Metrika (2016) 79:725–747
DOI 10.1007/s00184-016-0575-y
On the records of multivariate random sequences
Ismihan Bayramoglu1
Received: 6 April 2015 / Published online: 3 February 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract Two types of records in multivariate sequences are considered in this paper.
According to the first definition, a multivariate observation is accepted as a record if it
is not dominated in at least one of the coordinates of previous record and the first observation is a record. Some basic straightforward results concerning the distributions of
record times and records according to this definition are given. The development of
distribution theory for these types of record and also providing examples with available
analytical results still involves challenging unsolved problems. Second, we consider
records of bivariate sequences according to conditionally N -ordering, introduced in
Bairamov (J Multivar Anal 97:797–809, 2006). The joint distributions of record times
and distributions of record values are derived. Some examples, with particular underlying distributions demonstrating the availability of obtained formulae are provided.
Keywords Bivariate random variables · Joint distribution function · Conditionally
N-ordered random variables · Record times · Record values
1 Introduction
Ordering of multivariate vectors is important for developments of statistical methods
in multivariate case, which is necessary in many areas of applications requiring high
dimensional observations. Univariate ordered random variables are important in statistical theory as fundamental symmetric functions of observations and have many useful
properties which gave rise to development of a beautiful theory of order statistics and
B
1
Ismihan Bayramoglu
[email protected]
Department of Mathematics, Izmir University of Economics, Izmir, Turkey
123
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I. Bayramoglu
record values. The theory of order statistics and record values are well described in
pioneering review papers (Nagaraja 1988; Nevzorov 1988) and fundamental books
(David and Nagaraja 2003; Arnold et al. 1992; Ahsanullah 1995; Galambos 1978;
Arnold et al. 1998). A concept of generalized order statistics, which includes all
models of ordered random variables was introduced by famous German statistician
Udo Kamps in his book Kamps (1995). In multivariate observations, the ordering
principles are different from those in univariate order statistics, because of lack of
many obvious ordering properties of univariate samples in the case of multivariate
observations.
Barnet (1976) presented a fourfold classification of sub-ordering principles for
multivariate random vectors (observations). These principles can be classified as follows: Marginal Ordering (M-ordering), Reduced (Aggregate) Ordering (R-ordering),
Partial Ordering (P-ordering) and Conditional (Sequential) Ordering (C-Ordering). By
developing the C-ordering principle Bairamov and Gebizlioglu (1998) introduced a
so-called norm-ordering in multivariate observations. Considering a probability space
(, , P) and Rm valued random vectors Xi (ω) = (X 1 (ω), X 2 (ω), . . . , X m (ω)),
i = 1, 2, . . . , n, ω ∈ , defined on this space, they defined a conditional ordering
with respect to a norm defined on Rm . According to the definition of norm-ordering,
if X 1 (ω) ≤ X 2 (ω), for all ω ∈ , then X 1 is said to be less than X 2 in a
norm sense and this is shown as X 1 ≺n X 2 . Bairamov (2006) extended an initial
work Bairamov and Gebizlioglu (1998) and introduced the concept of N -conditional
ordering of multivariate random vectors. Arnold et al. (2009a) showed that n independent and identically distributed k-dimensional random variables can be ordered by
viewing them as concomitants of an auxiliary random variable. This definition then
extends the previously proposed norm ordering and N -conditional ordering. Similar
concepts of multivariate record values and multivariate generalized order statistics are
also described in Arnold et al. (2009a). Arnold et al. (2009b) discussed two new concepts of order statistics for multivariate observations and their applications in ranked
set sampling.
According to different ordering rules of multivariate observations, several types
of multivariate records can be considered. Goldie and Resnick (1995) considered
a sequence of independent and identically distributed (i.i.d.) random vectors (r.v’s)
Xn , n = 1, 2, . . . and call Xn a record if there is a record in all coordinates and
studied the asymptotic behavior of the records in a fixed rectangle. Gnedin (1998)
(1)
(2)
(d)
considered a sequence of i.i.d. Rd −valued r.v.’s Xn = (X n , X n , . . . , X n ),
n = 1, 2, . . . and similar to Goldie and Reznick (1995) identified Xn as a record if
(i)
) for all i = 1, 2, . . . , d, i.e. Xn is being classified as
X n(i) > max(X 1(i) , X 2(i) , . . . , X n−1
a record, if there is a record in all coordinates, at index n, simultaneously. Gnedin (1998)
has also investigated the probability pn , that a record appears at index, n, for the multivariate normal distribution. Hashorvaa and Hüssler (2005), considering the component
wise ordering for Rd -valued r.v.’s X1 , X2 , . . . , Xn and investigated the asymptotic
behavior of the probability that the multiple maximum occurs among these sample
points. An alternative definition of multivariate records is described in Naiman and
Torcaso (2009) (internet source http://www.ams.jhu.edu/~dan/multivariate_records).
Let x = (x1 , x2 , . . . , xd ) and y = (y1 , y2 , . . . , yd ) be two vectors in Rd . If xi ≤ yi ,
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On the records of multivariate random sequences
727
for all i = 1, 2, . . . , d, then we write x≺ y. According to Naiman and Torcaso (2009)
a multivariate record occurs at index i if Xi ⊀ X j , for all j < i.
In this paper, we consider two different new record schemes for multivariate observations. Firstly, in Sect. 2 we consider a record scheme in which a new record occurs
at time n if, assuming the most recent record time is K , that Xn ⊀ X K . We make
an attempt to investigate the distributions of record times and record values of multivariate records according to this definition. In Sect. 2, we present some simple and
straightforward results concerning the distributions of record times and records. The
distribution theory and properties of this type of records are still challenging problems
worthy of investigation. Secondly, we study the records of multivariate observations
in the sense of N -conditional ordering. The main results are presented in Sect. 3. The
distributions of multivariate record values are given in terms of integrals involving
joint probability density functions and structural functions. Examples presenting the
distributions of multivariate records for some particular underlying distributions are
provided.
