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Multivariate discrete scalar hazard rate
Statistical Methodology 27 (2015) 39–50
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Multivariate discrete scalar hazard rate N. Unnikrishnan Nair, P.G. Sankaran ∗ Department of Statistics, Cochin University of Science and Technology, Cochin 682022, Kerala, India
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Article history: Received 24 January 2015 Received in revised form 27 April 2015 Accepted 5 May 2015 Available online 28 May 2015 Keywords: Multivariate distributions Scalar hazard rate Multivariate mean residual life Product moment of residual life Characterizations
abstract In the present paper, we study the properties of the multivariate discrete scalar hazard rate. Its continuous analogue introduced in the early seventies did not attract much attention because it could not be used to identify the corresponding life distribution. We find the conditions under which an n-variate discrete scalar hazard rate can determine the distribution uniquely. Several other properties of this hazard rate which can be employed in modelling lifetime data are discussed. Some ageing classes based on the scalar hazard function are suggested. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The concept of hazard rate is fundamental in reliability modelling and analysis. Apart from uniquely determining the lifetime distribution, the hazard rate provides more information than the survival function about the pattern of failure. It also enables the study of various ageing classes that classify life distributions. Further, the distribution of lifetime can be identified using the functional form of the hazard rates through characterizations. When the hazard rate definition is extended to the multivariate case, there are different ways in which it can be proposed depending on the manner in which the univariate notion is generalized. If (X1 , X2 ) is a random vector in the support of N 2 , N = {0, 1, 2, . . .} with survival function, S (x1 , x2 ) = P [X1 ≥ x1 X2 ≥ x2 ].
(1.1)
Nair and Nair [8] defined the bivariate hazard rate as the vector b(x1 , x2 ) = (b1 (x1 , x2 ), b2 (x1 , x2 )) ,
∗
Corresponding author. E-mail addresses: [email protected] (N.U. Nair), [email protected] (P.G. Sankaran).
http://dx.doi.org/10.1016/j.stamet.2015.05.003 1572-3127/© 2015 Elsevier B.V. All rights reserved.
(1.2)
40
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
where, bi (x1 , x2 ) = P [Xi = xi |X1 ≥ x1 , X2 ≥ x2 ],
i = 1, 2.
(1.3)
Kotz and Johnson [7] proposed c(x1 , x2 ) = (c1 (x1 , x2 ), c2 (x1 , x2 )) ,
(1.4)
in which ci (x1 , x2 ) = P [Xi = xi |Xi ≥ xi , Xj = xj ],
i, j = 1, 2, i ̸= j
(1.5)
as an alternative. In a slightly different framework, Sun and Basu [13] used a three component vector (d(x), d1 (x1 |x2 ), d2 (x2 |x1 )) in presenting their bivariate total hazard rate with d(x) =
P [min(X1 , X2 ) = x] S (x1 , x2 )
f (x1 , x2 ) d1 (x1 |x2 ) = ∞ , f (t , x2 )
,
(1.6)
x1 > x2
(1.7)
x1 < x2 ,
(1.8)
t =x 1
and f (x1 , x2 ) , d2 (x2 |x1 ) = ∞ f (x1 , t ) t =x 2
where, f (x1 , x2 ) is the probability mass function of (X1 , X2 ). A fourth version of the bivariate hazard rate given by Shaked et al. [12] has five components. The extension of the above definitions to the n-variate case is straightforward. The earliest definition of bivariate hazard rate in continuous time, given by Basu [2], is the scalar quantity h(x1 , x2 ) =
g ( x1 , x2 )
¯ (x1 , x2 ) G
,
(1.9)
¯ respectively denote the density and survival function of (X1 , X2 ) of a continuous nonwhere g and G negative random vector. A major limitation of the above definition pointed out by several researchers, e.g. Galambos and Kotz [5], Finkelstein and Esaulova [4] is that h(x1 , x2 ) cannot determine the distribution of (X1 , X2 ). As a consequence, modelling bivariate data by the functional form of h(x1 , x2 ) becomes difficult. Recently, Navarro [9] has obtained some general conditions under which (1.9) determines the corresponding distribution. In the present work, we study the properties of the discrete n-variate version of (1.9), which does not appear to have been considered earlier. The role of reliability models in discrete time has been well established and the existence of complex equipments with several components whose reliability studies are essential motivates our study. Secondly, we demonstrate that under some mild conditions, the multivariate scalar hazard rate can determine the life distribution, which overcomes the main disadvantage of this concept. In modelling multivariate data, the dependence relation between the constituent variables is of primary importance. The hazard rate considered here can be used to infer the dependence relation that reduces considerably the list of candidate distributions sought for modelling. Identification of appropriate distributions can also be accomplished through some characterizations based simple functional forms of the scalar hazard rates. Certain reservations about the vector representation of hazard functions [3] is not applicable to our results as the hazard rate considered here is a scalar quantity. The paper is organized into four sections. In Section 2, we define and present the properties of the multivariate discrete scalar hazard rate. This is followed by suggesting how the hazard rate can be employed to study the ageing criteria. Finally in Section 4, we give a short conclusion of the present study.
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
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Table 1 Bivariate discrete scalar hazard rates. In all cases, (x1 , x2 ) ∈ N 2 except 8.
1 2
Survival function
a2 (x1 , x2 )
x x q11 q22 x x q11 q22
(1 − q1 )(1 − q2 ); 0 < q1 , q2 < 1 1 − q1 θ x2 +1 1 − q2 θ x1 +1 + θ − 1 θ −1 ; 0 < q1 , q2 < 1; 1 −θ ≤ (1 − q1 θ)(1 − q2 θ); 0 ≤ θ ≤ 1.
θ x1 x2
x +x 2
(1−p)q11
3
1−
x +x 2
1−pq11
(x1 +x2 )β
4
1 − 2q
q
1+e
1−
x1 +x2
σ
x1 +x2 −1
(1 − p + p π i=1 (m)x1 +x2 (m+n)x1 +x2
6 7
i
)
e−θ(x1 +x2 )
9
(x1 +x2 +1)β −(x1 +x2 )β x1 +x2
σ 2 x1 +x2 +1 σ 1+e
1+e
+q
, 0 < q1 , p < 1.
