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United StatesNaval Postgraduate School
THESISA GOODNESS OF PIT TEST FOR
BIVARIATE NORMAL DISTRIBUTIONS
by
James Edward Miller
April 1970
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I 1 J y _t
A Goodness of Pit Test for
Bivariate Normal Distributions
by
James Edward MillerLieutenant Colonel, United States Marine Corps
B.S., University of Utah, 1964
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN OPERATIONS RESEARCH
from the
NAVAL POSTGRADUATE SCHOOLApril 1970
c
ABSTRACT
This paper is an investigation of a goodness of fit
test for bivariate normal distributions. The test pro-
cedure is based on random linear functions of bivariate
normal random variables. The test makes use of the maximum
Kolmogorov D(M) statistic over the linear functions which
are computed. An estimate of the distribution of M is
obtained by computer simulation. No attempt is made to
determine the power of the test.
IPTGPADUATE SCHOOD
[, CALIF. 9394Q
TABLE OF CONTENTS
I. INTRODUCTION 7
II. THEORY 10
III. EMPIRICAL RESULTS 18
IV. RANDOM VARIABLE GENERATION 23
V. SUMMARY AND CONCLUSIONS 27
COMPUTER PROGRAM 32
BIBLIOGRAPHY 39
INITIAL DISTRIBUTION LIST 40
FORM DD 1^73 41
LIST OF TABLES
I. Distribution (Relative Frequency) ofKolmogorov-Smirnov Statistics that Resultfrom 100 Linear Combinations of 100Bivariate Random Vectors from TwoBivariate Normal Distributions 29
II. Distributions (Relative Frequency) of 500Maximum Kolmogorov-Smirnov Statistics,Each of which was Derived from 25Linear Combinations of Samples fromVarious Bivariate Normal Distributions 30
III. Distribution (Relative Frequency) of 500Maximum Kolmogorov-Smirnov Statisticsfor Samples Drawn from Various BivariateNormal Distributions with IdenticalCorrelation Coefficient, p 31
I. INTRODUCTION
An important application of Statistics is to attempt
to find a specific probability distribution which fits an
observed sample of data. A well fitting distribution can
then be used to predict values of future occurrences,
relative frequencies of future occurrences, etc. There
are numerous methods available for testing the goodness
of fit of data in scalar form to a hypothesized univariate
probability distribution. Among these are the Chi-square
and the Kolmogorov-Smirnov tests, of which the Kolmogorov-
Smirnov test is considered the more powerful [1] . Further-
more the Kolmogorov-Smirnov test exhibits the very
attractive characteristic that it is based on a statistic
which has a distribution of the random variable being
sampled.
However, there appears to be a lack of methods for
testing the fit of hypothesized multivariate distributions
to multivariate data (observations in vector form).
Furthermore, no statistic which has the desirable character-
istic of being distribution free and which can be used in
multivariate goodness of fit tests has been found. In
fact, no such statistic may even exist. For instance,
Simpson [2] has shown an example of continuous bivariate
distributions for which the analog of the Kolmogorov-
Smirnov statistic is dependent on the underlying distribution.
Rosenblatt [3] discusses a possible test which involves
a transformation of an absolutely continuous k-variate
distribution into the uniform distribution on the
k-dimensional hypercube« The transformation is uniquely
determined by the theoretical distribution against which
the sample is to be tested. Then the transformed sample
may be tested against the uniform distribution in
k-dimensions . There are several disadvantages to this
procedure, however. For example, the results are influenced
by the manner in which the components of the observed
vectors are ordered.
The purpose of this paper is to describe a goodness of
fit test for testing a bivariate normal distribution (with
given mean and covariance matrix) against samples of bi-
component data. The test results in acceptance or rejection
of the hypothesis that F„ = F, where F„ is the cumulative
distribution function of the population of the bivariate
sample vectors and F is the hypothesized cumulative distri-
bution function. The notation in this paper closely follows
the notation used by Anderson [4], It Is expected that the
test developed here for the bivariate normal distribution
can be extended to the case of the k-variate normal distri-
bution. This paper is restricted to a consideration of
testing the fit of samples to a distribution which has
zero mean. This restriction causes no loss of generality
since any distribution with a finite mean can be translated
to mean zero by a linear transformation.