2 Multivariate records according to component wise ordering
Let Zn = (X n , Yn ), n = 1, 2, . . . be a sequence of i.i.d. R2 valued r.v.’s (independent
copies of a r.v. Z = (X, Y )) with absolutely continuous joint distribution function
(c.d.f.) FX,Y (x, y) = C(FX (x), FY (y)), where C(u, v) is a joining copula and
FX (x), FY (y) are corresponding marginal c.d.f’s. Denote by f X,Y (x, y) the joint probability density function (p.d.f.) of (X, Y ) and let f X (x), f Y (y) be the corresponding
2
p.d.f.’s, so that f X,Y (x, y) = c(FX (x), FY (y)) f X (x) f Y (y), where c(x, y) = ∂ ∂C(x,y)
x∂ y .
2
Let z1 = (x1 , y1 ) and z2 = (x2 , y2 ) be two vectors in R and let for any z =
(x, y) ∈ R2 , Bz = {(u, v) ∈ R2 : u ≤ x, v ≤ y}. We write z1 ≺ z2 if z1 ∈ Bz2 and
/ Bz2 . Note that z1 z2 ⇔ {x1 > x2 , y1 > y2 or x1 ≤ x2 , y1 > y2 or
z1 z2 if z1 ∈
x1 > x2 , y1 ≤ y2 }.
Definition 1 Define the sequence of bivariate record times as follows: U (1) = 1,
/ BZU (n−1) }, n > 1. The corresponding sequence
U (n) = min{i : i > U (n − 1), Zi ∈
of records is ZU (n) , n = 1, 2, . . . . The first record is Z1 = (X 1 , Y1 ) and ZU (n) ZU (n−1) , n = 2, 3, . . ..
Lemma 1 The probability mass function (p.m.f) of U (2) is marginal free, i.e. it does
not depend on marginal distributions FX (x) and FY (y); it depends only on a copula
C(u, v). The probability mass function of U (2) is
1 1
P{U (2) = k} =
C(u, v)k−2 (1 − C(u, v))c(u, v)dudv
0
(1)
0
123
728
I. Bayramoglu
Proof By conditioning with respect to X 1 = x and X 2 = y, we have
P{U (2) = k} = P{Z2 ∈ BZ1 , Z3 ∈ BZ1 , . . . , Zk−1 ∈ BZ1 , Zk ∈
/ BZ1 }
= P{Z2 ≺ Z1 , Z3 ≺ Z1 , . . . , Zk−1 ≺ Z1 , Zk Z1 }
∞ ∞
=
(FX,Y (x, y))k−2 (1 − P{Z k ∈ B(x,y) } f X,Y (x, y)d xd y
−∞ −∞
∞ ∞
=
[C(FX (x), FY (y))]k−2 [1 − C(FX (x), FY (y))]
−∞ −∞
× c(FX (x), FY (y)) f X (x) f Y (y)d xd y.
Example 1 Let C(u, v) = uv. Then from the Lemma 1 we have
1 1
P{U (2) = k} =
[C(u, v)]k−2 [1 − C(u, v)]c(u, v)dudv
0
0
2k − 1
= 2
, k = 2, 3, . . .
k (k − 1)2
Therefore, the pmf of U (2) is
P{U (2) = k} =
2k − 1
, k = 2, 3, . . .
− 1)2
k 2 (k
For another example let,
C(u, v) = uv(1 + α(1 − u)(1 − v)), 0 ≤ u, v ≤ 1, −1 ≤ α ≤ 1
be the Farlie–Gumbel–Morgenstern (FGM) copula. The expression of p.m.f. of U (2)
for this distribution is cumbersome, so we provide below the graph of p.m.f. of U (2)
with respect to α, plotted in Maple 15, to illustrate the influence of association parameter α:
It can be seen that if α increases from −1 to 1, then the probability P{U (2) = k}
increases. It is known that the Pearson”s correlation coefficient for FGM distribution
is ρ = α3 . The value α = −1 corresponds to maximal negative quadrant dependence
(NQD) and α = 1 corresponds to maximal positive quadrant dependence (PQD). If
α increases from −1 to 1, then the positive dependence increases and the negative
dependence decreases. Intuitively, random variables X and Y are positive quadrant
dependent if the probability that they are large (or small) at the same time is great than
if would be were they independent. It seems that the increase of positive dependence
results in increase of the probability of record time (Fig. 1).
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On the records of multivariate random sequences
729
Fig. 1 The graph of
P{U (2) = k}, for different
values of k, plotted versus α
Lemma 2 The joint p.m.f. of the sequence of record times U (2), U (3), . . . , U (n) is
P{U (2) = j2 , . . . , U (n) = jn }
=
···
C(u 1 , v1 ) j2 −2 C(u 2 , v2 ) j3 − j2 −1 · · ·
(u n−1 ,vn−1 )(u n−2 ,vn−2 ),...,(u 2 ,v2 )(u 1 ,v1 )
× C(u n−2 , vn−2 ) jn − jn−1 −1 (1 − C(u n−1 , vn−1 )c(u 1 , v1 ) · · ·
× c(u n−1 , vn−1 )du 1 dv1 · · · du n−1 dvn−1
(2)
Proof It is clear that the probabilities of type
P{Z2 ≺ Z1 , Z3 Z2 } = P{(X 1 ≤ X 2 , Y1 ≤ Y2 ),
(X 3 > X 2 , Y3 > Y2 ∪ X 3 ≤ X 2 , Y3 > Y2 ∪ X 3 > X 2 , Y3 ≤ Y2 )}
are marginal free, i.e.
