2
and
1
OO
(10)
f(X)
= J f(X,
Z)dZ
= BS
foo -Z(X/S l +l/S 2 )
Ze
dZ
012 0
=
Sl~2
-Z(X/S +l/S )
2
l
•
=e~(X-/~S-1+-1-'/~S-'2)~2-
00
(-Z(X/S l +l/S 2 )-1)
J
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-3-
1
1
= 13 13
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1
1 2
where
This leads us to a "second" generalization of (1).
variables with parameters 8., 13., i=1,2.
~
That is,
~
8 . -1 -y. / 13 •
~
~
~
y.
e
(11)
f (y .)
~
~
= --=---.,,--8.
f(8.)I3.
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Suppose that Yi' are gamma
~
~
~
(Note that when 8.=1, f(y.) is the exponential distribution).
~
~
In this case the
distribution of X~Yl/Y2 is given by
(12)
f (x)
where a=l3 /13 .
l 2
It is easy to show that the distribution of V = X/X+a is a beta
distribution with parameters 8
1
and 8 .
2
This fact, of course, is well known.
This relationship allows us to calculate the percentiles of the distribution of X
easily.
8
1
If t
p
is the pth percentile of the beta distribution with parameters
and 8 , then the pth percentile of the distribution of X is atp/(l-t ).
p
2
this point we observe that if 8 =1, equation (12) becomes
1
At
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f (X)
(13)
= 62
6
e +1
a. 2/ (X+a.) 2
which is identical to equation (2).
Next, we briefly consider estimation of parameters for (12).
Maximum likeli-
hood leads to the following set of equations:
(14)
where
~(z)
d
= dz In f(z+C) , which is the digamma function.
equations cannot be solved without the aid of a computer.
moment estimators may be used.
(15)
When moments exist,
For the special case 6 =6 =1, we have
1 2
"&n -
2
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1
X.+Q
~
If n=l, then &=X .
l
Needless to say, these
If n=2, &=VX X .
l 2
=0
The geometric mean is not the MLE in general.
However, it seems to be a good starting value for an iterative procedure.
Other
methods of estimation use the median of the observations or the median of the
midranges.
The last method calculates all possible midranges
j=l, ••• ,n
k=j, j+l, ••• ,n
where X(j) is the jth order statistic from the sample.
midranges is then used as an estimate of a..
(2).
The median of the n(n+l)/2
This method was suggested by Moses
Some sampling experiments were done using these four methods.
Table 1 gives
the combinations of n and a. that were used in the study of the estimates .
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Table 1
Values of n for given ex.
ex.
1
5
5
5
10
10
20
20
50
50
Twenty estimates of ex. were computed for each (ex.,n) pair.
given in Table 2.
Some results are
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Table 2
Results of Sampling Experiment(l)
Notes:
a
n
Method (2)
Mean
S.D.
Skewness
Kurtosis
1
5
1
2
3
4
1.25
1.66
1.62
2.30
1.17
1.12
1.16
2.08
2.39
.83
1.21
1.84
8.8
2.94
4.18
5.55
5
5
1
2
3
4
3.94
4.51
4.83
8.29
3.67
2.77
3.09
7.53
2.23
.72
.87
2.25
8.01
2.06
2.94
8.34
1
10
1
2
3
4
1.23
1.16
1.25
2.24
.86
.60
.72
1.58
.79
.91
1.05
1.14
2.07
2.88
3.01
3.26
5
10
1
2
3
4
4.82
6.35
5.95
10.66
4.49
4.86
4.73
11.40
2.08
1.06
1.31
2.50
6.87
3.04
3.73
9.44
1
20
1
2
3
4
.79
.95
.94
1.46
.28
.35
.36
.57
-.16
.59
.95
.82
1.77
2.31
3.43
3.53
5
20
1
2
3
4
4.19
4.38
4.55
7.27
1.99
2.12
2.18
3.58
.85
1.42
1.50
1.24
3.59
5.06
5.22
4.35
1
50
1
2
3
4
.96
1.04
1.05
1.69
.23
.29
.25
.45
-.09
1.15
.36
-.12
2.00
4.03
2.41
1.90
5
50
1
2
3
4
5.00
5.02
5.03
8.17
1.40
1.19
1.22
2.17
.41
.30
.01
-.17
2.25
4.15
2.46
2.02
(1) These results are based on 20 samples at each (a,n) combination
(2) The methods are coded as
1 - Median
2 - Geometric Mean
3 - Maximum Likelihood
4 - Median of Midranges
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These results indicate the following:
1)
The median of the midranges is not a good estimator since it has larger
bias and larger standard deviation than the other three estimators.
The
positive bias is to be expected since the parent distribution is skewed
to the right.
2)
For samples of size 5, 10 and 20, the expectation of the geometric mean
estimator and the maximum likelihood estimator are approximately equal,
whereas the expectation of the median is slightly below these two.
For
samples of size 50, these three estimates have virtually the same expectation.
3)
The standard deviation of all the estimators except the median of the
midranges are approximately equal, regardless of sample size.
4)
For samples of size 5 and 10, the median is generally more skewed than the
geometric mean or the maximum likelihood estimators.
For larger sample
sizes the median is less skewed or has approximately the same skewness as
the other two estimators.
5)
For samples of size 5 or 10, the geometric mean and the maximum likelihood
estimators seem to have minimum kurtosis.
For samples of size 20 and 50,
the median seems to have less kurtosis than the other estimators.
One of us (P.A.L.) has found the distribution (1) useful as a teaching example
of a distribution with no moments, but whose percentiles are extremely simple to
obtain.
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References
1.
Gross, A.
Letter to the Editor, The American Statistician, Dec. 1969,
V. 23, No.5.
2.
Moses, L.
Query 10
v. 7, No.2.
Confidence Limits from Rank Tests,
Technometrics, 1965,