(x1 +x2 +2)−(x1 +x2 )β
; 0 < q < 1; β > 0.
x 1 +x 2
σ x1 +x2 +2 σ 1+e
1+e
k(k−1) ; (k+n−x1 −x2 )(k+n−x1 −x2 −1)
> 0.
x1 , x2 = 0, 1, 2 . . . n; k, n > 0.
θ (x1 +x2 )−2(1+θ (x1 +x2 +1))e + (1 + θ(x1 + x2 + 2))e−2θ ; θ > 0. 1+θ (x1 +x2 ) ( p − p )( 1 − p ) 1 1 : x1 > x2 p1 ; 0 < p ≤ p1 , p2 ; 1 + p > p1 + p2 . (p2 − p)(1 − p2 ) : x1 < x2 p 2 (1 + p − p 1 − p 2 ) : x 1 = x 2
1+θ +θ(x1 +x2 ) 1+θ
−θ
1+
x −x
px2 p11 2 : x1 ≥ x2 x −x px1 p22 1 : x1 ≤ x2 .
10
x +x q21 (1−pq11 2 ) x +x +2 1−pq11 2
n(n+1) ; m, n (m+n+x1 +x2 )(m+n+x1 +x2 +1)
k + n − x1 − x2 n − x1 − x2 k+n n
8
+
+ ; σ > 0. 1 − (1 − p) + pπ x1 +x2 1 + p − pπ x1 +x2 +1 ; 0 < p ≤ 1, 0 ≤ π < 1.
2
5
x +x 2q1 (1−pq11 2 ) x +x +1 1−pq11 2
2. Multivariate hazard rate Let X = (X1 , X2 , . . . , Xn ) be a discrete random vector defined on N n , N = {0, 1, 2 . . .} with probability mass function, fn (x) = P [X = x],
x = (x1 , x2 , . . . , xn )
and survival function Sn (x) = P [X ≥ x], where the ordering ≥ among vectors is taken to be component-wise. The multivariate scalar hazard rate (MSHR) of X is defined as an (x) = P [X = x|X ≥ x] =
fn (x) Sn (x)
(2.1)
at points x for which Sn (x) > 0. Note that an (x) is a straight forward generalization of the univariate f (x) hazard rate h(x) = S (x) of a discrete random variable X and it is the n-variate discrete version of the bivariate hazard rate defined in (1.9). We interpret (2.1) as the conditional probability that in a n-component device, the ith component fails at age xi given that it has survived age xi for all i = 1, 2, . . . , n. Note that 0 ≤ an (x) ≤ 1 and that it cannot reproduce the univariate hazard rates of the components. The expressions of an (x) for some bivariate distributions are presented in Table 1. Some important properties of an (x) are given below. 1. In general an (x) does not determine the distribution of X uniquely. Example 2.1. Let x
x
S2 (x1 , x2 ) = p11 p22 ,
0 < p1 , p2 < 1; x1 , x2 = 0, 1, 2 . . .
(2.2)
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N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
and T2 (x1 , x2 ) =
1 2
x
x
x
x
p11 p22 + p21 p12 .
(2.3)
Both have the same MSHR a2 (x1 , x2 ) = (1 − p1 )(1 − p2 ). 2. The conditions required to construct the survival function S (x) from an (x) are given in the following theorem. Theorem 2.1. The distribution of X is uniquely determined by
(a1 (x1 ), a2 (x2 ), . . . , an (xn )).
(2.4)
Proof. Since a1 (x1 ) is the hazard rate of X1 , it is well known that x 1 −1
S 1 ( x1 ) =
(1 − a1 (t )),
(2.5)
t =0
where Sr (xr ) denotes the survival function of (X1 , . . . , Xr ). From (2.1), a2 (x2 )S2 (x2 ) = f2 (x2 )
= P [X1 = x1 , X2 ≥ x2 ] − P [X1 = x1 , X2 ≥ x2 + 1] or P [X1 = x1 , X2 ≥ x2 + 1] = P [X1 = x1 , X2 ≥ x2 ] − a2 (x2 )S2 (x2 ). Summation from x1 to ∞ yields S2 (x1 , x2 + 1) = S2 (x2 ) −
∞
a2 (t , x2 )S2 (t , x2 ).
(2.6)
t =x 1
Setting x2 = 0, S2 (x1 , 1) = S1 (x1 ) −
∞
a2 (t , 0)S1 (t ).
(2.7)
t =x1
We can find S2 (x2 ) on using the recurrence relation (2.6) with (2.7) as the starting value, since S1 (x1 ) is known from (2.5). Similarly, S3 (x1 , x2 , x3 + 1) = S3 (x3 ) −
∞ ∞
a3 (t1 , t2 , x3 )S3 (t1 , t2 , x3 ).
(2.8)
t1 = x 1 t2 = x 2
Now work with x3 = 0 and determine S3 (x3 ) from (2.8). Finally, we have to use recursively Sn (xn−1 , xn + 1) = Sn (xn ) −
∞
...
t1 =x1
in the above manner to reach Sn (xn ).
∞
an (t1 , t2 . . . , tn−1 xn )Sn (t1 , . . . , tn−1 xn )
(2.9)
tn−1 =xn−1
We illustrate the procedure to characterize the multivariate Waring distribution and negative hyper-geometric law.