8
The goodness of fit test was developed according to the
following procedures
:
1) a characterization of the bivariate normal distri-
bution is used to develop a test statistic M for use in a
goodness of fit test,
2) the distributional properties of M are investigated
by computer simulation.
II. THEORY
Since there appeared to be no widely known statistic
for a reasonable goodness of fit test for multivariate
distributions in general, and the multivariate normal
distribution in particular, it seemed plausible that a
statistic suitable for a goodness of fit test might be
found by considering characterizations of the multivariate
normal distribution.
One property which characterizes a multivariate normal
distribution is given in the following theorem [4]
:
Theorem 1, A p-dimensional random variable X has a
p-variate normal distribution, if an only if every
linear function of X has a univariate normal distri-
bution .
The parameters of the univariate normal distribution can be
computed according to theorem 2.
Theorem 2. Let X (a column vector with p components)
be distributed according to N (y,E) a multivariate
normal distribution with mean (vector) u and covari-
ance (matrix) Z, and let C be a row vector of p
constants. Then
Y = CX
is distributed as univariate normal with mean Cy and
variance CZC (C is the transpose of C).
10
(NOTE: CX can be described as a linear combination of the
components of X.)
Prom the characterization of the multivariate normal
distribution given in theorem 1, it was felt that a suitable
goodness of fit test procedure might be to test the result
of a linear combination of the sample vectors (whose distri-
bution had been hypothesized as a specific multivariate
normal distribution) against the hypothesized theoretical
univariate normal distribution which has been computed for
the particular linear combination. Thus the problem is
reduced to the univariate level and use can be made of well
known univariate statistics which provide acceptable good-
ness of fit tests.
However, theorem 1 states that every linear combination
of multivariate normal random variables must be univariate
normal. Obviously one linear combination will not suffice
for a reasonable test. It is not difficult to envision
that there exists some linear function of nearly any vector
sample which will transform that vector sample into one
which is accepted as univariate normal. In fact, if the
marginal distribution of the components are univariate
normal, but the joint distribution is not multivariate
normal, the linear combination consisting of one component
(e.g. y = X-, + OX2
. . . + 0XN= X, ) is univariate normal. Thus
a test which uses only one linear combination might be
manipulated by the tester to give any results he desires.
11
On the other hand, it is clearly impossible to compute
every linear combination of a sample. As a compromise, it
was felt that a number (to be determined) of randomly
selected linear combinations would serve as a representa-
tive sample upon which an overall test statistic might be
based. To produce random linear combinations, the (column)
sample vectors were multiplied by a (row) vector of random
constants. The random components of the 'multiplying
vectors' were drawn from the uniform (0,1) distribution.
A uniform (0,1) distribution for the random multipliers
was used because:
1) Up to multiplicative constants, essentially any
linear combination of the components of the multivariate
vector could be produced using coefficients from the uniform
(0,1) distribution, and
2) A component of a random multiplier was equally
likely to be contained in any one interval in (0,1) as in
any other interval, provided the intervals were of equal
length. Thus there should be no specific interval contain-
ing a 'concentration' of the multipliers which might
adversely influence the performance of the goodness of fit
test
.
NOTE: The results of the goodness of fit test described
in this paper using random multipliers from a uniform (0,1)
distribution were the same as results obtained using random
multipliers from a uniform (-2,2) distribution.
12
The Kolmogorov-Smirnov test was employed to determine
acceptance (or rejection) of the hypothesis that the
linear combinations of bivariate sample vectors are from
the (computed) theoretical univariate distributions noted
in theorem 2. As noted previously, the Kolmogorov-Smirnov
test is considered more powerful than the Chi-square test.
Of course, the distribution free characteristic of the
Kolmogorov D statistic applies in particular to linear
combinations of the components of multivariate normal
random variables. A description of the Kolmogorov-
Smirnov test is presented in the following paragraphs.
One method of testing the simple hypothesis, H: F., = F,
where F^ is the cumulative distribution of the population
sampled and F is the theoretical continuous distribution
proposed for the population, is the Kolmogorov D statistic
[5]. The asymptotic distribution of D was investigated by
Kolmogorov and tabulated by Smirnov [6] and, for small
sample sizes, by Massey [7].
