Transcript Dependence and Measuring of Its Magnitudes Boyan Dimitrov

Dependence and
Measuring
of Its Magnitudes
Boyan Dimitrov
Math Dept. Kettering University,
Flint, Michigan 48504, USA
Outline
What I intend to tell you here, can not be found in
any contemporary textbook.
I am not sure if you can find it even in the older
textbooks on Probability & Statistics.
I have read it (1963) in the Bulgarian textbook on
Probability written by Bulgarian mathematician
Nikola Obreshkov .
Later I never met it in other textbooks, or
monographs.
There is not a single word about these basic
measures in the Encyclopedia on Statistical
Science published more than 20 years later.
1.

1.
2.
3.
4.
5.


Introduction
All the prerequisites are:
What is probability P(A) of a random event A,
When we have dependence between random events,
What is conditional probability P(A|B) of a random event
A if another event B occurs, and
Several basic probability rules related to pairs of events.
Some interpretations of the facts, related to probability
content.
Who are familiar with Probability and Statistics, will find
some known things, and let do not blame the textbooks
for the gaps we fill in here.
For beginners let the written here be a challenge to get
deeper into the essence of the concept of dependence.
2. Dependent events. Connection between
random events
– Let А and B be two arbitrary
random events.
– А and B are independent only when
the probability for their joint
occurrence is equal to the product
of the probabilities for their
individual appearance,
i.e. when it is fulfilled
P( A  B)  P( A) P( B).
(1)
2. Connection between random events
(continued)

the independence is equivalent to the fact, that the conditional
probability of one of the events, given that the other event occurs,
is not changed and remains equal to its original unconditional
probability, i.e.
P( A | B)  P( A).


(2)
The inconvenience in (2) as definition of independence is that it
requires P(B)>0
i.e. B has to be a possible event.
Otherwise, when P(B)=0, the conditional probability
P( A  B)
P( A | B) 
P( B)
is not defined
(3)
2. Connection between random events
(continued)


A zero event, as well as a sure event is independent with any
other event, including themselves.
The most important fact is that when equality in P( A  B)  P( A) P( B).
does not hold, the events А and В are dependent.




The dependence in the world of the uncertainty is a complex concept.
The textbooks do avoid any discussions in this regard.
In the classical approach equation (3) is used to determine the
conditional probabilities and as a base for further rules in operations
with probability.
We establish a concept of dependence and what are the ways of its
interpretation and measuring when А and В are dependent events.
2. Connection between random events
(continued)

Definition 1. The number
 ( A, B)  P( A  B)  P( A) P( B)
is called connection between the events А and В.


Properties
δ1) The connection between two random events
equals zero if and only if these events are
independent. This includes the cases when some of
the events are zero, or sure events.
δ2) The connection between events А and В is
symmetric, i.e. it is fulfilled
 ( A, B)   ( B, A).
2. Connection between random events
(continued)

δ3)
 ( A, B)   ( B, A).

δ4)
 ( Aj , B)   ( Aj , B)

δ5)
 ( A  C , B)   ( A, B)   (C , B)   ( A  C , B)

δ6)
 ( A, A)   ( A, A )  P( A)[1  P( A)]
These properties show that the connection function
between events has properties, analogous to properties
of probability function (additive, continuity).
2. Connection between random events
(continued)


δ7) The connection between the complementary events
and B is the same as the one between А and В;
A
δ8) If the occurrence of А implies the occurrence of В, i.e. if
A  B, then  ( A, B)  P( A) P( A) and the connection between А
and В is positive. The two events are called positively
associated.