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Theorem 2.2. The random vector X follows the multivariate Waring distribution Sn (x) =
(m)x1 +x2 +···+xn , (m + p)x1 +x2 +···+xn
m, p > 0, x ∈ N n ,
(2.10)
where (t )r = t (t + 1) . . . (t + r − 1) if and only if for r = 1, 2, . . . , n, ar (xr ) =
(p)r . (m + p)x1 +x2 +···+xr
(2.11)
Proof. Assume that (2.11) holds. From (2.5), x 1 −1
S1 (x1 ) =
1−
t =0
t
p
=
m+p+t
(p)x1 . (m + p)x1
Thus the distribution is (2.10) for n = 1. Now let the result be true for n = b. Then, Sb (xb ) =
(m)x1 +x2 +···+xb . (m + p)x1 +x2 +···+xb
Employing Eq. (2.10), ∞
Sb+1 (xb , xb+1 + 1) = Sb (xb ) −
...
t1 =x1
∞
ab (tb , xb+1 )Sb+1 (tb , xb+1 ).
tb = x b
When xb+1 = 0, Sb+1 (xb , 1) = Sb (xb ) −
∞
∞
...
t1 = x 1
tb =xb
(m)t1 +t2 +···+tb (p)b . (m + p)t1 +t2 +···+tb (m + p)t1 +t2 +···+tb
(2.12)
To proceed with further calculations, we note that p m + p + t1
=1−
m + t1
m + p + t1 p(p + 1)
.
(m + p + t1 + t2 )(m + p + t1 + t2 + 1)
= 1−2 +
m + t1 + t2 m + p + t1 + t2 (m + t1 + t2 )(m + t1 + t2 + 1)
(m + p + t1 + t2 )(m + p + t1 + t2 + 1)
.
By induction, m + t1 + · · · + tb (m + t1 + t2 + · · · + tb )2 (p)b = 1−b + (b − 1) (p + m)t1 +t2 +···+tb m + p + t1 + · · · + tb (m + p + t1 + · · · + tb )2
− · · · + (−1)b−1
(m + t1 + · · · + tb )b . (m + p + t1 + · · · + tb )b
Inserting (2.13) into (2.12), Sb+1 (xb , 1) = Sb (xb ) −
∞ t1 = x 1
...
∞ (m)t1 +t2 +···+tb (m + p)t1 +···+tb tb = x b
b(m)t1 +···+tb +1 (m)t1 +···+tb +2 + (b − 1) − ··· (m + p)t1 +···+tb +1 (m)t1 +···+tb +2 (m)x1 +x2 +···+xb +1 = . (m + p)x1 +x2 +···+xb +1
−
The same formula (2.13) will work for evaluating S (xb , 2) and so on.
(2.13)
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N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
Finally, Sb+1 (xb+1 ) =
(m)x1 +x2 +···+xb+1 . (m + p)x1 +x2 +···+xb+1
Thus the distribution (2.10) is obtained by induction. The converse can be obtained by direct calculation of ar (xr ) from (2.10). Theorem 2.3. The random vector X has multivariate negative hyper-geometric distribution
Sn (x) =
k + p − x1 − x2 − · · · − xn p − x1 − x2 − · · · − xn
k+p p
k, p > 0, x1 , . . . , xn = 0, 1, 2, . . . , p; ar (xr ) =
n
i =1
,
(2.14)
xi < p if and only if
(k)(r ) (k + p − x1 − x2 − · · · x1 )(r )
(2.15)
for all x, r = 1, 2 . . . , n and t (r ) = t (t − 1) . . . (t − r + 1) is the descending factorial expression. The proof being similar to that of Theorem 2.2, we give only the main steps. Assume that (2.14) holds. Then for n = 1, a1 (x1 ) = k−pk−x , 1 and
S 1 ( x1 ) =
x −1 t =0
1−
k k−p−t
=
k + p − x1 p − x1
k+p p
so that the result is true for n = 1. We notice that p − x1 − x2 (p − x1 − x2 )(p − x1 − x2 − 1) (k)(2) = 1−2 + . (k − p − x1 − x2 )(2) k − p − x1 − x2 (k + p − x1 − x2 )(k + p − x1 − x2 − 1) By induction
(k)(b) p − x1 − x2 − · · · − xb = 1−b + ··· (k − p − x1 − x2 − · · · − xb )(b) k − p − x1 − x2 − · · · − xb (p − x1 − · · · − xb )(b) . + (−1)b−1 (k − p − x1 − · · · − xb )(b) Using this
x (k)(b) S (k − p − x1 − · · · − xb )(b) b t1 = x 1 tb = x b k + p − x1 − · · · − xb − 1 p − x1 − · · · − xb − 1 = . k+p
S (xb , 1) = Sb (xb ) −
p
···
p
p
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
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Now calculate S (xb , 2), S (xb , 3), . . . successively
S (xb+1 ) =
k + p − x1 − · · · − xb+1 p − x1 − · · · − xb+1
k+p p
for all positive integers b. This proves the ‘if’ part. The ‘only if’ part is obtained by direct calculation. 3. If X1 , X2 . . . , Xn are independent lifetimes, then an (x) = h1 (x1 )h2 (x2 ) . . . hn (xn )
(2.16)
and conversely, where hi (xi ) is the hazard rate of Xi . Proof. The proof of the first part is obvious and that of the second part is obtained by the method of Theorem 2.1, by starting from S2 (x1 , 1) = S1 (x1 ) −
∞
h1 (t )h2 (0)S1 (t1 )
t =x 1
= S1 (x1 ) − h2 (0)
∞
f1 (t1 )
t =x 1
= S1 (x1 )[1 − h2 (0)] = S1 (x1 )P [X2 ≥ 1] and then arriving at S2 (x2 ) = P [X1 ≥ x1 ]P [X2 ≥ x2 ]. The case of n variables now follows from (2.8) and (2.9).
4. In modelling multivariate data, the dependent relation between the constituent variables play a crucial role. The knowledge of whether the association is positive or negative may limit the choice of models considered for the data. We prove a result in this connection. Theorem 2.4. When E (Xi Xj ) < ∞, i ̸= j, Cov(Xi , Xj ) = E
1 a2 (Xi , Xj )
−E
1 a1 ( X i )
E
1
a1 (Xj )
.
(2.17)
Proof.