The Kolmogorov D statistic is derived from the sample
cumulative distribution function, SN , and the proposed
theoretical cumulative distribution function, F, as follows;
Let Y,,...YM be a random sample from a continuous population
with cumulative distribution function F. Let Z-,,...Z„ be
the ordered statistics of Y, so that
-co < Zl < z2
< ... < zN
<
13
The sample cumulative distribution, then, is
SN(x) = J/N
1
< Z.
< Z1
< x < Z , j = 1,...,N-1
> ZN
The Kolmogorov D statistic is defined as
D = Sup|
SN(x) - P(x)
x
and can be described roughly as the maximum deviation of
the sample cumulative distribution function from the pro-
posed theoretical cumulative distribution function. The
D statistic has the property that its distribution does
not depend upon the underlying distribution P. Clearly,
it is dependent on the sample size N, because the sample
cumulative distribution functions, S„, takes as values only
multiples of 1/N. Naturally the D statistic approaches
zero, almost surely, as N becomes large without limit, pro-
viding the sample is actually from a population with
distribution F. Critical values, T, of the D statistic are
obtained from the tabulated distributions and are used with
a sample to determine acceptance or rejection (reject if
D > T) of the hypothesis that FN
= F.
One value of the Kolmogorov D statistic is derived from
each linear combination of a bivariate sample of vectors.
For example, let X = (X,,...XM ) where X. = (x,^,x.p) and
x . . is a scalar, be a random sample of size N of bivariate
14
random vectors. A linear combination Y = CX, where
C = (c-,,Cp) and c. is a scalar, is a vector of N scalars,
(Y., s . . . Y„) . Let Z be the hypothesized covariance matrix
of the distribution of X. To obtain a rough test of the
hypothesis that the distribution of X is bivariate normal
with covariance matrix E (and mean zero), one may test the
hypothesis that Y = CX is distributed as univariate normal
with variance CZC' (and mean zero). A Kolmogorov-Smirnov
test to determine the acceptance of the hypothesis when
one particular value of C, say C, is used to compute a
linear combination will yield one value of D, that is
D = Sup|FT(y) - S*(y)|
y
where Fy is the hypothesized univariate normal cumulative
distribution of Y and S^ is the sample cumulative distribu-
tion of the transformed sample. When the procedure listed
above is repeated for a different value of C, say C., then
another value of D, which may or may not be identical to
the first value, is obtained. If every value of D obtained
with various values of C is less than the critical value of
D for the given sample size and level, then the hypothesis
is accepted. Likewise if every value of D obtained from
using various values of C is greater than the critical
value, then the hypothesis is rejected. However, some
linear combinations of typical samples from a bivariate
normal population can be expected to give values of D
15
exceeding the critical value while other linear combinations
may give values of D less than the critical value. To
eliminate the ambiguity of such results, another statistic
must be used, preferably one whose distribution function
can be readily tabulated or computed. For this purpose we
use the maximum of the D statistics, M, derived from
Kolmogorov-Smirnov tests of a large number, m, of linear
combinations of the sample. That is,
M=max sup |SC j (y) - F*(y)| i = 1,2,. ..m
i y 1
where S„ y is the sample cumulative distribution functioni
of the linear combination C.X and F.(y) is the hypothesized
theoretical cumulative distribution of the linear combina-
tion C.X.1
The values of the D statistics derived from linear com-
binations of a particular sample appear to be more highly
dependent on the sample than on the random multiplying
vectors (see Section III, Empirical Results). Therefore
the maximum D might be expected to be a result of only the
sample so that rejection or acceptance of the hypothesis
H: F.. = F , where FN is the cumulative distribution function
of the population sampled, and F is the proposed theoretical
cumulative distribution function, would depend only on the
bivariate sample.
The distribution of M, as defined above, appeared to
be intractable to get in closed mathematical form. However,
16
the properties of the distribution of F were investigated
for several cases by examining empirical data obtained from
computer simulation. The data was produced by generating
samples of a specified bivariate normal distribution, com-
puting random linear combinations of the sample vectors,
and recording the maximum of the resulting Kolmogorov D
statistics. This procedure was repeated to give several
lists of 500 values of M. Each list of 500 values of M
was derived from bivariate sample vectors with different
underlying bivariate normal distributions. The empirical
results and generating techniques are discussed in Sections
III and IV.