P{X 1 ≤ X 2 , Y1 ≤ Y2 , (X 3 > X 2 , Y3 > Y2 ∪ X 3
≤ X 2 , Y3 > Y2 ∪ X 3 > X 2 , Y3 ≤ Y2 )}
= P{FX (X 1 ) ≤ FX (X 2 ), FY (Y1 ) ≤ FY (Y2 ), (FX (X 3 )
> FX (X 2 ), FY (Y3 ) > FY (Y2 ) ∪ FX (X 3 ) ≤ FX (X 2 ), FY (Y3 )
> FY (Y2 ) ∪ FX (X 3 ) > FX (X 2 ), FY (Y3 ) ≤ FY (Y2 )}
123
730
I. Bayramoglu
= P{U1 ≤ U2 , V1 ≤ V2 , U3 > U2 , V3 > V2 ∪ U3
≤ U2 , V3 > V2 ∪ U3 > U2 , V3 ≤ V2 },
where (U1 , V1 ), (U2 , V2 ) and (U3 , V3 ) are independent random vectors with uniform
marginals and with the copula C(u, v). Then (2) follows from
P{U (2) = j2 , . . . , U (n) = jn }
= P{U2 ≺ U1 , U3 ≺ U1 , . . . , U j2 −1 ≺ U1 , U j2 U1 ,
U j2 +1 ≺ U j2 , . . . , U j3 −1 ≺ U j2 , U j3 U j2 . . . ,
U jn−1 +1 ≺ U jn−1 , . . . , U jn −1 ≺ U jn−1 , U jn U jn−1 },
by conditioning with respect to U1 = u 1 , V1 = v1 , . . . , Un−1 = u n−1 , Vn−1 = vn−1 ,
where Ui = (Ui , Vi ), i = 1, 2, . . . , n are i.i.d. r.v.’s with a copula C(u, v).
The sequence of ordinary record times from i.i.d. sequence of random variables
forms Markov chain. However, the sequence of bivariate record times does not seem
to be a Markov chain.
Proposition 1 The c.d.f. of the second record value is
FZ U (2) (x, y) = P{ZU (2) ∈ Bx,y }
y x
=
−∞ −∞
∞ x
F(x, y) − F(x1 , y) − F(x, y1 ) + F(x1 , y1 )
d F(x1 , y1 )
1 − F(x1 , y1 )
+
−∞ −∞
y ∞
+
−∞ −∞
F(x, min(y1 , y)) − F(x1 , min(y1 , y))
d F(x1 , y1 )
1 − F(x1 , y1 )
F(min(x1 , x), y) − F(min(x1 , x), y1 )
d F(x1 , y1 )
1 − F(x1 , y1 )
(3)
Proof Conditioning with respect to X 1 = x1 , Y1 = y1 , X k = xk , Yk = yk , we can
write
P{ZU (2) ∈ Bx,y } =
∞
P{ZU (2) ∈ Bx,y , U (2) = k}
k=2
=
∞ ∞
∞ P{(X k , Yk ) ∈ Bx,y , (X 2 , Y2 ) ∈ B(x1 ,y1 ) , . . . , (xk−1 , yk−1 )
k=2 −∞ −∞
∈ B(x1 ,y1 ) , (X k , Yk ) ∈
/ B(x1 ,y1 ) |
X 1 = x1 , Y1 = y1 }d F(x1 , y1 ).
123
(4)
On the records of multivariate random sequences
731
Let
A1 = {x1 < X k ≤ x, y1 < Yk ≤ y, C},
A2 = {x1 < X k ≤ x, Yk ≤ min(y1 , y), C},
A3 = {X k ≤ min(x1 , y), y1 < Yk ≤ y, C},
where C is any event. It is clear that A1 , A2 , A3 are mutually exclusive. Since,
P{(X k , Yk ) ∈ Bx,y , (X k , Yk ) ∈
/ B(x1 ,y1 ) , C} = P(A1 ) + P(A2 ) + P(A3 ),
then taking into account of independence of (X 1 , Y1 ), . . . , (X n , Yn ) we have from (4)
P{ZU (2) ∈ Bx,y }
∞ ∞
∞ =
F(x1 , y1 )k−2 [P{x1 < X k ≤ x, y1 < Yk ≤ y}
k=2 −∞ −∞
+ P{x1 < X k ≤ x, Yk ≤ min(y1 , y)}
+ P{X k ≤ min(x1 , y), y1 < Yk ≤ y}]d F(x1 , y1 )
y x
1
[P{x1 < X k ≤ x, y1 < Yk ≤ y}d F(x1 , y1 )
=
1 − F(x1 , y1 )
−∞ −∞
∞ x
+
−∞ −∞
y ∞
+
−∞ −∞
y x
=
−∞ −∞
∞ x
1
[P{x1 < X k ≤ x, Yk ≤ min(y1 , y)}d F(x1 , y1 )
1 − F(x1 , y1 )
1
[P{X k ≤ min(x1 , x), y1 < Yk ≤ y}d F(x1 , y1 )
1 − F(x1 , y1 )
F(x, y) − F(x1 , y) − F(x, y1 ) + F(x1 , y1 )
d F(x1 , y1 )
1 − F(x1 , y1 )
+
−∞ −∞
y ∞
+
−∞ −∞
which leads to (3).
F(x, min(y1 , y)) − F(x1 , min(y1 , y))
d F(x1 , y1 )
1 − F(x1 , y1 )
F(min(x1 , x), y) − F(min(x1 , x), y1 )
d F(x1 , y1 ),
1 − F(x1 , y1 )
Finding the distributions of r th record value ZU (l) , 2 < l ≤ n and the joint distributions of records Z U (k1 ) , . . . , Z U (kr ) , 2 ≤ r ≤ n still remain as challenging problems.
123
732
I. Bayramoglu
3 Multivariate records according to N-conditional ordering
Let {Zn , n ≥ 1} ≡ {(X n , Yn ), n ≥ 1} be a sequence of independent and identically
distributed (i.i.d.) random variables, copies of a random vector Z = (X, Y ) having joint distribution function FZ (x, y) = FX,Y (x, y) = C(FX (x), FY (y)), where
C is a joining copula and FX (x) and FY (y) are marginal distributions of X and Y ,
respectively. Denote Bx,y = {(u, v) : u ≤ x, v ≤ y}, (x, y) ∈ R2 . We say that
the random vector Z1 = (X 1 , Y1 ) precedes Z2 = (X 2 , Y2 ), (or Z1 is conditionally less than Z2 ) in the sense of conditional ordering with respect to a function
N (x, y), if N (X 1 , Y1 ) ≤ N (X 2 , Y2 ), and write (X 1 , Y1 ) ≺ N (X 2 , Y2 ) or in vector
form Z1 ≺ N Z2 . The function h(x, y) = P{N (X, Y ) ≤ N (x, y)}, (x, y) ∈ R2 is
called the structure function. The theory of conditionally N -ordered random variables
is described in Bairamov (2006).