E
1 a2 (Xi , Xj )
=
∞ ∞
S2 (xi , xj )
x i =0 x j =0
= S2 (0, 0) +
∞ x i =1
Si (xi ) +
∞
Sj (xj ) +
x j =1
∞ ∞
S2 (xi , xj )
i =1 j =1
= 1 + E (Xi ) + E (Xj ) + E (Xi Xj ). Since E
1 ai (Xi )
=
∞
xi =0
Si (xi ) = 1 + E (Xi ), the conclusion follows.
5. In the bivariate case, the vector hazard rate (1.2) is related to a2 (x1 , x2 ) by a2 (x1 , x2 ) = b1 (x1 , x2 ) − b1 (x1 , x2 + 1) + b1 (x1 , x2 + 1)b2 (x1 , x2 )
= b2 (x1 , x2 ) − b2 (x1 + 1, x2 ) + b2 (x1 + 1, x2 )b1 (x1 , x2 ).
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6. The hazard rate (2.1) can be related to the multivariate mean residual life function in the following way. In discrete set up, the multivariate mean residual life function of X is defined as the vector mn (x) = (m1 (x), m2 (x), . . . , mn (x)),
(2.18)
where, mi (x) = E (Xi − xi |X > x),
=
=
i = 1, 2 . . . , n; xi = −1, 0, 1 . . .
∞
1
Sn (x + en ) t =x +1 1 1 ∞
1
Sn (x + en ) t =x +1 i i
...
∞
(ti − xi )f (t )
tn = x n + 1
S (xi + 1, . . . , xi−1 + 1, ti , xi+1 + 1, . . . , xn + 1)
with en = (1, 1, . . . , 1). In the bivariate case, 1 − a2 (x1 , x2 ) =
m 1 ( x1 , x2 ) − 1 S2 (x1 + 1, x2 + 1)
+
+
m 2 ( x1 , x2 ) − 1 m2 (x1 , x2 + 1)
m1 (x1 , x2 ) − 1 m2 (x1 , x2 + 1) − 1 m1 (x1 + 1, x2 ) m2 (x1 + 1, x2 + 1)
.
(2.19)
This result can be extended to the n-variate case, but being lengthy, it is not given here. 7. By virtue of Theorem 2.1, all the functional forms of a2 (x1 , x2 ) in Table 1 along with the distribution of X1 (obtained by setting x2 = 0 in S2 (x1 , x2 )) characterize the respective bivariate models. 8. In reliability and survival analysis, the lifetime distribution of a device or organism after it has survived a specific age is of considerable interest. Instead of considering the usual covariance in such situations, the residual covariance, which is a time-dependent measure of association is more desirable. Therefore, we define the covariance of residual life as a measure in this context by C (x1 , x2 ) = M (x1 , x2 ) − m1 (x1 , x2 )m2 (x1 , x2 )
(2.20)
where M (x1 , x2 ) = E [(X1 − x1 )(X2 − x2 )|X1 > x1 , X2 > x2 ] is the product moment of residual life. We see that M (x1 , x2 ) =
=
1
∞
∞
(t1 − x1 )(t2 − x2 )f (t1 , t2 )
S (x1 + 1, x2 + 1) t =x +1 t =x +1 1 1 2 2 1
∞
∞
S (x1 + 1, x2 + 1) t =x +1 t =x +1 1 1 2 2
S (t1 , t2 ).
(2.21)
It is easy to see that M (x1 , x2 ) = E
1 a2 (X1 , X2 )
|X1 > x1 , X2 > x2 .
The constancy of the hazard rate is of general importance in the univariate and multivariate cases. From the definitions of a2 (x1 , x2 ) and M (x1 , x2 ), we have the following result. Theorem 2.5. The bivariate scalar hazard rate a2 (x1 , x2 ) = k, a constant, for all x1 , x2 in N if and only if M (x1 , x2 ) = 1k . Remark 2.1. The relationship a2 (x1 , x2 ).M (x1 , x2 ) = 1 is satisfied by several distributions as is evident from Example 2.1. The analogous results for continuous distributions were studied in Gupta and Sankaran [6], Navarro et al. [10] and Navarro and Sarabia [11].
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
47
9. The equilibrium distribution of a discrete random variable X with probability mass function f (x) and survival function S (x) is defined as f1,E (x) =
S1 (x + 1) E (X )
,
x = 0, 1, 2 . . . .
(2.22)
A direct generalization provides that for the random vector X, the equilibrium distribution is fn,E (x) =
Sn (x + en ) E (X1 X2 . . . Xn )
.
(2.23)
By summation, fn,E (x) Sn,E (x)
Sn (x + en )
=
∞ t1 = x 1 + 1
∞
...
. Sn (t )
tn = x n + 1
This leads to the following result. Theorem 2.6. The scalar hazard rate of the multivariate equilibrium distribution is an,E (x) =
1 Mn (x + en )
(2.24)
where Mn (x) = E [(X1 − x1 )(X2 − x2 ) . . . (Xn − xn )|X > x] . The converse of this result is also true. 3. Ageing properties In this section, we discuss various ageing properties based on the multivariate scalar hazard rate. 3.1. No-ageing property We say that the lifetime X has multivariate no-ageing property if and only if it satisfies P [Xr = xr |Xr ≥ xr ] = P [Xr = 0r |X ≥ 0r ]
(3.1)
for all xr in N and for r = 1, 2, . . . , n and 0r = (0, 0 . . . , 0). This means that the conditional probability of failure of a n-component system at age xr upon survival of age xr , r = 1, 2 . . . , n is the same as the failure of the system at the starting point of its operation. Equivalently one can also express it as the probability of failure is the same at all ages. In terms of MSHR, (3.1) is equivalent to the statement r
ar (xr ) = kr ,
a constant for all xr , 0 < kr < 1, for r = 1, 2, . . . , n.