δ9) When А and В are mutually exclusive, then the
connection between А and В is negative.
δ10) When  ( A, B)  0, the occurrence of one of the two
events increases the conditional probability for the occurrence
of the other event. The following is true:
P( A | B)  P( A) 
 ( A, B)
P( B)
.
2. Connection between random events
(continued)

δ11) The connection between any two events А
and В satisfies the inequalities
max{  P( A) P( B),[1  P( A)][1  P( B)]}   ( A, B)
 ( A, B)  min{ P( A)[1  P( B)], [1  P( A)]P( B)}

We call it Freshe-Hoefding inequalities. They also
indicate that the values of the connection as a
measure of dependence is between – ¼ and + ¼ .
2. Connection between random events
(continued)

One more representation for the connection between the
two events А and В
 ( A, B)  [ P( A | B)  P( A)]P( B).
If the connection is negative the occurrence of one event
decreases the chances for the other one to occur.

Knowledge of the connection can be used for calculation of
posteriori probabilities similar to the Bayes rule!
We call А and В positively associated when  ( A, B)  0,
and negatively associated when
 ( A, B)  0
.
2. Connection between random events
(continued)

Example There are 1000 observations on the stock
market. In 80 cases there was a significant increase in
the oil prices (event A). Simultaneously it is registered
significant increase at the Money Market (event В) в
50 cases. Significant increase in both investments
(event ) is observed in 20 occasions.
The frequency
estimated probabilities produce:
P ( A) 


80
 .08
1000
Definition 1 says
, P( B) 
50
 .05 ,
1000
P( A  B) 
20
 .02
1000
 ( A, B) =.02 – (.08)(.05) = .016.
If it is known that there is a significant increase in the
investments in money market, then the probability to
see also significant increase in the oil price is
P ( A | B ) = .08 + (.016)/(.08) = .4.
2. Connection between random events
(continued)

Analogously, if we have information that
there is a significant increase in the oil prices
on the market, then the chances to get also
significant gains in the money market at the
same day will be:
P ( B | A) = .05 + (.016)/(.08) = .25.

Here we assume the knowledge of the
connection  ( A, B) , and the individual
prior probabilities Р(А) and Р(В) only.
It seems much more natural in the real
life than what Bayes rule requires.
2. Connection between random events
(continued)

Remark 1.
If we introduce the indicators of the
random events, i.e.
IA  0
then
I A  1 , when event A occurs, and
when the complementary event occurs,
E( I A )  P( A)
and
Cov( I A , I B )  E( I A .I B )  E( I A ) E( I B )  E( I AB )  P( A) P( B)   ( A, B)

Therefore, the connection between two
random events equals to the covariance
between their indicators.
2. Connection between random events
(continued)


Comment: The numerical value of the connection does not speak
about the magnitude of dependence between А and В.
The strongest connection must hold when А=В. In such cases we
have
 ( A, B)  P( A)  P 2 ( A).


Let see the numbers. Assume А = В, and P(A)= .05. Then the event
A with itself has a very low connection value .0475. Moreover, the
value of the connection varies together with the probability of the
event A.
Let P(A)=.3, P(B)=.4 but А may occur as with В, as well as with
and P(A|B)=.6. Then
 ( A, B)  (.6  .3)(.4)  .12

The value of this connection is about 3 times stronger then the
previously considered, despite the fact that in the firs case the
occurrence of В guarantees the occurence of А.
3. Regression coefficients as
measure of dependence between
random events.



P ( A | B ) is the conditional
The conditional probability
measure of the chances for the event A to occur, when it is
already known that the other event B occurred.
When В is a zero event the conditional probability can not be
defined. It is convenient in such cases to set P ( A | B ) =P(A).
Definition 2.
Regression coefficient of the event А with
respect to the event В is
rB (A)

=
P( A | B)
--
P( A | B)
.
The regression coefficient is always defined, for any pair of
events А and В (zero, sure, arbitrary).
3. Regression coefficients
(continued)

Properties (r1) The equality to zero rB (A) = rA (B) = 0

takes place if and only if the two events are independent.
rA ( A)  1
(r2) rA ( A)  1 ;
.