We consider bivariate records of a sequence {Zn , n ≥ 1} with respect to N -ordering.
Hereafter for simplicity we use “≺” instead of “≺ N ”.
Definition 2 Let U (1) = 1 and U (n) = min{i : i > U (n − 1), ZU (n−1) ≺ Zi },
n > 1. The sequence {U (n), n ≥ 1} is called the sequence of record times of {Zn ,
n ≥ 1}. We say Zk = (X k , Yk ) is upper bivariate record if,
N (X k , Yk ) ≥ max(N (X 1 , Y1 ), . . . , N (X k−1 , Yk−1 )), k > 1
and Z1 = (X 1 , Y1 ) is the first record. The sequence of bivariate records is {ZU (n) ,
n ≥ 1}.
Example 2 According to a public opinion in Turkey, a critical dispute between the
government and Central Bank starting from 20 January 2015 resulted in increased rate
of the US Dollar (USD) versus Turkish Lira (TL), breaking records almost every day.
This has also resulted in an increased of the Euro (EUR) against TL. Economists in
Turkey do not consider only the rate of EUR or USD versus TL, they consider the
average of both exchange rates. This helps to be more informative in estimating of
the effect of political dispute on exchange rates. If the USD breaks sharp records
versus TL and EUR/TL also increases, then the economists make a conclusion that
the rate is changing due to the effect of the international market. If the records of the
USD are more frequent than EUR, then this rate change is artificial and it is result
of political dispute. Table of 1 US Dollar and 1 EURO to Turkish Lira Exchange
Rate during the period 25 January 2015-4 March 2015 is given below. The average of
(USD/TL+EUR/TL)/2 is also presented in Table 1. The data are taken from the web site
http://www.exchangerates.org.uk/USD-TRY-exchange-rate-history.html The graphs
are given in Fig. 2, below. The records considered according to Definition 1 and
Definition 2 are starred. We denote X ≡ U S D/T L , Y ≡ EU R/T L and N (X, Y ) =
X +Y
2 .
It can be seen from the table that, for example, the value of (X, Y ), at 26 January
is a record according to a Definition 1, considered in Chapter 1. The value of (X, Y )
at 25 February is also a record, while the value at 19 February is not a record, etc. The
value of (X, Y ) at 25 February, for example, is a record according to Definition 2. The
value of (X, Y ) at 26 February is a record in the sense to Definition 1, while it is not a
123
2.8430
2.8535*
2.8189
2.7982
2.8082
2.8082
2.8074
2.8274*
2.7941
2.8037
2.7962
2.7921
2.7921
2.7908
2.787
2.7977
2.7838
2.801
2.7992
2.7992
4 Mar 2015
3 Mar 2015
2 Mar 2015
1 Mar 2015
28 Feb 2015
27 Feb 2015
26 Feb 2015
25 Feb 2015
24 Feb 2015
23 Feb 2015
22 Feb 2015
21 Feb 2015
20 Feb 2015
19 Feb 2015
18 Feb 2015
17 Feb 2015
16 Feb 2015
15 Feb 2015
14 Feb 2015
13 Feb 2015
* denotes the record value
X
Date
Table 1 USD, EUR exchange rates against TL
2.457
2.457
2.4574
2.4566
2.4533
2.4448
2.4554
2.4535
2.4535
2.457
2.474
2.4642
2.488
2.5058*
2.508*
2.508
2.5072
2.5221*
2.5352*
2.5510*
Y
2.6281
2.6281
2.6292
2.6202
2.6255
2.6159
2.6231
2.6228
2.6228
2.6266
2.6388
2.6291
2.6577*
2.6566
2.6581*
2.6581
2.6527
2.6705*
2.69435*
2.6976*
X +Y
2
25 Jan 2015
26 Jan 2015
27 Jan 2015
28 Jan 2015
28 Jan 2015
29 Jan 2015
30 Jan 2015
30 Jan 2015
31 Jan 2015
1 Feb 2015
2 Feb 2015
3 Feb 2015
4 Feb 2015
5 Feb 2015
6 Feb 2015
7 Feb 2015
8 Feb 2015
9 Feb 2015
10 Feb 2015
11 Feb 2015
12 Feb 2015
Date
2.6321*
2.6511*
2.6881*
2.6949*
2.6949
2.7389*
2.756*
2.756
2.756
2.7635*
2.7575
2.7529
2.7852*
2.7964*
2.7986*
2.7986
2.7975
2.8041*
2.8251*
2.8267*
2.8151
X
2.3567*
2.3646*
2.3646
2.3879*
2.3879
2.4165*
2.4416*
2.4416
2.4416
2.4438*
2.4316
2.4027
2.4609*
2.4376
2.4734*
2.4734
2.4727
2.4762*
2.4958*
2.4991*
2.4686
Y
2.4944*
2.5039*
2.5263*
2.5414*
2.5414
2.5777*
2.5988*
2.5988
2.5988
2.6036*
2.5945
2.5778
2.6230*
2.6170
2.6360*
2.6360
2.6351
2.6401*
2.6604*
2.6629*
2.6418
X +Y
2
On the records of multivariate random sequences
733
123
734
I. Bayramoglu
Fig. 2 The graphs of exchange rate of USD, EUR and (USD/TL + EUR/TL)/2 versus TL
record in the sense of Definition 2. According to these records it can be concluded that
the increase of rate of USD against TL is not artificial and supported with the actions
in international money market.
Economists are interested in prediction of future records of the (USD +EUR)/2
versus TL by using the data of realized past records. It is known that with respect
to the squared error loss, the best unbiased predictor for U (m), given U (k), k < m,
is E{U (m) | U (k)}. Therefore, to predict the future record times and record values
given past records, we need the joint probability mass functions of records times and
record values.
123
On the records of multivariate random sequences
735
Fig. 3 The graph of a p.d.f. of
second record vector given in
(13)
Fig. 4 The graph of c.d.f.
FZU(2) (t, s) given in (14)
Here we present some results on the joint distributions of record times and records
(Figs. 3, 4).