17
III. EMPIRICAL RESULTS
In order to obtain the empirical data to study the
distribution of M, the maximum D statistic, a computer
program was written to accomplish the following for
specific selections of the covariance matrix, E:
1. Generate a sample of desired size of bivariate
normal random vectors from the given distribution,
2. Generate the desired number of multiplying vectors,
each of which would produce one linear combination of the
sample vectors,
3. Compute the linear combinations of the sample
vectors by vector multiplication,
4. Perform a Kolmogorov-Smirnov test on each univariate
sample obtained as a result of a linear combination and
record the resulting value of D. The values of
D = sup |SN(x) - P(x)
x
when S^ and F are functions previously defined, were
determined at values of .OIK (K = 1,...100) for the proposed
theoretical distribution P. For example, the value of the
sample cumulative distribution, SN(x), was evaluated at each
point x. where F(x.) was multiple of .01, D being assumed
to be the maximum value of the 100 differences
sN ( Xl ) - P( Xl )
18
5. Record the maximum value, over the linear combina-
tions performed on each sample, of the D statistics pro-
duced by each particular sample.
The initial simulation procedure generated sets of 100
random vectors from a bivariate normal distribution with
covariance matrix E. One hundred 'random' linear combina-
tions of each set of vectors were computed and the D
Statistic derived from each linear combination was recorded.
The results of this simulation indicated that the D sta-
tistics for a given sample were grouped within an interval
approximately .05 units in length, but the location of the
interval was dependent upon the particular sample. This
phenomenon suggested that the value of D is dependent on
the sample to a higher degree than it is on the linear
function used. The results of five such simulations for
samples from each of two different bivariate normal distri-
butions are summarized in Table I. Note that with sample
number lb, several D values exceeded the univariate
Kolmogorov-Smirnov critical value at the .05 level of sig-
nificance. Thus, using the univariate Kolmogorov-Smirnov
critical value, the hypothesis that the sample was from the
underlying distribution from which it was generated would
have been rejected for some linear combinations and
accepted for others. But using a critical value (determined
by level of significance and sample size) for maximum D
would have eliminated the ambiguity.
It was also found that the relative frequency, within
each interval of length .01, of the D statistics remained
nearly constant as the number of linear combinations was varied.
19
The data produced by the simulation procedure described
above led to the consideration of using the maximum D
Statistic for testing a bivariate sample against the pro-
posed bivariate normal distribution. This simulation data
indicates that the maximum D statistic derived from random
linear combinations of samples from a bivariate normal
population had the desirable characteristics that:
1) A unique maximum is obtained for each sample,
independent of the random multipliers, provided a sufficient
number of linear combinations are computed, and
2) The maximum value is obtained from various linear
combinations, at least one of which could be randomly
selected, with high probability, in as few as 25 trials
(selections) of multiplying vectors. In all cases investi-
gated, including those listed in Table I, the same value of
M was achieved over 25 linear combinations as was achieved
in 100 linear combinations for each particular sample.
As noted in Section II, the exact distribution of M was
found to be intractable. Therefore, in order to study
some of the characteristics of the distribution of M,
another computer simulation procedure was used to produce
a large sample of M. The simulation procedure may be
described as follows:
1) A sample of 25 vectors was generated from a pre-
determined bivariate normal distribution. Since D, and
therefore M, are dependent on sample size, it was recog-
nized that data obtained by this simulation would pertain
20
to samples of size 25 only. However, one might expect the
characteristics of the distribution of M to be similar for
all sample sizes.
2) Twenty-five randomly selected multiplying vectors
were generated so that 25 linear combinations of each
sample were produced. The maximum D over the resulting 25
univariate samples was recorded. (From the initial simula-
tion, it was expected that 25 linear combinations would
produce the maximum D for any sample.) A total of 500 N
statistics, all derived from the same underlying bivariate
normal distribution, were produced.
The simulation procedures were repeated for different
parameter values of the underlying bivariate normal distri-
bution to produce five sets of 500 statistics. Thus, each
set of 500 values was derived from linear combinations of
samples drawn from a different bivariate normal distribution.