(r3)
rB ( A j )   rB ( A j )

(r4)
rS ( A)  rø ( A)  0

(r5) The regression coefficients and are numbers with equal
signs - the sign of their connection . However, their numerical
values are not always equal. To be valid
rB (A) = rA (B)
it is necessary and sufficient to have
P( A)[1  P( A)] = P( B)[1  P( B)]
3. Regression coefficients
(continued)
(r6) The regression coefficients and are numbers between –1 and
1, i.e. they satisfy the inequalities
;
 1  r ( A)  1
B
 1  rA ( B)  1
(r6.1) The equality rB (A)= 1 holds only when the random event А
coincides (is equivalent) with the event В. Тhen is also valid the
equality r (B) =1;
A
(r6.2) The equality rB (A) = - 1 holds only when the random
event А coincides (is equivalent) with the event
complement of the event В.
Тhen is also valid
rA (B)= - 1, and respectively
B - the
AB .
3. Regression coefficients
(continued)
(r7) It is fulfilled
rB ( A)
= - rB (A) ,
rB ( A)
(r8) Particular relationships
 ( A, B)
rB ( A) 
,
P( B)[1  P( B)]
(r9)
=-
rB (A)
rA ( B) 
,
rB ( A)  rB ( A).
 ( A, B)
P( A)[1  P( A)]
3. Regression coefficients
(continued)

For the case where we have
A B
 ( A, B) = P( A)[1  P( B)]  P( A) P( B)
rB (A)
= P( A) P( B)
rA (B) = P ( B ) P ( A)
3. Regression coefficients (continued)

For the case where we have
 ( A, B) =  P( A) P( B)
rB (A) =  P( A)
P( B)
rA (B) =  P( B) P( A)
A B  
3. Regression coefficients (continued)



A  B   the measures
In the general case
of dependence may be positive, or negative.
If Р(А)=Р(В)=.5, and P ( A  B )=.3, then the
connection and both regression coefficients are
positive; if P ( A  B ) =.1, these all measures are
negative.
The sign and magnitude of the dependence
measured by the regression coefficients could be
interpreted as a trend in the dependence toward
one of the extreme situations A  B , A  B ,
or A  B   and the two events are independent
3. Regression coefficients (continued)

Example 1 (continued): We calculate here the
rA (B)
two regression coefficients rB (A) and
according to the data of Example 1.

The regression coefficient of the significant increase of
the gas prices on the market (event A), in regard to
the significant increase in the Money Market return
(event B) has a numerical value
rB (A) =(.016)/[(.05)(.95)] = .3368

In the same time we have
rA (B) = (.016)/[(.08)(.92)] = .2174 .
3. Regression coefficients (continued)









There exists some asymmetry in the dependence between random
events - it is possible one event to have stronger dependence on the
other than the reverse.
The true meaning of the specific numerical values of these
regression coefficients is still to be clarified.
We guess that it is possible to use it for measuring the magnitude of
dependence between events.
In accordance with the distance of the regression coefficient from
the zero (where the independence stays) the values within .05
distance could be classified as “one is almost independent on the
other”;
Distances between .05 to .2 from the zero may be classified as
weakly dependent case
Distances is between .2 and .45 could be classified as moderately
dependent;
Cases with | rB ( A) | from.45 to .8 to be called as in average
dependent,
above .8 to be classified as strongly dependent.
This classification is pretty conditional, made up by the author.
3. Regression coefficients
(continued)

The regression coefficients satisfy the inequalities
 P( A)
 P( A) 1  P( A) 
1  P( A) 
max 
,
'
  rB ( A)  min 

P( B) 
 1  P( B)
 P( B) 1  P( B) 
 P( B)
 P( B) 1  P( B) 
1  P( B) 
max 
,
,
  rA ( B)  min 

P( A) 
 1  P( A)
 P( A) 1  P( A) 

These also are called Freshe-Hoefding inequalities.
4. Correlation between two random events

Definition 3. Correlation coefficient between two
events A and B we call the number
R A, B =  rB ( A)  rA ( B)
Its sign, plus or minus, is the sign either of the two
regression coefficients.