Lemma 3 The joint probability mass function (p.m.f.) of U (2), U (3), . . . , U (n) is
distribution free and is the same as the p.m.f. of record times for i.i.d. univariate
random variables, i.e. the p.m.f. of U (2) is
P{U (2) = k} =
1
, k = 2, 3, . . .
k(k − 1)
(5)
123
736
I. Bayramoglu
and the joint p.m.f. of U (2), U (3), . . . , U (n) is
P{U (2) = j2 , U (3) = j3 , . . . , U (n) = jn }
1
=
,
( j2 − 1)( j3 − 1) · · · ( jn − 1) jn
ji > ji−1 , i = 3, 4, . . . , n.
j1 = 2, 3, . . . ,
(6)
Proof From Definition 1, for random i.i.d. random variables N (Z1 ),N (Z2 ), . . . using
probability integral transformation we have
P{U (2) = k} = P{Z2 ≺ Z1 , Z3 ≺ Z1 , . . . , Zk−1 ≺ Z1 .Z1 ≺ Zk }
= P{N (Z2 ) ≤ N (Z1 ), N (Z3 ) ≤ N (Z1 ), . . . ,
N (Zk−1 ) ≤ N (Z1 ), N (Z1 ) ≤ N (Zk )}
and (5) follows. Considering i.i.d. random variables N (Z1 ),N (Z2 ), . . . and we have
P{U (2) = j2 , U (3) = j3 , . . . , U (n) = jn }
= P{Z2 ≺ Z1 , Z3 ≺ Z1 , . . . , Z j2 −1 ≺ Z1 , Z1 ≺ Z j2 ,
Z j2 +1 ≺ Z j2 , Z j2 +2 ≺ Z j2 , . . . , Z j2 −1 ≺ Z j2 , Z j2 ≺ Z j3 . . . ,
Z jn−1 +1 ≺ Z jn−1 , . . . , Z jn −1 ≺ Z jn−1 , Z jn−1 ≺ Z jn }
= P{N (Z2 ) ≤ N (Z1 ), N (Z3 ) ≤ N (Z1 ), . . . , N (Z j2 −1 ) ≤ N (Z1 ),
N (Z1 ) ≤ N (Z j2 ), N (Z j2 +1 ) ≤ N (Z j2 ), . . . ,
N (Z j3 −1 ) ≤ N (Z j2 ), N (Z j2 ) ≤ N (Z j3 ), . . . ,
N (Z jn−1 +1 ) ≤ N (Z jn−1 ), . . . , N (Z jn −1 ) ≤ N (Z jn−1 ), N (Z jn−1 ) ≤ N (Z jn )}
and (6) follows.
The record times of conditionally ordered random vectors possess all properties of
record times from univariate i.i.d. sequence of random variables, described in Ahsanullah (1995) and Galambos (1978).
Remark 1 It is clear that in the case where the considered sequence is p ( p > 2)
dimensional, then Lemma 1 will keep its assertion unchanged.
The distribution of bivariate records are given in the following theorem.
Theorem 1 The p.d.f. of ZU (2) is
f ZU (2) (x, y) = f (x, y)
N (x1 ,y1 )≤N (x,y)
123
f (x1 , y1 )
d x1 dy1
1 − h(x1 , y1 )
(7)
On the records of multivariate random sequences
737
and the p.d.f. of ZU (n) , n > 2 is
f ZU (n) (x, y) =
f (x, y)
···
n−1
N (x1 ,y1 )≤N (x2 ,y2 )≤···≤N (xn−1 ,yn−1 )≤N (x,y) i=1
×
n−1
f (xi , yi )
1 − h(xi , yi )
d xi dyi
(8)
i=1
Proof The c.d.f. of the second bivariate record ZU (2) is
P{ZU (2) ∈ Bx,y } =
=
∞
k=2
∞
P{X k ≤ x, Yk ≤ y, U (2) = k}
P{X k ≤ x, Yk ≤ y, Z2 ≺ Z1 ,
k=2
Z3 ≺ Z1 , . . . , Zk−1 ≺ Z1 .Z1 ≺ Zk }
∞
=
P{X k ≤ x, Yk ≤ y, N (X 2 , Y2 ) ≤ N (X 1 , Y1 ), . . . ,
k=2
N (X k−1 , Yk−1 ) ≤ N (X 1 , Y1 ), N (X 1 , Y1 ) ≤ N (X k , Yk )}.
(9)
Conditioning with respect to X 1 = x1 , Y1 = y1 , X k = xk , Yk = yk , we obtain from
(9)
P{ZU (2) ∈ Bx,y }
∞ ∞ ∞
∞ ∞ =
P{X k ≤ x, Yk ≤ y, N (X 2 , Y2 ) ≤ N (X 1 , Y1 ), . . . ,
k=2 −∞ −∞ −∞ −∞
N (X k−1 , Yk−1 ) ≤ N (X 1 , Y1 ), N (X 1 , Y1 ) ≤ N (X k , Yk )} |
X 1 = x1 , Y1 = y1 , X k = xk , Yk = yk } f (x1 , y1 ) f (k , yk )d x1 dy1 d xk dyk
∞ ∞
∞ x y =
P{N (X 2 , Y2 ) ≤ N (x1 , y1 )} × · · ·
k=2 −∞ −∞ −∞ −∞
× P{N (X k−1 , Yk−1 ) ≤ N (x1 , y1 )}I (N (x1 , y1 ) ≤ N (xk , yk ))
f (x1 , y1 ) f (k , yk )d x1 dy1 d xk dyk
∞ ∞
∞ x y =
(h(x1 , y1 ))k−2
k=2 −∞ −∞ −∞ −∞
× I (N (x1 , y1 ) ≤ N (xk , yk )) f (x1 , y1 ) f (k , yk )d x1 dy1 d xk dyk ,
(10)
123
738
I. Bayramoglu
where I (N (x1 , y1 ) ≤ N (xk , yk )) is an indicator function, i.e.
I (N (x1 , y1 ) ≤ N (xk , yk )) =
1 i f N (x1 , y1 ) ≤ N (xk , yk )
.