The results of the simulation described above are sum-
marized in Table II. Unfortunately, it appears that there
is not a simple statistical relationship between the distri-
butions of the M statistics obtained with the samples drawn
from different bivariate distributions. And, of course,
if each different underlying distribution (of the sample)
produces a different distribution of the M statistic, it
would be impossible to tabulate values of all distribution
functions of M.
21
It did not seem unlikely that a similarity or other
relationship existed between the distributions of M
statistics obtained from samples which were derived from
bivariate normal distributions with identical correlation
coefficients. Therefore, a final simulation procedure was
repeated, using samples drawn from several non-identical
bivariate normal distributions with constant correlation
coefficients. The results are summarized in Table III.
Although there is a notable similarity between the
distributions of M statistics derived from bivariate normal
distributions with identical correlation coefficients (p),
the hypothesis that the distributions are identical was
rejected by a Kolmogorov-Smirnov test at the .05 level
of significance. This is also readily apparent for the
case in which p = .3162. Note that the difference in the
means of the samples of M is .0059, whereas one standard
deviation of the mean (computed from the sample standard
deviation) is approximately .0022. Thus the means are
nearly three standard deviations apart, which suggests that
the distributions are not the same.
An interpretation of these results and how they may be
applied to a possible goodness-of-fit test for the bivariate
normal distribution is discussed in Section V, Summary and
Conclusions
.
22
IV. RANDOM VARIABLE GENERATION TECHNIQUES
In order to study the distribution of the M statistics
described in Sections II and III, it was necessary to
produce a large number of random variables from various
bivariate normal distributions. There are several possible
methods which might be used to generate the bivariate
normal random variables on a computer. One method would
be to generate independent normal random variables and
perform an appropriate transformation on them which will
produce a bivariate normal random vector. For example, to
generate random vectors from a bivariate normal distribu-
tion with mean zero and covariance matrix Z, where Z is
symmetric and positive definite, one could use the follow-
ing procedure.
1) Generate two independent random variables, from a
normal (0,1) distribution, so that X = (X-.,X„) is bivariate
normal (0,1) where I is the identity matrix.
2) Perform the transformation Z = CX, where C satisfies
CC = Z. Then Z = (Z,,Z2
) is bivariate normal (0,Z).
In this study the bivariate normal random vectors were
generated using a conditional distribution approach. It
is a well known characteristic of the bivariate normal
distribution that if X = (X, ,X?
) is distributed bivariate
normal (u,Z) where y=(u,,Up) and
23
a12
a12
a21
a22
then Xp is distributed univariate normal (Up,<jpp). It
is also well known that the conditional distribution of
X.. , given Xp = Xp is univariate normal
[u-,+a-,papp (Xp-yp ) ,a.. .. -a.. pOp" a p-i] • Therefore, after
generating X?
from a univariate normal (Up,cjpp) distribu-
tion, the conditional distribution of X, , given Xp = x? ,
was computed and X-. was then generated from that univariate
normal distribution.
To verify that this produces a random vector with the
characteristics of the given bivariate normal distribution,
consider the following:
Let y = (0,0) = (\i1,U
2)
Generate X?
= x?
from its marginal distribution, N(0, Opp).
Then,
E(X2
) =
and
E[(X2
- y 2)
2]
= E[(X2
)
2] = c
22
Generate V, independent of Xp, from univariate normal
(0,1) distribution.
Now
E(V) = 0, and
E(V 2) = 1
24
-1 3* -1Now let X
1 = ( a-i -i
— an p
a22 a 21^
2 V + °12 a22 ^ X 2^*
X, is univariate normal (o-.~°?n~ (X? ), a-.^-a^~a~ a
?1 ) , and
E(X1
) = E[(o11
-o12
o2
~ 1o21 ) V + a
12a2
~ 1(X
2)] = = y
1
Similarly, the covariance between X.. and X?
is
Cov(X]_,X
2) = E[(X
1-y
1)(X
2-y
2)] = EU-^)
-1 h -1= E{ [(o
11-a
12o22
o21 )
2 V + cj12
q22
(X2)]X
2>
—
1
h -1= ( a 11
-a12
a22
a 21')
2 E ( v ' x2
) +a i2a22
E ^ X2^*
Since V and X?
are independent,
E(V • X2
) = E(V)E(X2
) = 0.