(3.2)
Theorem 3.1. A lifetime X satisfies the multivariate no-ageing property (3.2) if and only if Sn (x) = (1 − k1 )x1 (1 − k2 )x2 . . . (1 − kn )xn ,
(3.3)
a multivariate geometric distribution with independent geometric marginals. The proof of the above theorem is easily obtained by applying Theorem 2.1. It may also be observed that in this case, ar (xr ) is the product of the marginal hazard rates of X1 , X2 , . . . , Xr . Further, the condition an (x) = k for some constant k, 0 < k < 1 neither guarantees the independence of the Xi′ s nor geometric marginals. Another result of interest in which the bivariate hazard rate is constant is given by Asha and Nair [1].
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N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
Theorem 3.2. A bivariate lifetime (X1 , X2 ) on N 2 has bivariate geometric distribution S (x1 , x2 ) =
x −x
px2 p11 2 : x1 ≥ x2 x −x px1 p22 1 : x1 ≤ x2
(3.4)
0 < p < pi < 1, i = 1, 2; 1 + p ≥ p1 + p2 if and only if the form of the bivariate hazard rate c1 : x1 > x2 c2 : x1 < x2 c3 : x1 = x2
a2 (x1 , x2 ) =
(3.5)
1 −1 c1 = (p1 − p)(1 − p1 )p− 1 , c2 = (p2 − p)(1 − p2 )p2 and c3 = 1 + p − p1 − p2 .
3.2. Positive and negative ageing Definition 3.1. We say that X has increasing (decreasing) multivariate scalar hazard rate, MIHR (MDHR) if and only if ar (xr ) is increasing (decreasing) in x1 , x2 , . . . , xr for r = 1, 2, . . . , n. In this definition, we mean increasing (decreasing) to be non-decreasing (non-increasing). Example 3.1. (a) The multivariate Waring distribution (2.10) is MIHR and the multivariate hypergeometric distribution in Theorem 2.3 is MDHR. (b) S (x1 , x2 ) =
(m)x1 (m+n)x1
a2 (x1 , x2 ) =
k + p − x1 p − x1
k+p p
, x1 = 0, 1, 2 . . . ; x2 = 0, 1, 2 . . . n for which nk
(m + n + x1 )(k + p − x2 )
is neither MIHR nor MDHR. Definition 3.2. A random vector X on N p is multivariate increasing (decreasing) scalar hazard rate average, MIHRA (MDHRA) if and only if Hr (xr ) =
1 x1 x2 . . . xr
x 1 −1
x r −1
...
t1 = 0
ar (tr )
(3.6)
tr = 0
is increasing (decreasing) in x1 , x2 , . . . , xr ; r = 1, 2 . . . , n. We call (x1 x2 . . . xr )Hr (xr ), the cumulative hazard rate of Xr . Theorem 3.3. X is MIHR (MDHR) ⇒ X is MIHRA (MDHRA). Proof. Since the proof of the n-variate case is apparent from that of the bivariate case, we give proof of the latter case only. Since (X1 , X2 ) is MIHR, a1 (x1 ) is increasing in x1 and a2 (x1 , x2 ) is increasing in x1 , x2 . x 1 −1 Using H1 (x1 ) = x1 t1 =0 a1 (t ), 1
H1 (x1 + 1) ≥ H1 (x1 ). Hence, H1 (x1 + 1) − H1 (x1 ) ⇐⇒
1
x1
x 1 + 1 t =0 1
a1 (t ) − H1 (x1 )
⇐⇒ a1 (x1 ) ≥ H1 (x1 ).
(3.7)
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
49
Also, H1 (x1 ) =
x 1 −1 1
x 1 t =0
a1 (t ) ≤
x 1 −1 1
x t =0
a( t )
(3.8)
= a1 (x1 ),
(3.9)
since a1 (x1 ) ≥ a1 (t ) for all t = 0, 1, 2 . . . x − 1. From (3.7) and (3.8), H1 (x1 ) is increasing if a1 (x1 ) is increasing. In the case of two variables, assume that a2 (x1 , x2 ) is increasing in both x1 and x2 . H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) =
1
x1 x2
x1 x2 (x1 + 1)(x2 + 1)
x1 x2
x 1 −1 x 2 −1
− (x1 + 1)(x2 + 1)
a2 (t1 , t2 )
t1 =0 t2 =0
a2 (t1 , t2 ) .
t1 = 0 t2 = 0
Write x1 x2
x 1 −1 x 2 −1
a2 (t1 , t2 ) =
t1 = 0 t2 = 0
x 1 −1
a2 (t1 , t2 ) −
t1 = 0 t2 = 0
x2 −1
a2 (t1 , x1 ) −
t1 =0
a2 (x1 , t2 ) + a2 (x1 , x2 ).
t2 = 0
We have,
H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) = x1 x2
a2 (t1 , x2 ) −
+ x2 x1
x 2 −1
a2 (t1 , t2 )
t1 = 0 t2 = 0
t1 = 0
x1 −1 x2 −1
x 1 −1
x1 −1 x2 −1
a2 (x1 , t2 ) −
t2 = 0
+ x 1 x 2 a2 ( x 1 , x 2 ) −
a2 (t1 , t2 )
t1 = 0 t2 = 0 x 1 −1 x 2 −1
a2 (t1 , t2 ) .
(3.10)
t1 = 0 t2 = 0
Since a2 (x1 , x2 ) is increasing in x1 and x2 , a2 (x1 , t2 ) ≥ a2 (t1 , t2 ) for all t1 in [0, x1 − 1] and a2 (t1 , x2 ) ≥ a2 (t1 , t2 ) for all t2 in [0, x2 − 1]. Hence all the terms in the square braces in (3.10) remain non-negative. This means that H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) ≥ 0 for all x1 and x2 and hence (X1 , X2 ) is MIHRA. Definition 3.3. We say that X is multivariate new better than used in hazard rate, MNBUHR (MNWUHR) if ar (xr ) ≥ ar (0r ) for all xr and r = 1, 2, . . . , n. From the definition, it follows that MIHR⇒ MNBUHR. Thus we see that the notions of ageing in the univariate case using hazard rate can be extended to the multivariate case employing the scalar hazard rate. 4. Conclusion In this paper, we have studied the properties of a discrete multivariate hazard rate. It is seen through the various results that it can carry out all the functions prescribed for a hazard rate. The properties obtained here compare favourably with those of the competing alternative definitions of multivariate hazard rates. The major distinction of the scalar form from the others is that while the former is indicative of the joint variation of the lifetimes, the latter does the same thing about the mean of the observations. We have provided only an illustrative account of the role that the scalar hazard can carry out in reliability modelling. The modelling aspects based on the above results like estimation of the hazard rate, selection of models for real data, etc. need to be studied further, hopefully in a future work.