0,
other wise
Differentiating (10) with respect to x and y, we obtain the p.d.f. of ZU (2) as follows
f ZU (2) (x, y) =
∞ ∞
∞ (h(x1 , y1 ))k−2 I (N (x1 , y1 ) ≤ N (x, y))
k=2 −∞ −∞
× f (x1 , y1 ) f (x, y)d x1 dy1
f (x1 , y1 )
d x1 dy1 .
= f (x, y)
1 − h(x1 , y1 )
N (x1 ,y1 )≤N (x,y)
To prove (8) we follow a similar way and consider
P{ZU (n) ∈ Bx,y }
P{X k ≤ x, Yk ≤ y, U (2) = j2 , . . . , U(n) = jn }
=
2≤ j2 <, j3 <···< jn
=
P{X jn ≤ x, Y jn ≤ y, Z2 ≺ Z1 , Z3 ≺ Z1 , . . . ,
2≤, j2 <···< jn−1 ≤ jn
Z j2 −1 ≺ Z1 , Z1 ≺ Z j2 , Z j2 +1 ≺ Z j2 ,
Z j2 +2 ≺ Z j2 , . . . , Z j3 −1 ≺ Z j2 , Z j2 ≺ Z j3 . . . ,
Z jn−1 +1 ≺ Z jn−1 , . . . , Z jn −1 ≺ Z jn−1 , Z jn−1 ≺ Z jn }.
P{X jn ≤ x, Y jn ≤ y, N (X 2 , Y2 ) ≤ N (X 1 , Y1 ),
=
2≤, j2 <···< jn−1 ≤ jn
N (X 3, Y3 ) ≤ N (X 1, Y1 ), , . . . , N (X j2 −1 , Y j2 −1 ) ≤ N (X 1 , Y1 ),
N (X 1 , Y1 ) ≤ N (X j2 , Y j2 ), N (X j2 +1 , Y j2 +1 ) ≤ (X j2 , Y j2 ), . . . ,
N (X j3 −1 , Y j3 −1 ) ≤ N (X j2 , Y j2 ), N (X j2 , Y j2 ) ≤ N (X j3 , Y j3 ), . . . ,
N (X jn−1 +1 , Y jn−1 +1 ) ≤ N (X jn−1 , Y jn−1 ), . . . ,
N (X jn −1 , Y jn −1 ) ≤ N (X jn−1 , Y jn−1 ), N (X jn−1 , Y jn−1 ) ≤ N (X jn , Y jn )} (11)
Conditioning with respect to X ji = xi , Y ji = yi , i = 2, . . . , n − 1 and X 1 = x1 , Y1 =
y1 from (11) one obtains
P{ZU (n) ∈ Bx,y }
=
∞
jn = jn−1
···
∞
∞
∞ x y j3 = j2 +1 j2 =2 −∞ −∞ −∞
∞
···
P{N (X 2 , Y2 ) ≤ N (x1 , y1 )} · · ·
−∞
P{N (X j2 −1 , Y j2 −1 ) ≤ N (x1 , y1 )}, I (N (x1 , y1 ) ≤ N (x2 , y2 ))
P{N (X j2 +1 , Y j2 +1 ) ≤ N (x2 , y2 )} · · · P{N (X j3 −1 , Y j3 −1 ) ≤ N (x2 , y2 )}
123
On the records of multivariate random sequences
739
I {N (x2 , y2 ) ≤ N (x3 , y3 )) · · · P{N (X jn−1 +1 , Y jn−1 +1 ) ≤ N (xn−1 , yn−1 )} · · ·
P{N (X jn −1 , Y jn −1 ) ≤ N (xn−1 , yn−1 )I (N (xn−1 , yn−1 ) ≤ N (xn , yn )) f (x1 , y1 )
× f (x2 , y2 ) · · · f (xn , yn )d x1 dy1 · · · d xn dyn
x y ∞
∞
∞
∞
∞
=
···
···
I (N (x1 , y1 ) ≤ N (x2 , y2 ))
j2 =2
jn−1 = jn−2 +1 jn = jn−1 +1 −∞ −∞ −∞
j2 −2
−∞
j3 − j2 −1
× (h(x1 , y1 ))
(h(x2 , y2 ))
I {N (x2 , y2 ) ≤ N (x3 , y3 )) · · ·
× (h(xn−1 , yn−1 )) jn − jn−1 −1 I (N (xn−1 , yn−1 ) ≤ N (xn , yn ))
× f (x1 , y1 ) f (x2 , y2 ) · · · f (xn−1 , yn−1 ) f (xn , yn )d x1 dy1 · · · d xn dyn
since
∞
(h(xn−1 , yn−1 )) jn − jn−1 −1 =
jn = jn−1 +1
1
, etc.
1 − h(xn−1 , yn−1 )
we have,
P{ZU (n) ∈ Bx,y }
x y ∞
∞
n−1
=
···
f (xn , yn )
I (N (xi , yi ) ≤ N (xi+1 , yi+1 ))
i=1
−∞ −∞ −∞
−∞
n
f (xi , yi )
i=1
1 − h(xi , yi )
d xi dyi .
(12)
Differentiating (12) with respect to x and y (xn ≤ x, yn ≤ y) we obtain the p.d.f. of
ZU (n) as in (8).
Example 3 Let F(x, y) = x y, 0 ≤ x, y ≤ 1, N (x, y) = max(x, y), then h(x, y) =
P{max(X, Y ) ≤ max(x, y)} = (max(x, y))2 . From ( 7) we have
1
d x1 d x2
f ZU (2) x, y) =
1 − (max(x, y))2
max(x1 y1 )≤max(x,y)
=
x1 ≤max(x,y)
y1 ≤max(x,y)
x1 ≤y1
1
d x1 d x2
1 − (max(x, y))2
+
x1 ≤max(x,y)
y1 ≤max(x,y)
x1 >y1
max(x,y)
y1
=
0
0
1
d x1 d x2
1 − (max(x, y))2
1
d x1 dy1 +
1 − y12
max(x,y)
x1
0
0
1
dy1 d x1
1 − x12
123
740
I. Bayramoglu
1
max(x,y)
max(x,y)
= − [ln(1 − y12 ) |0
+ ln(1 − x12 ) |0
]
2
= − ln(1 − (max(x, y))2 ).