Continuing from above,
Cov(X.,,X2
) = + o-.2
<j22 (a 22 ) = a-,- = o
2-.
The variance of X-. is
E[(X1
- yx
)
2]
= E(X2
)
"1"I Q
= E{[ (o 11
-o12
o?
~ ±o21
)^ V + cf
1 2a 22 ^ X2^ *
-1 2 —1 J"=
^ a il~°12a22
a2i^
E ^vL"^ + (
^ a ii"ai2
a22
a2i^
E^
(a12
a22
1) E(X
2) + (o
12a22
1)
2E(X
2
2)
= o11
-a12
a2 ;
1c21
+ °12
c2 l
1°12
= ai;L
,
since o^2
= o2
~ .
25
Incidentally, this technique can be extended to a method
of generation of p-component multivariate normal random
variables, since the distribution of X . (i=l ,2 , . . . p) given
any X. = X . ( j = l ,2 , . . . p , j¥i) , is also a normal distri-j J
bution whose parameters may be computed.
Standard computer routines were used to generate the
univariate normal and uniform random variables required for
the simulation procedure previously described. The routines
are shown in the computer program under Subroutine RANDU,
(for uniform random variables) and Subroutine GAUSS (for
univariate normal random variables).
26
V. SUMMARY AND CONCLUSIONS
The empirical data indicates that the distribution of
the maximum of the D statistics (M) , derived from Kolmogorov-
Smirnov tests of linear combinations of samples from
bivariate normal distributions, was dependent upon the
covariance matrix of the underlying distribution of the
sample. Therefore it would be impossible to tabulate the
distribution of M except for specific parameters of the
underlying distribution.
However, a goodness of fit test for the bivariate nor-
mal distribution can be constructed using the M statistic.
The test might consist of using a simulation procedure,
similar to that used in this paper, to produce a sample
distribution of M. This distribution of M would be derived
from samples which are from a bivariate distribution
identical to the proposed hypothesized bivariate distribu-
tion. Then a critical value of M, for a test with level
of significance a = a , may be established as the value at
which the (1 - a ) percentile point of the distribution
of M occurs. Obviously the number of linear combinations
and the number of M statistics for development of the
sample distribution of M must be determined by the experi-
mentor performing the test. (Note that the size of the
samples generated in the simulation procedure must be iden-
tical to the size of the sample to be tested.
27
For example, suppose one wishes to test the hypothesis
that a sample of N vectors was drawn from a population whose
5 2distribution is bivariate normal (0,1) where E = (_ _-)j
(one of the distributions for which M is tabulated in Table
II). If N = 25, then the distribution of the M statistics
is shown in Table II, listed under the appropriate covari-
ance matrix. For a = .05, the critical value of M is . 33 s
the value at which the .95 percentile point of the distri-
bution occurs. Then 25 random linear combinations of the
sample vectors would be computed and the 25 resulting uni-
variate samples tested against the computed univariate
distribution by a Kolmogorov-Smirnov test. If the maximum
of the 25 D statistics thus obtained is greater than . 33 s
the hypothesis is rejected. Otherwise the hypothesis Is
accepted.
There are obviously many interesting aspects concerning
this (and other) multivariate goodness of fit tests which
should be investigated. For example the power of the
test described in this paper, when applied to samples from
distributions other than the bivariate normal, might be
investigated. Also, a goodness of fit test based on a
statistic other than M (e.g., the mean or variance of D
obtained from linear combinations of the sample components)
might prove to be interesting. It is, of course, desirable
to find a "reasonable" statistic for which the distribution
may be found and tabulated.
28
TABLE I
Distribution (Relative Frequency) of Kolmogorov-SmirnovStatistics (D^) that Result from 100 Linear Combinationsof 100 Bivariate Random Vectors from Two Bivariate NormalDistributions
Valueof K-S
£ ="5
3
_3 5_E =
"5 2"
1 2
SAMPLE NUMBER
Statistic la lb :Lc Id le 2a 2b 2c 2d 2e
.04 2
.05 4 21 25 10 38
.06 11 50 66 35 41
.07 26 2 24 9 23 19 8
.08 6 22 5 17 9 28
.09 44 20 J46 22 14 36 6
.10 9 17 30 32 9 18 33)
.11 4 29 10 29|
.12 9 20
.13 39 14
.14 11
.15
NOTE: 1) The critical value of the univariate Kolmogorov-Smirnov statistic for this sample size anda = .05, is . 136.
