Statistica & Applicazioni
Vol. 2, n. 1, 2004
Some properties of the Arctangent Distribution
Angiola Pollastri§
Franco Tornaghi‡
Summary: In the present paper some characteristics of the Arctangent Distribution
are analysed. Arctangent was utilised in order to obtain simultaneous confidence
intervals of the probabilities of the multinomial distribution. In particular, the
asimmetry and the relations with the Folded Standard Normal and the Folded Skew
Normal are studied. Then the distribution of the maximum and of the minimum of
the absolute value of the component of a Bivariate Correlated Normal r.v are
studied and used to improve confidence region and corresponding tests of
hypotheses for the two means of a Bivariate Correlated Normal.
Keywords: Asymmetry, Folded Standard Normal, Skew–Normal distribution,
Bivariate Correlated Normal.
1. Introduction
The Arctangent distribution was proposed by Zenga (1979).
The Arctangent density function (d. f.) with parameter a>0, denoted by
Fg
i.1D
ensityfunctionsoftheA
rctangentr.v.
fora=0,2;0,8;1;5.
1
0,8
a=
0,2
0,6
a=
0,8
a=
1
0,4
a=
5
0,2
0
0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5
g(x;a) , is

1 2
 −2 x
 e
g ( x; a ) = 
 arc tan( a )

0
§
‡
ax
∫
0
1
− y2
e 2 dy ,
for
x ≥ 0,
(1)
elsewhere.
Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali
Università degli Studi di Milano Bicocca. Italy – Piazza dell’Ateneo Nuovo,
20126 MILANO (email: [email protected]).
Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali
Università degli Studi di Milano Bicocca. Italy – Piazza dell’Ateneo Nuovo,
20126 MILANO (email: [email protected]).
–
1
–
1
A. Pollastri, F. Tornaghi.
In Fig.1 we show the d.f. for a = 0.2, 1, 5, 10, 15.
This distribution has been utilised to obtain the simultaneous confidence
regions of the probabilities of a trinomial distribution (dn.) (Zenga, Fedrizzi,
1981), of a quadrinomial dn. (Brunazzo, 1979; Brunazzo, Fedrizzi, 1980), of
the differences between the probabilities of a trinomial dn. (Pollastri, 1980),
of the marginal probabilities of a 2x2 table (Pollastri, 1979) and of the
transition probabilities in a Markov Chain (Pollastri, 1982).
The aim of this paper is to study the skewness and the relations between
the Arctangent random variable (r.v.) with the Folded Standard Normal and
the Skew-Normal r.v.. Then we will show that the Arctangent distribution is
useful in providing confidence regions, or corresponding tests of hypotheses,
for the two means of a Bivariate Correlated Normal.
2. Characteristics of the Arctangent distribution1
The Cumulative Distribution Function (c. d. f.) of the Arctangent r.v. is
given by
F (h; a ) = 1 − (2π )
T (h , a )
arc tan(a )
(2)
where
1 2
1 2
1 + ∞ − 2 x ax − 2 y
T (h, a ) =
dxdy
∫ e
∫e
2π h
0
(3)
is a function tabulated by Owen (1956) helpful in finding probabilities
over regions of a Bivariate Normal Distribution. In Fig.2 we show the c.d.f.
for a = 0.2, 1, 5,10,15.
The c.d.f. of the Arctangent r.v. with parameter a is stochastically larger
than the Arctangent r.v. with parameter a' where a < a' , that is
F(h;a)<F(h;a’)
(see Zenga, 1979).
∀h ∈ R +
(4)
The r th moment of the Arctangent r. v. ( Zenga, 1979 ) is
1
The present paper is financially supported by MURST. The title of the project,
directed by Prof. Michele Zenga, is: Modelli Distributivi per fenomeni SocioEconomici”.
Section 2 is due to Tornaghi. Section 3 is due to Pollastri.
Some properties of the Arctangent Distribution
( )
E Xr =
1
θ
∞
ax
∫ ∫
0
xr e
−
(
1 2 2
x +y
2
)
dydx =
0
θ
r
r

e 2 Γ  + 1 ∫ cos r ( t )dt
θ
2  0
1
where θ = arc tan( a )
In particular, we have
E( X ) =
π sin(θ )
2 θ
( )
E X3 =
E( X 2 ) = 1 +
sin(θ ) cos(θ )
θ
π  sin (θ ) cos 2 (θ )
sin (θ ) 
+2


θ
θ 
2
The index of skewness γ 1 takes the form:
γ1 =
µ3
=
σ3
π sin 2 (θ )  sin (θ )