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Acknowledgements The authors wish to thank the associate editor and the referees for their valuable suggestions that helped them to improve the presentation. References [1] G. Asha, N.U. Nair, Bivariate failure rates in discrete time, in: Quality Improvement through Statistical Methods, Springer, 1998, pp. 339–350. [2] A.P. Basu, Bivariate failure rate, J. Amer. Statist. Assoc. 66 (333) (1971) 103–104. [3] S.E. Chick, M.B. Mendel, New characterizations of the no-ageing property and the ℓ1-isotropic model, J. Appl. Probab. 35 (4) (1998) 903–910. [4] M. Finkelstein, V. Esaulova, On the weak IFR ageing of bivariate lifetime distributions, Appl. Stoch. Models Bus. Ind. 21 (3) (2005) 265–272. [5] J. Galambos, Kotz S, Characterization of Probability Distributions, in: Lecture Notes in Mathematics, Springer, 1978. [6] R.P. Gupta, P.G. Sankaran, Bivariate equilibrium distribution and its application to reliability, Comm. Statist. Theory Methods 27 (1998) 385–394. [7] S. Kotz, N.L. Johnson, A note on renewal distributions for discrete distributions, Statist. Probab. Lett. 12 (1991) 229–231. [8] N.U. Nair, K.R.M. Nair, Characterizations of bivariate geometric distribution, Statistics 46 (1990) 247–253. [9] J. Navarro, Characterizations using the bivariate failure rate function, Statist. Probab. Lett. 78 (12) (2008) 1349–1354. [10] J. Navarro, T.M. Ruiz, Y. del Auguila, Multivariate weighed distributions: a review and some extensions, Statistics 40 (2006) 57–64. [11] J. Navarro, J.M. Sarabia, Alternative definitions of bivariate equilibrium distributions, J. Statist. Plann. Inference 140 (2010) 2046–2056. [12] M. Shaked, J.G. Shanthikumar, J.B. Valdez-Torres, Discrete hazard rate functions, Comput. Oper. Res. 22 (4) (1995) 391–402. [13] K. Sun, A.P. Basu, A characterization of a bivariate geometric distribution, Statist. Probab. Lett. 23 (4) (1995) 307–311.
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Statistical Methodology journal homepage: www.elsevier.com/locate/stamet
Multivariate discrete scalar hazard rate N. Unnikrishnan Nair, P.G. Sankaran ∗ Department of Statistics, Cochin University of Science and Technology, Cochin 682022, Kerala, India
article
info
Article history: Received 24 January 2015 Received in revised form 27 April 2015 Accepted 5 May 2015 Available online 28 May 2015 Keywords: Multivariate distributions Scalar hazard rate Multivariate mean residual life Product moment of residual life Characterizations
abstract In the present paper, we study the properties of the multivariate discrete scalar hazard rate. Its continuous analogue introduced in the early seventies did not attract much attention because it could not be used to identify the corresponding life distribution. We find the conditions under which an n-variate discrete scalar hazard rate can determine the distribution uniquely. Several other properties of this hazard rate which can be employed in modelling lifetime data are discussed. Some ageing classes based on the scalar hazard function are suggested. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The concept of hazard rate is fundamental in reliability modelling and analysis. Apart from uniquely determining the lifetime distribution, the hazard rate provides more information than the survival function about the pattern of failure. It also enables the study of various ageing classes that classify life distributions. Further, the distribution of lifetime can be identified using the functional form of the hazard rates through characterizations. When the hazard rate definition is extended to the multivariate case, there are different ways in which it can be proposed depending on the manner in which the univariate notion is generalized. If (X1 , X2 ) is a random vector in the support of N 2 , N = {0, 1, 2, . . .} with survival function, S (x1 , x2 ) = P [X1 ≥ x1 X2 ≥ x2 ].
(1.1)
Nair and Nair [8] defined the bivariate hazard rate as the vector b(x1 , x2 ) = (b1 (x1 , x2 ), b2 (x1 , x2 )) ,
∗
Corresponding author. E-mail addresses: [email protected] (N.U. Nair), [email protected] (P.G. Sankaran).
http://dx.doi.org/10.1016/j.stamet.2015.05.003 1572-3127/© 2015 Elsevier B.V. All rights reserved.
(1.2)
40
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
where, bi (x1 , x2 ) = P [Xi = xi |X1 ≥ x1 , X2 ≥ x2 ],
i = 1, 2.