2) £ = Covariance matrix of the distribution fromwhich the sample was drawn.
29
TABLE II
Distribution (Relative Frequency) of 500 Maximum Kolmogorov-Smirnov (M) Statistics Each of which was Derived from 25Linear Combinations of Samples from Various Bivariate NormalDistributions
Covariance Matrix of Distribution of
Range ofMax. K-
Statistic (M)
Population from which Samples were Drawn
^ 313 5
"4 2
2 1
"4 o"
1
"4 -1,8"-1.8 1
"5 l"
1 2V--il__Jl__Jl_ _l l_ _J
.06-. 07 1
.07-. 08 3
.08-. 09 7
.09-. 10 22
.10-. 11 22 1
.11-. 12 5 40 4
.12-. 13 8 44 5 6
.13-. 14 21 26 7 5 9
.14-, 15 31 55 18 12 21
.15-. 16 28 34 19 19 32
.16-. 17 38 37 31 23 43
.17-. 18 34 32 24 30 59
.18-. 19 41 22 41 35 35
.19-. 20 42 31 42 42 38
.20-. 21 39 20 50 40 31
.21-. 22 30 25 41 36 26
. 2 ?- . 2 3 27 16 33 32 31
.23-. 24 28 14 34 49 21
.24-. 25 25 12 31 38 33
.25-. 26 29 8 30 23 21
.26-. 27 20 6 20 21 20
.27-. 28 8 5 15 21 14
.28-. 29 18 6 15 21 15
.29-. 30 11 4 10 13 15
.30-. 31 7 3 7 8 8
.31-. 32 8 2 7 9 7
.32-. 33 2 7 8 7
.33-. 34 6 17 10 11
.34-. 35 2 12 3
.35-. 36 3 3 3 5
.36-. 37 2 2 2
.37-. 38 1
Wean* .2033 .1624 .2151 .2237 .2061
Variance* .0028 .0027 .0024 .0027 .0028
NOTE: Sample mean and Sample variance of M statistics
2) Each sample size was 25 bivariate vectors.
30
TABLE III
Distribution (Relative Frequency) of 500 Maximum Kolmogorov-Smirnov Statistics (M) for Samples Drawn from VariousBivariate Normal Distributions with Identical CorrelationCoefficient, p
Corr.Coeff.(p) p = .6 P = o P
=S±°_ =. 3162w
10
Range^\^of M ^s
5 3 1 .6 4 1 5 1 1 . 3162
3 5 .2 1 1 1 1 2 .3162 1
.10-. 11 1 1
.12 5 3 4
.13 8 8 2 5 6 4
.14 21 18 3 7 9 8
.15 31 27 19 12 21 21
.16 28 51 31 31 32 29
.17 38 41 26 33 43 31
.18 34 38 25 46 59 26
.19 4l 51 39 55 35 44
.20 42 31 35 29 38 39
.21 39 26 38 37 31 41
.22 30 28 43 34 26 35
.23 27 27 45 31 31 33
.24 28 21 37 23 21 44
.25 25 26 23 24 33 15
.26 29 16 30 23 21 28
.27 28 25 22 26 20 16
.28 8 9 10 19 14 17
.29 18 16 21 13 15 18
.30 11 6 14 16 15 11
.31 7 9 11 7 8 9
.32 8 5 12 9 7 9
.33 j3 4 3 7 3
.34 6 1 1 2 11 4
.35 2 2 4 6 3 3
.36 3 5 4 6 5 3
.37 1 3 2 3 2
.38 1
Mean* .2032 .1998 .2168 .2130 .2061 .2120
Variance .0028 .0028 .0026 .0027 .0028 .0026
NOTE: 1) Mean and variable of M.
2) Sample size - 25 vectors.
3) E = covariance matrix or distribution of popula-tion from which samples were drawn.