An equivalent representation
R A, B =
 ( A, B)
P( A) P( A) P( B) P( B)
=
P( A  B)  P( A) P( B)
P( A) P( A) P( B) P( B)
4. Correlation (continued)






Remark 3. The correlation coefficient R A, B between the events
 I A ,I B
А and В equals to the correlation coefficient
between the indicators of the two random events A and B .
Properties
R1. It is fulfilled
R A, B = 0 if and only if the two events А and В
are independent.
R2. The correlation coefficient always is a number between –1
and +1, i.e. it is fulfilled
-1≤ R A, B ≤ 1.
R2.1. The equality to 1 holds if and only if the events А and В
are equivalent, i.e. when А = В.
R2.2. The equality to - 1 holds if and only if the events А and B
are equivalent
4. Correlation (continued)


R3. The correlation coefficient has the same sign as the
other measures of the dependence between two random
events А and В , and this is the sign of the connection.
R4. The knowledge of R A, B allows calculating the posterior
probability of one of the events under the condition that the
other one is occurred. For instance, P(B | A) will be
determined by the rule
P ( B | A) = P(B) +

R A, B
P( A) P( B) P( B)
P( A)
The net increase, or decrease in the posterior probability
compare to the prior probability equals to the quantity added
to P(B), and depends only on the value of the mutual
correlation.
4. Correlation
P ( B | A)

(continued)
= P (B ) - R A, B
P( A) P( B) P( B)
P ( A)

R5. It is fulfilled R A, B= R A, B = - R A, B ; R = R A, B
A, B

R6. R A, A  1; RA, A  1;

R7. Particular Cases. When A  B, then
R A, B 
P ( A) P( B)
P ( A) P( B)
RA, S  RA, 0
; If A  B   then R A, B
P( A) P( B)

P( A) P( B)
4. Correlation



(continued)
The use of the numerical values of the
correlation coefficient is similar to the use of
the two regression coefficients.
As closer is R A, B to the zero, as “closer”
are the two events А and В to the
independence.
Let us note once again that R A, B= 0 if and
only if the two events are independent.
4. Correlation




(continued)
As closer is R A, B to 1, as “dense one within the other” are the
events А and В, and when R A, B = 1, the two events coincide
(are equivalent).
As closer is R A, B to -1, as “dense one within the other” are
the events А and B , and when R
= - 1 the two events
A, B
coincide (are equivalent).
These interpretations seem convenient when conducting
research and investigations associated with qualitative (nonnumeric) factors and characteristics.
Such studies are common in sociology, ecology, jurisdictions,
medicine, criminology, design of experiments, and other similar
areas.
4. Correlation

(continued)
Freshe-Hoefding inequalities for the Correlation
Coefficient


P( A) P( B) 
 P( A) P( B)

 P( A) P( B) P( A) P( B) 

max 
,

R
(
A
,
B
)

min
,



P
(
A
)
P
(
B
)
P
(
A
)
P
(
B
)
P
(
A
)
P
(
B
)
P
(
A
)
P
(
B
)








4. Correlation




(continued)
Example 1 (continued): We have the numerical values of the
two regression coefficients and from the previous section. In
this way we get
R A, B = (.3368)(.2174) = .2706.
Analogously to the cases with the use of the regression
coefficients, could be used the numeric value of the
correlation coefficient for classifications of the degree of the
mutual dependence.
Any practical implementation will give a clear indication about
the rules of such classifications.
The correlation coefficient is a number in-between the two
regression coefficients. It is symmetric and absorbs the
misbalance (the asymmetry) in the two regression
coefficients, and is a balanced measure of dependence
between the two events.
4. Correlation