(1.3)
Kotz and Johnson [7] proposed c(x1 , x2 ) = (c1 (x1 , x2 ), c2 (x1 , x2 )) ,
(1.4)
in which ci (x1 , x2 ) = P [Xi = xi |Xi ≥ xi , Xj = xj ],
i, j = 1, 2, i ̸= j
(1.5)
as an alternative. In a slightly different framework, Sun and Basu [13] used a three component vector (d(x), d1 (x1 |x2 ), d2 (x2 |x1 )) in presenting their bivariate total hazard rate with d(x) =
P [min(X1 , X2 ) = x] S (x1 , x2 )
f (x1 , x2 ) d1 (x1 |x2 ) = ∞ , f (t , x2 )
,
(1.6)
x1 > x2
(1.7)
x1 < x2 ,
(1.8)
t =x 1
and f (x1 , x2 ) , d2 (x2 |x1 ) = ∞ f (x1 , t ) t =x 2
where, f (x1 , x2 ) is the probability mass function of (X1 , X2 ). A fourth version of the bivariate hazard rate given by Shaked et al. [12] has five components. The extension of the above definitions to the n-variate case is straightforward. The earliest definition of bivariate hazard rate in continuous time, given by Basu [2], is the scalar quantity h(x1 , x2 ) =
g ( x1 , x2 )
¯ (x1 , x2 ) G
,
(1.9)
¯ respectively denote the density and survival function of (X1 , X2 ) of a continuous nonwhere g and G negative random vector. A major limitation of the above definition pointed out by several researchers, e.g. Galambos and Kotz [5], Finkelstein and Esaulova [4] is that h(x1 , x2 ) cannot determine the distribution of (X1 , X2 ). As a consequence, modelling bivariate data by the functional form of h(x1 , x2 ) becomes difficult. Recently, Navarro [9] has obtained some general conditions under which (1.9) determines the corresponding distribution. In the present work, we study the properties of the discrete n-variate version of (1.9), which does not appear to have been considered earlier. The role of reliability models in discrete time has been well established and the existence of complex equipments with several components whose reliability studies are essential motivates our study. Secondly, we demonstrate that under some mild conditions, the multivariate scalar hazard rate can determine the life distribution, which overcomes the main disadvantage of this concept. In modelling multivariate data, the dependence relation between the constituent variables is of primary importance. The hazard rate considered here can be used to infer the dependence relation that reduces considerably the list of candidate distributions sought for modelling. Identification of appropriate distributions can also be accomplished through some characterizations based simple functional forms of the scalar hazard rates. Certain reservations about the vector representation of hazard functions [3] is not applicable to our results as the hazard rate considered here is a scalar quantity. The paper is organized into four sections. In Section 2, we define and present the properties of the multivariate discrete scalar hazard rate. This is followed by suggesting how the hazard rate can be employed to study the ageing criteria. Finally in Section 4, we give a short conclusion of the present study.
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
41
Table 1 Bivariate discrete scalar hazard rates. In all cases, (x1 , x2 ) ∈ N 2 except 8.
1 2
Survival function
a2 (x1 , x2 )
x x q11 q22 x x q11 q22
(1 − q1 )(1 − q2 ); 0 < q1 , q2 < 1 1 − q1 θ x2 +1 1 − q2 θ x1 +1 + θ − 1 θ −1 ; 0 < q1 , q2 < 1; 1 −θ ≤ (1 − q1 θ)(1 − q2 θ); 0 ≤ θ ≤ 1.
θ x1 x2
x +x 2
(1−p)q11
3
1−
x +x 2
1−pq11
(x1 +x2 )β
4
1 − 2q
q
1+e
1−
x1 +x2
σ
x1 +x2 −1
(1 − p + p π i=1 (m)x1 +x2 (m+n)x1 +x2
6 7
i
)
e−θ(x1 +x2 )
9
(x1 +x2 +1)β −(x1 +x2 )β x1 +x2
σ 2 x1 +x2 +1 σ 1+e
1+e
+q
, 0 < q1 , p < 1.
(x1 +x2 +2)−(x1 +x2 )β
; 0 < q < 1; β > 0.
x 1 +x 2
σ x1 +x2 +2 σ 1+e
1+e
k(k−1) ; (k+n−x1 −x2 )(k+n−x1 −x2 −1)
> 0.
x1 , x2 = 0, 1, 2 . . . n; k, n > 0.
θ (x1 +x2 )−2(1+θ (x1 +x2 +1))e + (1 + θ(x1 + x2 + 2))e−2θ ; θ > 0. 1+θ (x1 +x2 ) ( p − p )( 1 − p ) 1 1 : x1 > x2 p1 ; 0 < p ≤ p1 , p2 ; 1 + p > p1 + p2 . (p2 − p)(1 − p2 ) : x1 < x2 p 2 (1 + p − p 1 − p 2 ) : x 1 = x 2
1+θ +θ(x1 +x2 ) 1+θ
−θ
1+
x −x
px2 p11 2 : x1 ≥ x2 x −x px1 p22 1 : x1 ≤ x2 .
10
x +x q21 (1−pq11 2 ) x +x +2 1−pq11 2
n(n+1) ; m, n (m+n+x1 +x2 )(m+n+x1 +x2 +1)
k + n − x1 − x2 n − x1 − x2 k+n n
8
+
+ ; σ > 0. 1 − (1 − p) + pπ x1 +x2 1 + p − pπ x1 +x2 +1 ; 0 < p ≤ 1, 0 ≤ π < 1.
2
5
x +x 2q1 (1−pq11 2 ) x +x +1 1−pq11 2
2. Multivariate hazard rate Let X = (X1 , X2 , . . . , Xn ) be a discrete random vector defined on N n , N = {0, 1, 2 . . .} with probability mass function, fn (x) = P [X = x],
x = (x1 , x2 , . . . , xn )
and survival function Sn (x) = P [X ≥ x], where the ordering ≥ among vectors is taken to be component-wise. The multivariate scalar hazard rate (MSHR) of X is defined as an (x) = P [X = x|X ≥ x] =
fn (x) Sn (x)
(2.1)
at points x for which Sn (x) > 0. Note that an (x) is a straight forward generalization of the univariate f (x) hazard rate h(x) = S (x) of a discrete random variable X and it is the n-variate discrete version of the bivariate hazard rate defined in (1.9). We interpret (2.1) as the conditional probability that in a n-component device, the ith component fails at age xi given that it has survived age xi for all i = 1, 2, . . . , n. Note that 0 ≤ an (x) ≤ 1 and that it cannot reproduce the univariate hazard rates of the components. The expressions of an (x) for some bivariate distributions are presented in Table 1. Some important properties of an (x) are given below. 1. In general an (x) does not determine the distribution of X uniquely. Example 2.1. Let x
x
S2 (x1 , x2 ) = p11 p22 ,
0 < p1 , p2 < 1; x1 , x2 = 0, 1, 2 . . .