31
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BIBLIOGRAPHY
1. Ostle, B., Statistics in Research , low State, 1963*
2. Simpson, P. B., "Note on the Estimation of a BivariateDistribution Function,: Annals of MathematicalStatistics , v. 22, p. 476-478, 1951.
3. Rosenblatt, M.
, "Remarks on a Multivariate Transfor-mation," Annals of Mathematical Statistics , v. 22p. 470.472, 1952.
4. Anderson, T. W. , An Introduction to MultivariateStatistical Analysis , Wiley"] I960
.
5. Kolmogorov, A., "Sulla Determinazione Empirica diUna Legge di Distribuzione ,
" Giornale dell ' Istitutedegi Attuari , v. 4, p. 1-11, 1933-
6. Smirnov. H. , "Sur les Ecarts de la Courbe de Distri-bution Empirique," Recueil Mathematique , v. 6,p. 3-26, 1939.
7. Massey, F. J., "The Kolmogorov-Smirnov Test forGoodness of Fit," Journal of the American StatisticalAssociation , v. 46, p. 68-78, March 1951.
39
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 20Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 93940
3. Asst. Professor D. R. Barr 1
Department of Operations AnalysisNaval Postgraduate SchoolMonterey, California 93940
4. LCOL James E. Miller, 067320, USMC 1
Third Marine Aircraft WingMarine Corps Air StationEl Toro, California
5. Commandant of the Marine Corps (Code A03C) 1
Headquarters, U.S. Marine CorpsWashington, D.C. 20380
6. James Carson Breckinridge Library 1
Marine Corps Development and Educational CommandQuantico, Virginia 22134
7. Department of Operations Analysis (Code 55) 1
Naval Postgraduate SchoolMonterey, California 93940
8. Commandant of the Marine Corps (Code AX) 1
Headquarters, U.S. Marine CorpsWashington, D.C, 20380
40
UNCLASSIFIEDSecurity Classification2
DOCUMENT CONTROL DATA -R&D(Security classification of title, body ol abstract and indexing annotation must be entered when the overall report Is classified)
I ORIGINATING ACTIVITY (Corporate author)
Naval Postgraduate SchoolMonterey, California 939^0
la. REPORT SECURITY CLASSIFICATION
Unclassified2b. GROUP
3 REPORT TITLE
A GOODNESS OF FIT TEST FOR BIVARIATE NORMAL DISTRIBUTION
4 descriptive NO T E S (Type of report and, inclusi ve dates)
Master's Thesis; April 19708 AUTHORISI (First name, middle initial, laal name)
James Edward Miller, Lieutenant Colonel, United States Marine Corps
6 REPOR T D A TE
April 1970
la. TOTAL NO. OF PAGES
40
76. NO. OF REFS
78a. CONTRACT OR GRANT NO.
b. PROJEC T NO
9a. ORIGINATOR'S REPORT NUMBER(S)
96. OTHER REPORT NOIS) (Any other numbers that may ba assignedthis report)
10 DISTRIBUTION STATEMENT
This document has been approved for public release and sale;its distribution is unlimited.
II SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate SchoolMonterey, California 939^0
13. ABSTR AC T
This paper is an investigation of a goodness of fittest for bivariate normal distributions. The test proced-ure is based on random linear functions of bivariatenormal random variables. The test makes use of the maximumKolmogorov D(M) statistic over the linear functions whichare computed. An estimate of the distribution of M isobtained by computer simulation. No attempt is made todetermine the power of the test.
DD """ 1473t NOV 68 I "T # WS/N 0101 -807-681
1
(PAGE 1)UNCLASSIFIED
41 Security ClassificationA-31408
UNCLASSIFIEDSecurity Classification
KEY WORDSROLE W T ROLE AT ROLE W T
GENERATION OF BIVARIATE NORMALRANDOM VARIABLES
GOODNESS OF FIT TEST
BIVARIATE NORMAL DISTRIBUTION
I
DD F ° R:.,1473 back)
a 2
UNCLASSIFIEDSecurity Classification a - 3 i 4 01
ThesisM5873c.l
U:; 8
Mi HerA goodness of fit
test for bivariatenormal distributions.
ThesisM5873c.l
A J v^
MillerA aoodness of f i t
test for bivariatenormal distributions.
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