(continued)
Examples can be given in variety areas of our life. For
instance:
Consider the possible degree of dependence between
tornado touch downs in Kansas (event A), and in
Alabama (event B).
T A sociology a family with 3, or more children (event
In
A), and an income above the average (event B);
In medicine someone gets an infarct (event A), and
a stroke (event B).
More examples, far better and meaningful are
expected when the revenue of this approach is
assessed.
5. Empirical estimations
 The
measures of dependence
between random events are made
of their probabilities. It makes
them very attractive and in the
same time easy for statistical
estimation and practical use.
5. Empirical Estimations (contd)

Let in N independent experiments (observations)
the random event А occurs k A times, the
random event В occurs k B times, and the event
A  B occurs k AB times. Then statistical
estimators of our measures of dependence will be
respectively:
k AB k A k B
ˆ
 ( A, B) 


N
N N
5. Empirical Estimations (contd)

The estimators of the two regression coefficients
are
k A B k A k B


N
N N
kA
k
(1  A )
N
N
k A B k A k B


N
N N
rˆA ( B) 
; rˆB ( A) =
kB
kB
(1  )
N
N
 The correlation coefficient has estimator
Rˆ ( A, B ) =
k A B k A k B


N
N N
kA
k A kB
kB
(1  ) (1  )
N
N N
N
5. Empirical Estimations (contd)



Each of the three estimators may be simplified
when the fractions in numerator and denominator
are multiplied by , we will not get into detail.
The estimators are all consistent; the estimator of
the connection δ(А,В) is also unbiased, i.e. there is
no systematic error in this estimate.
The proposed estimators can be used for practical
purposes with reasonable interpretations and
explanations, as it is shown in our discussion, and
in our example.
6. Some warnings
and some recommendations





The introduced measures of dependence between random
events are not transitive.
It is possible that А is positively associated with B, and this
event В to be positively associated with a third event С, but
the event А to be negatively associated with С.
To see this imagine А and В compatible (non-empty
intersection) as well as В and С compatible, while А and С
being mutually exclusive, and therefore, with a negative
connection.
Mutually exclusive events have negative connection;
For non-exclusive pairs (А, В) and (В, С) every kind of
dependence is possible.
6. Some warnings and some
recommendations (contd)



One can use the measures of dependence
between random events to compare degrees of
dependence.
We recommend the use of Regression Coefficient
for measuring degrees of dependence.
For instance, let
| rB ( A) || rC ( A) |
then we say that the event А has stronger association

with С compare to its association with B.
In a similar way an entire rank of association of any fixed
event can be given with any collection of other events.
7. An illustration of possible
applications


Alan Agresti Categorical Data Analysis, 2006.
Table 1: Observed Frequencies of Income and Job
Satisfaction Pi , j , Pi .. , P. j
Job Satisfaction
Income US $$
< 6,000
Very
Little
Moderate
Dissatisfied Satisfied
ly
Satisfied
Very
Total
Satisfied Marginal
ly
20
24
80
82
206
6,000–15,000
22
38
104
125
289
15,000-25,000
13
28
81
113
235
> 25,000
7
18
54
92
171
Total
Marginally
62
108
319
412
901
7. An illustration … (contd)

Table 2: Empirical Estimations of the probabilities for each
Pi , j , Pi.. , P. j
particular case
Job Satisfaction
Income US
$$
Very
Dissati
sfied
Little
Satis
fied
Moderat
ely
Satisf
ied
< 6,000
.02220
.02664
.08879
.09101
.22864
6,000–15,000
.02442
.04217
.11543
.13873
.32075
15,000-25,000
.01443
.03108
.08990
.12542
.26083
> 25,000
.00776
.01998
.05993
.10211
.18978
.06881
.11987
.35405
.45727
1.00000
Total
Very
Satisfi
ed
Total
7. An illustration … (contd)