(π − θ 2 ) − 3cos (θ ) 

2
θ  θ

2
3
sin(θ )cos (θ ) π sin (θ ) 2
(1 +
)
−
θ
2 θ2
(5)
The index γ 1 as a function of a is reported in Fig. 3. The index of
skewness increases when the parameter a increases until a≈10.26 and then
decreases. It is possible to demonstrate that:
lim γ 1 = 0.6311
lim γ 1 = 0.9953
a →0
a →∞
The asymmetry of the Arctangent r.v. is then analyzed with the help of
computer.
First of all, we consider the index proposed by Bowley in 1901
(Brentari,1990)
w(0.25)=
Q3 + Q1 − 2Me
Q3 − Q1
(6)
A. Pollastri, F. Tornaghi.
where Me is the median, Q1 and Q3 are respectively the first and the third
quartile. The index w(0.25) varies in the interval [-1,1] and so it is easily
interpetable.
Observing the diagram of w(0.25) reported in Fig.4, we note that the index
increases until a≈5.452.
Then we use a more analytic index based on the asymmetry of points
suggested by David F.N. and Johnson in 1956 (see, f.i., Brentari,1990); for a
continuous r.v. it is defined as
w(p) =
x(1 − p ) + x( p ) − 2 Me
x(1 − p ) − x( p )
0≤ p<
1
2
(7)
where x( p ) = F −1 ( p ) is the pth percentile.
The index w(p) is the sum of the distance of the (1-p)th percentile from the
median and the distance of the median from the pth percentile divided by the
distance from the
(1 -p)th percentile and the pth percentile. So, having -1 ≤ w( p ) ≤ 1 , the
index w( p) is normalized.
Particular functions w( p ) of the Arctangent r.v. with parameter a=0.2, 1,
5, 10, 15 are given in Fig. 5.
For small value of a, if a ' < a ' '
wa ' ( p ) < w a ' ' ( p )
0≤ p<
1
2
For some couples of values of a, the relation above is not always true; for
instance, if we take a’=5 and a’’=10 we can observe that wa ' ( p ) < wa '' ( p )
until p≈0.172 and then wa ' ( p ) > wa '' ( p ) . That is, the skewness for the
Arctangent distribution of parameter a’ may be greater in some part of the
distribution and in other part inferior with respect to the skewness of the
Arctangent of parameter a” . This fact is shown in Fig. 6.
Some properties of the Arctangent Distribution
0.9
a=15.00
0.8
a=10.00
a=5.00
0.7
a=1.00
0.6
0.5
0.4
0.3
a=0.2
0.2
0.1
0
0
1
2
3
4
5
6
Figura 1. Density functions of the Arctangent for a=0.2; 1; 5; 10 ;15
1.2
1
0.8
a=5
0.6
a=1
a=15
0.4
a=0.20
0.2
a=10
0
0
0.5
1
1.5
2
2.5
3
Figura 2. Cumulative Distribution Functions for a=0.2; 1; 5; 10 ;15
3.5
A. Pollastri, F. Tornaghi.
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
Figura 3: The index
6
γ1
8
10
12
14
16
18
20
12.00
14.00
16.00
18.00
20.00
as a function of a
0.16
0.15
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.00
2.00
4.00
6.00
8.00
10.00
Figura 4: Diagram of w(0,25) as a function of a
Some properties of the Arctangent Distribution
0.5
0.45
a 10
a =15
0.4
0.35
a =5
0.3
0.25
a =1
0 01002
0.2
a =0,20
0 01 002
0.15
0.1
0.05
0
0.000
0.100
0.200
0.300
0.400
0.500
0.600
Figura 5: Diagram of w(p) for a=0.2; 1; 5; 10 ;15
0.45
a=15
0.4
a=10
a=5
0.35
0.3
0.25
0.2
0.15
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Figura 6. Diagram of w(p) ( 0.02<p<0.2) for a= 5; 10 ;15
It is interesting to note the extreme case of the Arctangent distribution
g ( x; a ) , that is the function limit, if a → +∞.
THEOREM 1.
The Arctangent distribution having parameter a for a → +∞ tends to the
Folded Standard Normal distribution.
A. Pollastri, F. Tornaghi.
PROOF:
We denote with ϕ ( x ) and with Φ (⋅) respectively the d.f. and the c.d.f. of
the Standard Normal r.v. .
+∞
∫e
Given that
−
y2
2
2π
, we can write
2
dy =
0
lim g ( x; a ) = lim
a → +∞
a → +∞
e
−
x 2 ax
2
∫e
−
y2
2
dy
0
arc tan(a )
e
−
x 2 +∞
y2
−
2
2
=
∫e
dy
0
π
= 2ϕ ( x )
x≥0
2
Observation
For a Skew-normal r.v. of parameter λ having d.f.
f ( z; λ ) = 2ϕ ( z )Φ(λz )
Azzalini (1985) obtained a similar result:
lim f ( z ; λ ) = 2ϕ ( z )
λ →+∞
3. Relations with other distributions
THEOREM 2.
Let us suppose that the r.v. (X,Y) has Standardized Bivariate Normal (SBN)
Distribution, that is
 0   1
X
  ~ N  ; 
Y 
 0   ρ
{
}
a1
and
The d.f. of the r.v. T = max X , Y
densities
with
parameters
ρ 