Therefore,
− ln(1 − (max(x, y))2 ) i f 0 ≤ x, y ≤ 1
.
0,
other wise
f ZU (2) x, y) =
The graph of this p.d.f. is given below:
The c.d.f. of ZU (2) is
s t
FZU (2) (t, s) =
[− ln(1 − (max(x, y))2 )]d xd y
0
0
s min(t,y)
=−
t
min(t,y)
(ln(1 − y )d xd y −
ln(1 − x 2 )d yd x
2
0
0
0
s
=−
0
t
min(t, y) ln(1 − y 2 )dy −
0
min(t, y) ln(1 − x 2 )d x
0
= I1 + I2 .
First we calculate I1 as follows:
s
I1 = −
min(t, y) ln(1 − y 2 )dy
0
min(t, y) ln(1 − y )dy −
=−
min(t, y) ln(1 − y 2 )dy
2
0≤y≤s
y<t
0≤y≤s
y>t
min(t,s)
max(t,s)
y ln(1 − y )dy −
=−
t ln(1 − y 2 )dy
2
0
0
Similarly,
min(t,s)
I2 = −
x ln(1 − x )d x −
0
123
max(t,s)
s ln(1 − x 2 )d x.
2
0
(13)
On the records of multivariate random sequences
741
Evaluating, I1 and I2 , we obtain
FZU(2) (t, s) =
⎧
ln(1 − t 2 ) − t 2 − ts ln(1 − s 2 )
⎪
⎪
⎪
⎨
+2ts + t ln (1−s)(1+t)
(1−t)(1+s)
if t < s
⎪
ln(1 − s 2 ) − s 2 − ts ln(1 − t 2 )
⎪
⎪
⎩
+2ts + s ln (1−t)(1+s)
(1−s)(1+t)
if t > s
, 0 ≤ t, s ≤ 1.
(14)
The graph of is FZU(2) (t, s) presented below:
The marginal p.d.f.’s of this distribution are
1+x
− 2x,
1−x
1+y
− 2y.
f 2 (y) = 2 − 2 ln 2 − ln
1−y
f 1 (x) = 2 − 2 ln 2 − ln
(15)
The c.d.f.’s are
F1 (x) = 2x(1 − ln(2)) + (1 − x) ln(1 − x) + (1 + x) ln(1 + x) − x 2
0 ≤ x ≤ 1.
F2 (y) = 2y(1 − ln(2)) + (1 − y) ln(1 − y) + (1 + y) ln(1 + y) − y 2
0 ≤ y ≤ 1.
(16)
The graph of marginal p.d.f. and marginal c.d.f. given in (15) and (16) are given in
Fig. 5.
Example 4 Let the joint p.d.f. of (X, Y ) is
f (x, y) =
and N (x, y) =
1
π R2
0,
i f x 2 + y2 ≤ R2
(17)
other wise
x 2 + y 2 . Then,
P{ X 2 + Y 2 ≤ t} = P{X 2 + Y 2 ≤ t 2 }
πt2
f (x, y)d xd y =
=
π R2
x 2 +y 2 ≤t 2
=
t2
, 0 ≤ t ≤ R2.
R2
and the structure function is
x 2 + y2
h(x, y) = P{ X 2 + Y 2 ≤ x 2 + y 2 } =
, x 2 + y2 ≤ R2.
R2
(18)
123
742
I. Bayramoglu
Fig. 5 The graph of marginal p.d.f. and c.d.f. given in (15) and (16)
From (7) he p.d.f. of ZU (2) is
f ZU (2) (x, y) =
x12 +,y12 ≤x 2 +,y 2 ≤R 2
=
x12 +,y12 ≤x 2 +,y 2 ≤R 2
1
π 2 R4
1
1−
x12 +y12
R2
d x1 dy1
1
1
d x1 dy1
π 2 R 2 R 2 − x12 − y12
Using polar coordinates x1 = ρ cos θ , y1 = ρ sin θ , ρ 2 = x12 + y12 , we have
1
f ZU (2) (x, y) = 2 2
π R
2
π R2
x2 +y 2
−π
√
=
√
π
x 2 +y 2
0
0
ρ
dρdθ
R2 − ρ2
ρ
2
dρ =
2
2
R −ρ
π R2
√
1
x 2 +y 2
− ln(R 2 − ρ 2 ) |0
2
R2
1
ln
, x 2 + y2 ≤ R2.
=
π R2
R2 − x 2 − y2
123
On the records of multivariate random sequences
743
Fig. 6 Graph of the p.d.f. given in (19) for different values of R, R = 1 (top left), R = 2 (top right), R = 3
(bottom left), R = 8 (bottom right)
Therefore,
f ZU (2) (x, y) =
1
π R2
0,
ln
R2
R 2 −x 2 −y 2
if x 2 + y 2 ≤ R 2
other wise
(19)
The graph of this p.d.f. is given below:
The graphs of f ZU (2) (x, y) for R = 2, 3, 8 given in Fig. 5 show that this p.d.f. may
be used as a good example for bathtub shaped bivariate density (Figs. 6, 7).
Example
5 Let (X, Y ) has the p.d.f. given in (17) with R = 1 and N (x, y) =
x 2 + y 2 . Then h(x, y) = x 2 + y 2 . Using formula (8) we can calculate the p.d.f. of
ZU (n) . We have
123
744
I. Bayramoglu
Fig. 7 Graph of the p.d.f. of nth record vector given in (22)
f ZU (n) (x, y) =
n−1
···
2 +y 2 ≤x 2 +y 2
x12 +y12 ≤x22 +y22 ≤···≤xn−1
n−1
i=1
n−1
1 d xi dyi .