Table 3: Empirical Estimations of the connection function for
each particular category of Income and Job Satisfaction
Job Satisfaction
Income US
$$
Very
Dissatisfied
Little
Satisfied
Moderate
ly
Satisfied
Very
Satisfied
Total
< 6,000
0.006467
-.00077
0.00784
-0.01354
0
6,000–15,000
0.0023491
0.003722 0.001868 -0.00794
0
15,000-25,000
-0.003517
-0.00019
-0.00245
0.00615
0
> 25,000
-0.005298
-0.00277
-0.00726
0.015329
0
0
0
0
0
0
Total
7. An illustration … (contd)

Surface of the Connection Function (Z variable),
between Income level (Y variable) and the Job
Satisfaction Level (X variable) , according to Table 3.
Connection Function between Income Levesl and
Satisfaction Levels
0.02
0.015
0.01
Connection 0.005
0
values
-0.005
-0.01
-0.015
Series1
Series2
Series3
S3
1
2
Income Level
S1
3
4
Satisfaction
Level
Series4
7. An illustration … (contd)

Table 4: Empirical Estimations of the regression coefficient
between each particular category of income with respect to
the job satisfaction rSatisfaction j ( IncomeGroupi )
Job Satisfaction
Income US
$$
Very
Dissatisfied
Little
Satisfied
< 6,000
0.109327
-.00727 0.034281 -.05456
NA
0.035276 0.00817
-0.03199
NA
-0.0107
0.024782
NA
-0.03175
0.061761
NA
0
0
6,000–15,000
0.036603
15,000-25,000 -0.054899 -0.00176
> 25,000
Total
-0.005298 -0.02625
0
0
Moderatel
y Satisfied
0
Very
Satisfied
Total
7. An illustration … (contd)

Surface of the Regression Coefficient
rSatisfaction j ( IncomeGroupi )
Function
Regression Coefficients of Sattisfaction w.r. to
Income Level
0.15
0.1
Regr. Coeff. 0.05
values
0
Series1
Series2
-0.05
-0.1
1
2
Satisfaction
Level
S3
Income Level
S1
3
4
Series3
Series4
7. An illustration … (contd)

Table 5: Empirical Estimations of the regression coefficient
between each particular category of the job satisfaction with
respect to the income groups rIncomeGroup ( Satisfaction j )
i
Job Satisfaction
Income US
$$
< 6,000
Very
Dissatisfied
0.036670
Little
Satisfied
Moderatel
y Satisfied
Very
Satisfied
Total
-0.00435
0.04445
-.07677
0
0.010783 0.017082
0.008576
-0.03644
0
15,000-25,000 -0.018246 -0.00096
-0.01269
0.0316
0
-0.034460 -0.01801
-0.04723
0.099594
0
NA
0
6,000–15,000
> 25,000
Total
NA
NA
NA
7. An illustration … (contd)


Surface of the Regression Coefficient
Function, according to Table 5.
rIncomeGroupi ( Satisfaction j )
Regression Coefficients surface for Satisfaction w.r.to
Income
0.1
0.05
Regr. Coeff.
values
Series1
0
-0.05
Series2
S4
-0.1
1 2
Satisfaction
Level
S1
3
4
Income Level
Series3
Series4
7. An illustration … (contd)

Table 6: Empirical Estimations of the correlation coefficient
between each particular income group and the categories of
the job satisfaction
Job Satisfaction
Income US
$$
< 6,000
6,000–15,000
Very
Dissatisfied
0.060838
Little
Satisfied
Very
Satisfied
Total
-0.005623 0.03904 -.06472
NA
0.019883 0.024548
15,000-25,000 -0.031649 -0.001302
> 25,000
Total
-0.053363 -0.02174
NA
NA
Moderate
ly
Satisfied
0.008371 -0.03414
NA
-0.01165
0.028117
NA
-0.03872
0.078742
NA
NA
NA
NA
7. An illustration … (contd)
R( IncomeGroupi , Satisfaction j )
Surface of the Correlation
Coefficient Function according to Table 6.