1 
is a mixture of two Arctangent
a2
and
with
proportions
Some properties of the Arctangent Distribution
π1 =
a1 =
2
π
arc tan(a1 )
and
π2 =
2
π
arc tan(a 2 ) respectively, where
1+ ρ
1− ρ
and a 2 =
.
1− ρ
1+ ρ
PROOF.:
The r.v. T has c.d.f. given by
FT (t ) = P{T ≤ t} = P{ X ≤ t , Y ≤ t} = P{− t ≤ X ≤ t ,−t ≤ Y ≤ t}
Note that FT (t ) corresponds to the integration of the d.f. of the S.B.N.
over a square region. Using the formula indicated by B (h, k ; ρ ) and
introduced by Owen (1956) for finding volumes over rectangles in a SBN
over the lower-hand quadrant divided at x=h and y=k we can write
FT ( t ) = B(t , t ; ρ ) − B(− t , t ; ρ ) − B(t ,−t ; ρ ) + B(− t ,−t ; ρ )
where:





 1 Φ (h ) − T  h, k − ρh  + 1 Φ (k ) − T  k , h − ρk 
 k 1− ρ 2 
 h 1− ρ 2  2
 2





 if hk > 0 or if hk = 0 or k ≥ 0
B(h, k ; ρ ) = 




1
 h, k − ρh  + 1 Φ (k ) − T  k , h − ρk  − 1
(
)
h
T
Φ
−

 h 1− ρ 2  2
 k 1− ρ 2  2
2





if hk < 0 or if hk = 0, h or k > 0
By applying algebra and T (h ,− a ) = T (h , a ) , we find
FT (t ) = 1 − 4{T (t , a1 ) + T (t , a 2 )}
From (2.1) we can write
T (h , a ) = [1 − F ( h; a )]arc tan( a ) 2π .
A. Pollastri, F. Tornaghi.
The c.d.f. of the r.v. T is
FT (t ) =
2
π
{F (x; a1 )arc tan(a1 ) + F (x; a 2 )arc tan( a 2 )}
By differentiating FT (t ) , the d.f. of the r.v. T is immediately obtained:
f T (t ) = g (t ; a1 )
arc tan(a1 )
arc tan(a 2 )
+ g (t ; a 2 )
π 2
π 2
Bearing in mind the relation
arc tan(a ) + arc tan(1 a ) =
π
a>0
2
1
, the sum of the proportions is:
a1
and observing that a 2 =
2
π
arctan(a1 ) +
COROLLARY.
If ρ = 0 , the r.v. T = max X , Y
{
2
π
arctan(a 2 ) = 1
q.e.d.
} has Arctangent density with parameter
a=1.
REMARK 1.
Let us select a simple random sample of s=2 elements from the Folded
Standardized
Normal r.v. T = X , where X∼N(0,1). The Folded Standardized Normal
d. f. is
h(t ) =
ϕ (t )
Φ (0)
=
2
π
e
1
− t2
2
t ≥ 0.
The observations are denoted by t(1) and t(2) where t (1) ≤ t (2 ) .
If
we
consider
the
absolute
value
of
Y = {max(Ti ), i = 1,2} , the relative d. f. is given by
T(2 ) ,
indicated
by
Some properties of the Arctangent Distribution
f (y) = 2
where c=
π
4
2
π
e
y
1
− y2
2
∫
0
2
π
e
1
− t2
2
e
y
1
− y2
2
dt =
∫e
0
c
−t 2 2
dt
y ≥0
= arc tan(1) .
The meaning of this result is that the Arctangent r.v. with parameter a=1
corresponds to the maximum of the absolute value of a sample for s=2 from
Standardized Normal r.v..
REMARK 2.
Theorem 2 has been proposed with a different proof by Pollastri (1979). It
has been used for the asymptotic confidence regions for the marginal
probabilities p1. and p .1 in a Bivariate Binomial.
It can be useful in finding simultaneous confidence regions or
simultaneous test procedures for the means of a Bivariate Correlated
Normal.
Confidence region for the means of a Bivariate correlated Normal
Tukey in 1953 (see Miller, 1981) proposed a confidence region based on
Maximum modulus, that is
{
}
P max[ Y1 , Y2 ] ≤ c = 1 − α