1 − xi2 − yi2 π n
1
i=1
(20)
Changing variables in (20) as xi = ρi cos(θi ), yi = ρi sin(θi ), we have xi2 + yi2 =
i = 1, 2, . . . , n − 1. The Jacobian of this transformation is
ρ2,
J (ρ1 , . . . , ρn−1 , θ1 , . . . , θn−1 )
⎡ ∂x
∂x
∂x
∂x
1
∂ρ1
∂ x2
∂ρ1
⎢
⎢
= det ⎢
⎢
⎣ ···
⎡
∂ xn−1
∂ρ1
1
∂θ1
∂ x2
∂θ1
1
∂ρ2
∂ x2
∂ρ2
1
∂θ2
∂ x2
∂θ2
···
∂ x1
∂ x1
∂ρn−1 ∂θn−1
∂ x2
∂ x2
∂ρn−1 ∂θn−1
···
··· ··· ··· ··· ··· ···
∂ xn−1 ∂ xn−1 ∂ xn−1
∂ xn−1 ∂ xn−1
∂θ1
∂ρ2
∂θ2 · · · ∂ρn−1 ∂θn−1
cos θ1 −ρ1 sin θ1
0
0
⎢ sin θ1 ρ1 cos θ1
0
0
⎢
⎢ 0
0
cos θ2 −ρ2 sin θ2
⎢
0
sin θ2 ρ2 cos θ2
= det ⎢
⎢ 0
⎢ ···
···
···
···
⎢
⎣ 0
0
0
0
0
0
0
0
= ρ1 ρ2 · · · ρn−1 .
123
⎤
⎥
⎥
⎥
⎥
⎦
⎤
···
0
0
⎥
···
0
0
⎥
⎥
···
0
0
⎥
⎥
···
0
0
⎥
⎥
···
···
···
⎥
0 cos θn−1 −ρn−1 sin θn−1 ⎦
0 sin θn−1 ρn−1 cos θn−1
On the records of multivariate random sequences
745
Then from (20) we have
(2π )n−1
πn
f ZU (n) (x, y) =
···
√
ρ1≤ ρ2 ≤···≤ρn−1 ≤
√
2n−1
=
π
n−1
x 2 +y 2
x2 +y 2 ρ
n−1
···
0
i=1
n−1
ρi dρi
1 − ρi2 i=1
ρ3 ρ2 n−1
0
0
0 i=1
ρi
dρ1 dρ2 · · · dρn−1 . (21)
1 − ρi2
Evaluating integral in (21) as follows
ρ2
0
ρ3
0
ρ4
0
ρ1
1 2
ln
1
−
ρ
,
dρ
=
−
1
2
2
1 − ρ12
2
1 ρ2
1 2
2
−
ln(1
−
ρ
ln
1
−
ρ
dρ
=
)
,
2
2
3
2
2·4
1 − ρ22
ρ3
1 − ρ32
2 3
1 1
2
dρ3 = −
ln(1 − ρ3 )
ln(1 − ρ32 ) ,
2×4
2·4·6
···
√
x2 +y 2
=
(−1)n+1
0
=
ρn−1
2
1 − ρn−1
n−2 1
2
dρn−1
ln(1 − ρn−1
)
2 · 4 · 6 · · · 2(n − 2)
n−1
(−1)n−1
ln(1 − x 2 − y 2 )
2 · 4 · 6 · · · 2(n − 1)
we have
f ZU (n) (x, y) =
n−1
n−1
2n−1 (−1)n−1 1 ln(1 − x 2 − y 2 )
π
2·i
i=1
=
2n−1 (−1)n−1
1
π
2n−1
n−1
i=1
n−1
1
ln(1 − x 2 − y 2 )
i
n−1
1
(−1)n−1
ln(1 − x 2 − y 2 )
=
.
π
(n − 1)!
Therefore,
f ZU (n) (x, y) =
(−1)n−1
1
π
(n−1)!
0
n−1
ln(1 − x 2 − y 2 )
i f x 2 + y2 ≤ 1
.
other wise
(22)
123
746
I. Bayramoglu
Remark 2 It can be easily verified that f ZU (n) (x, y) given in (22) satisfies
∞ ∞
f ZU (n) (x, y)d xd y = 1.
−∞ −∞
In fact, using x = ρ cos ϕ and y = ρ sin ϕ one has
(−1)n−1
1
π
(n − 1)!
ln(1 − x 2 − y 2 )
n−1
d xd y
x 2 +y 2 ≤1
1
(−1)n−1
=
π
(n − 1)!
=
(−1)n−1 2π
π
(n − 1)!
π 1
n−1
ρ ln(1 − ρ 2 )
dρdϕ
−π 0
1
n−1
ρ ln(1 − ρ 2 )
dρ
0
2(−1)n−1 1
=
(−1)n (n, − ln(1 − x 2 )) |x=1
x=0 = 1,
(n − 1)! 2
where →
∞
(x, y) =
e−t t x−1 dt and (n, − ln(1 − x 2 )) |x=1
x=0 = (n, ∞) − (n, 0)
y
= −
(n) = −(n − 1)!,
∞
(x) = e−t t x−1 dt is a Gamma function.
0
4 Conclusion
The record values of univariate random sequence of i.i.d. random variables were first
introduced by Chandler (1962) and intrigued many researchers dealing with theory and
applications of probability and statistics. There appeared numerous papers and several
monographs devoted to record values. There are many papers investigating characterizations of distributions through the properties of records. There exist numerous papers
devoted to inferential statistical problems involving records and many papers dealing
with applications of records in fields such as sports, extreme weather conditions and
flooding etc. The theory of multivariate records has been studied rarely and only in
the sense of coordinate wise ordering. In this work, we make an attempt to investigate
the distributions of two types bivariate record times and record values. First, assuming
that K is the most recent record time, we say that a new record occurs at time n, if
123
On the records of multivariate random sequences
747
(X n , Yn ) ⊀ (X K , Y K ). This type of record is different than presented in Naiman and
Torcaso (2009). More precisely, in Naiman and Torcaso (2009) the record occurs at
time i, if (X n , Yn ) ⊀ (X i , Yi ), for i = 1, 2, . . . , n − 1. Thus, to determine whether a
new record occurs at time n one needs to check that the value (X n , Yn ) is not dominated by some former record. However, in definition given and used in this paper
a new record occurs at time n, if (X n , Yn ) is not dominated by most recent record.
Second, we consider records defined by using conditional N -ordering. In this paper,
we present some distributional results for two types of new records for multivariate
observations and provide some examples.
Acknowledgments The author are thankful to anonymous reviewer for very valuable comments and suggestions which resulted in improvement of the presentation of this paper.
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