Correlation Function for the Income Level and the Job
Satisfaction Levels
0.1
0.05
Correlation
values
Series1
0
Series2
-0.05
S4
-0.1
1
2
Income Level
S1
3
4
Job
Satisfaction
Series3
Series4
7. An illustration … (contd)
A prediction of the Income group

…
Table 7: Forecast of the probabilities P(A|B) =P(A)+
δ(A,B)/P(B) of particular income group given the
categories of the job satisfaction
Job Satisfaction
Income US
$$
Very
Dissatisfied
Little
Satisfied
Moderately
Satisfied
Very
Prior
Satisfied P(A)
< 6,000
0.32258
0.22223
0.25079 0.19903 .22864
6,000–15,000
0.35484
0.35184
0.32601 0.30339 .32075
15,000-25,000
0.20968
0.16668
0.25393
> 25,000
0.11289
0.16666
0.16927
Total
1.00
1.00
1.00
0.27428
.26083
0.22329 .18978
1.00
1.00
7. An illustration … (contd)
A prediction of the Income group

…
Table 8: Forecast of the probabilities P(B|A) =P(B)+
δ(A,B)/P(A) of particular income group given the
categories of the job satisfaction
Job Satisfaction
Income US
$$
Very
Dissatisfied
Little
Satisfied
Moderately
Satisfied
Very
Satisfied
Total
< 6,000
0.09708 0.11651
0.38835 0.39806
1.00
6,000–15,000
0.07612 0.13149
0.35861 0.43253
1.00
15,000-25,000
0.55317
0.07660
0.34468
> 25,000
0.02093
0.10526
0.31579
.06881
.11987
Total
.35405
0.48085
1.00
0.53802
1.00
.45727
1.00
6. CONCLUSIONS






We discussed four measures of dependence between two
random events.
These measures are equivalent, and exhibit natural properties.
Their numerical values may serve as indication for the
magnitude of dependence between random events.
These measures provide simple ways to detect independence,
coincidence, degree of dependence.
If either measure of dependence is known, it allows better
prediction of the chance for occurrence of one event, given
that the other one occurs.
If applied to the events A=[X<x], and B=[Y<y], these
measures immediately turn into measures of the LOCAL
DEPENDENCE between the r.v.’s X and Y associated with the
point (x,y) on the plane.
References




[1] A. Agresti (2006) Categorical Data Analysis, John Wiley
& Sons, Hew York.
[2] B. Dimitrov, and N. Yanev (1991) Probability and
Statistics, A textbook, Sofia University “Kliment Ohridski”,
Sofia (Secоnd Edition 1998, Third Edition 2007).
[3] N. Obreshkov (1963) Probability Theory, Nauka i
Izkustvo, Sofia (in Bulgarian).
[4] Encyclopedia of Statistical Sciences (1981 – 1988), v. 1
– v. 9. Editors-in-Chief S. Kotz and N. L. Johnson, John
Wiley & Sons, New York .
References (contd)




Genest, C. and Boies, J. (2003). Testing
dependence with Kendall plots. The American
Statistician 44, 275-284.
Kolev, N., Goncalves, M. and Dimitrov, B. (2007).
On Sibuya's dependence function. Submitted to
Annals of the Institute of Statistical Mathematics
(AISM).
Kotz, S. and Johnson, N., Editors-in-Chief (1982 1988). Encyclopedia of Statistical Sciences, v. 1 - v.
9. Wiley & Sons.
Nelsen, R. (2006). An Introduction to Copulas. 2nd
Edition, Springer, New York.
References (contd)




Schweitzer, B. and Wol, E.F. (1981). On nonparametric measures of dependence, Ann.
of Statistics 9, 879-885.
Sibuya, M. (1960). Bivariate extreme
statistics. Annals of the Institute of
Statistical Mathematics 11, 195-210.
Sklar, A. (1959). Fonctions de repartition a n
dimensions et leurs marges. Publ. Inst.
Statist. Univ. Paris 8, 229-331.