If (1- α )=0,95, when Y1 and Y2 are independent, c=2,236 (see Miller
(1981), page 14).
This corresponds to the 95th percentile of the r.v. T when ρ = 0
(Pollastri,1979).
When Y1 and Y2 are dependent, the Bonferroni inequality is used in order to
find a region with a probability greater than (1- α ) fixed. The critical point is
given by ± 2,241 . Using the exact distribution of T we can find, for
instance, that the value c is ± 2,1799 if ρ = 0,7 or ± 2,1081 when
ρ = 0,9 .The improvement is really considerable.
A. Pollastri, F. Tornaghi.
REMARK 3.
Another application is suggested by Jang (1997) and Loperfido (2002). Let
us consider the hypothesis H0: µ = 0 against H1: µ ≠ 0. Researcher who
applies two statistics to the data and then chooses the most significant one
uses a level α * greater than the α fixed. Let us suppose the investigator
uses the statistics T1 and T2 . If T1 and T2 tend to the Normal, when the
sample size increases, then the real level is given by
α * = lim P[max( T1 , T2 ) > tα ]
n →∞
*
The level α may be computed through the Arctangent distribution if T1
and T2 are
independent or through a mixture of two Arctangent
distributions if the correlation coefficient between T1 and T2 is equal to
ρ ≠ 0.
Loperfido (2002) shows that the distribution of the random variables max
(X,Y) and
min (X,Y) are Skew-Normal. He uses this interesting result in order to find
the real level α when the researcher uses two statistics and reports only the
most significant one in testing H0: µ = 0 against H1: µ > 0.
THEOREM 3.
Let us suppose that the r.v. (X,Y) has SBN Distribution.
The d.f. of the r.v. V = min X , Y is a function of Folded Standard
{
}
Normal densities and of a mixture of two Arctangent densities.
PROOF:
P (V > x) =
= P[( X > x, Y > x) ∪ (Y > x, X < − x) ∪ ( X < − x, Y > x) ∪ ( X < − x, Y < − x)]
= 1 − 2Φ( x) + 2Φ(− x) + B( x, x; ρ ) + B(− x,− x; ρ ) − B( x,− x; ρ ) − B(− x, x; ρ )
= 3 − 4Φ ( x) + FT ( x)
Then the c.d.f. of the r.v. V is
FV ( x) = 4Φ ( x) − FT ( x) − 2
Some properties of the Arctangent Distribution
and the d.f. is
fV ( x ) = 2(2ϕ ( x )) − fT ( x ) .
q.e.d.
REMARK 4.
The researcher who applies two statistics to the data and then chooses the
least significant one uses a level α ’ smaller than the α fixed. Let us suppose
the investigator uses the statistics T1 and T2 . If T1 and T2 tend to the
Normal as n → ∞ then the real level is given by
α ' = lim P[min( T1 , T2 ) < vα ]
n →∞
THEOREM 4.
The Skew-Normal truncated at c (c ≥ 0) is a mixture of a Standardized
Normal truncated at c and of an Arctangent r.v. folded at c with weights
respectively equal to
2T (c , a )
1 − Φ (c )
and
where S (c; a ) is the c.d.f.
1 − S (c; a )
1 − S (c; a )
of a Skew-Normal of parameter a.
PROOF:
The c.d.f. of a Skew-Normal of parameter a (Azzalini,1985) is
S (x; a ) = Φ ( x ) − 2T ( x, a )
The d.f. of a Skew-Normal truncated at c is
l ( x; a, c ) =
2
=
2π
2ϕ ( x )Φ (ax )
1 − S (c; a )
1

ax − y 2 
 1 + 1 ∫ e 2 dy 
2

2π 0


1 − Φ(c ) + 2T (c, a )
1
− x2
e 2
A. Pollastri, F. Tornaghi.
=
[1 − Φ(c )]
ϕ (x )
+ 2T (c, a )
1 − Φ (c )
1 − Φ(c ) + 2T (c, a )
g ( x, a )
2T (c, a )
q.e.d.
The sum of the weights is equal to 1.
COROLLARY.
The Folded Skew – Normal r.v. is a mixture of a Folded Standardized
arc tan(a )
1
π
Normal and of an Arctangent r.v. with weights equal to 2 and
b
b
1 arc tan(a )
respectively, where b = +
.
π
2
PROOF:
The Skew – Normal r. v. truncated at 0 has d. f.:
p ( x; a ) =
2ϕ ( x )Φ (ax )
+∞
2 ∫ ϕ (t )Φ (at )dt
0
The denominator is equal to b. In fact
∞
2 ∫ ϕ (t )Φ (at )dt
0
= 1 − S (0; a ) = 1 − Φ (0 ) + 2T (0 , a )
=
1 arctan (a )
+
= b.
2
π
So we can write
2
p(x; a ) =
.
2π
e
1
− x2
2
1
1
 +
2π
2
b
ax
∫e
0
1
− y2
2

dy  1 h( x ) arc tan( a ) g ( x; a )
= 2
π
+
b
b
Some properties of the Arctangent Distribution
Note that the sum of the weights is equal to 1.
4. Conclusions
The Arctangent distribution has been used to give confidence regions for the
probabilities of the multinomial distribution shorter than the intervals
commonly used.
In the present paper we have shown further characteristics of this
distribution, in particular we have studied the asymmetry of it.
We have analysed the distribution of the absolute maximum and minimum
of the component of a correlated Standard Bivariate Normal. We have
shown that the results obtained may be very useful in multiple comparison.
Then we have analysed the relations with other distributions and, in
particular, with the Folded Standard Normal and with the Folded Skew
Normal.
References
Azzalini A. (1985), A Class of Distributions which includes the
Normal Ones, Scand. J. Statist., 12, pp. 171-178.
Azzalini A. (1986), Further results on a class of distributions which
includes the normal ones, Statistica, anno XLVI, n° 2, pp. 199-208.
Brentari E. (1990), Asimmetria e misure di Asimmetria, Giappichelli
Ed., Torino.
Brunazzo A. (1979), Distribuzione campionaria asintotica dello
scarto standardizzato assoluto massimo di una quadrinomiale,
Quaderni di Statistica e Matematica Applicata alle Scienze
Economico-Sociali, Vol. 2, nn. 1-2, 17-31.
Brunazzo A., Fedrizzi M. (1980), Ancora sulla distribuzione
campionaria asintotica dello scarto standardizzato assoluto massimo di
una quadrinomiale ( caso non simmetrico ), Quaderni di Statistica e
Matematica applicate alle Scienze Economico – Sociali, Vol. 3, pp. 316.
Jiang J. (1997), Sharp upper and lower bounds for asymptotic levels
of some statistical tests, Statistics & Probability Letters, Vol. 35,
pp.395-400.
Liseo B. (1990), La classe delle densità Normali Sghembe : Aspetti
Inferenziali da un punto di vista Bayesiano, Statistica, anno 50, n° 1,
pp. 71-79.
A. Pollastri, F. Tornaghi.
Loperfido N. (2002), Statistical implications of selectively reported
inferential result, Statistics and probability letters, n° 56, pp. 13-22.
Miller R.G.,Jr. (1981), Simultaneuos Statistical Inference, SpringerVerlag , Heidelberg and Berlin, second edition.
Owen D. B. (1956), Tables for computing bivariate normal
probabilities. Annals of Mathematical Statistics, Vol. 27, 1075-1090.
Pollastri A. (1979), Intervalli di confidenza simultanei asintotici per
le probabilità marginali in una tabella 2x2, Quaderni di Statistica e
Matematica applicata alle Scienze Economico – Sociali.
Pollastri A. (1982), Intervalli di confidenza simultanei asintotici per
le probabilità di transizione in una catena di Markov, Quaderni di
Statistica e Matematica applicata alle scienze Economico – Sociali,
Vol II n 1-2.
Pollastri A. (1980), Intervalli di confidenza simultanei asintotici
delle differenze fra probabilità in una trinomiale, Atti della XXX
Riunione Scientifica della Società Italiana di Statistica.
Zenga M. (1979), L’impiego della funzione arcotangente incompleta
nello studio della distribuzione asintotica dello scarto standardizzato
assoluto massimo di una trinomiale, Statistica, XXXIX, n° 2, pp. 269286.
Zenga M., Fedrizzi M. (1981), The Arctangent distribution and the
simultaneous confidence intervals for trinomial proportions, Statistica,
XLI, n° 3, pp. 411-419.