(2.2)
42
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
and T2 (x1 , x2 ) =
1 2
x
x
x
x
p11 p22 + p21 p12 .
(2.3)
Both have the same MSHR a2 (x1 , x2 ) = (1 − p1 )(1 − p2 ). 2. The conditions required to construct the survival function S (x) from an (x) are given in the following theorem. Theorem 2.1. The distribution of X is uniquely determined by
(a1 (x1 ), a2 (x2 ), . . . , an (xn )).
(2.4)
Proof. Since a1 (x1 ) is the hazard rate of X1 , it is well known that x 1 −1
S 1 ( x1 ) =
(1 − a1 (t )),
(2.5)
t =0
where Sr (xr ) denotes the survival function of (X1 , . . . , Xr ). From (2.1), a2 (x2 )S2 (x2 ) = f2 (x2 )
= P [X1 = x1 , X2 ≥ x2 ] − P [X1 = x1 , X2 ≥ x2 + 1] or P [X1 = x1 , X2 ≥ x2 + 1] = P [X1 = x1 , X2 ≥ x2 ] − a2 (x2 )S2 (x2 ). Summation from x1 to ∞ yields S2 (x1 , x2 + 1) = S2 (x2 ) −
∞
a2 (t , x2 )S2 (t , x2 ).
(2.6)
t =x 1
Setting x2 = 0, S2 (x1 , 1) = S1 (x1 ) −
∞
a2 (t , 0)S1 (t ).
(2.7)
t =x1
We can find S2 (x2 ) on using the recurrence relation (2.6) with (2.7) as the starting value, since S1 (x1 ) is known from (2.5). Similarly, S3 (x1 , x2 , x3 + 1) = S3 (x3 ) −
∞ ∞
a3 (t1 , t2 , x3 )S3 (t1 , t2 , x3 ).
(2.8)
t1 = x 1 t2 = x 2
Now work with x3 = 0 and determine S3 (x3 ) from (2.8). Finally, we have to use recursively Sn (xn−1 , xn + 1) = Sn (xn ) −
∞
...
t1 =x1
in the above manner to reach Sn (xn ).
∞
an (t1 , t2 . . . , tn−1 xn )Sn (t1 , . . . , tn−1 xn )
(2.9)
tn−1 =xn−1
We illustrate the procedure to characterize the multivariate Waring distribution and negative hyper-geometric law.
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
43
Theorem 2.2. The random vector X follows the multivariate Waring distribution Sn (x) =
(m)x1 +x2 +···+xn , (m + p)x1 +x2 +···+xn
m, p > 0, x ∈ N n ,
(2.10)
where (t )r = t (t + 1) . . . (t + r − 1) if and only if for r = 1, 2, . . . , n, ar (xr ) =
(p)r . (m + p)x1 +x2 +···+xr
(2.11)
Proof. Assume that (2.11) holds. From (2.5), x 1 −1
S1 (x1 ) =
1−
t =0
t
p
=
m+p+t
(p)x1 . (m + p)x1
Thus the distribution is (2.10) for n = 1. Now let the result be true for n = b. Then, Sb (xb ) =
(m)x1 +x2 +···+xb . (m + p)x1 +x2 +···+xb
Employing Eq. (2.10), ∞
Sb+1 (xb , xb+1 + 1) = Sb (xb ) −
...
t1 =x1
∞
ab (tb , xb+1 )Sb+1 (tb , xb+1 ).
tb = x b
When xb+1 = 0, Sb+1 (xb , 1) = Sb (xb ) −
∞
∞
...
t1 = x 1
tb =xb
(m)t1 +t2 +···+tb (p)b . (m + p)t1 +t2 +···+tb (m + p)t1 +t2 +···+tb
(2.12)
To proceed with further calculations, we note that p m + p + t1
=1−
m + t1
m + p + t1 p(p + 1)
.
(m + p + t1 + t2 )(m + p + t1 + t2 + 1)
= 1−2 +
m + t1 + t2 m + p + t1 + t2 (m + t1 + t2 )(m + t1 + t2 + 1)
(m + p + t1 + t2 )(m + p + t1 + t2 + 1)
.
By induction, m + t1 + · · · + tb (m + t1 + t2 + · · · + tb )2 (p)b = 1−b + (b − 1) (p + m)t1 +t2 +···+tb m + p + t1 + · · · + tb (m + p + t1 + · · · + tb )2
− · · · + (−1)b−1
(m + t1 + · · · + tb )b . (m + p + t1 + · · · + tb )b
Inserting (2.13) into (2.12), Sb+1 (xb , 1) = Sb (xb ) −
∞ t1 = x 1
...
∞ (m)t1 +t2 +···+tb (m + p)t1 +···+tb tb = x b
b(m)t1 +···+tb +1 (m)t1 +···+tb +2 + (b − 1) − ··· (m + p)t1 +···+tb +1 (m)t1 +···+tb +2 (m)x1 +x2 +···+xb +1 = . (m + p)x1 +x2 +···+xb +1
−
The same formula (2.13) will work for evaluating S (xb , 2) and so on.
(2.13)
44
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
Finally, Sb+1 (xb+1 ) =
(m)x1 +x2 +···+xb+1 . (m + p)x1 +x2 +···+xb+1
Thus the distribution (2.10) is obtained by induction. The converse can be obtained by direct calculation of ar (xr ) from (2.10). Theorem 2.3. The random vector X has multivariate negative hyper-geometric distribution
Sn (x) =
k + p − x1 − x2 − · · · − xn p − x1 − x2 − · · · − xn
k+p p
k, p > 0, x1 , . . . , xn = 0, 1, 2, . . . , p; ar (xr ) =
n
i =1
,
(2.14)
xi < p if and only if
(k)(r ) (k + p − x1 − x2 − · · · x1 )(r )
(2.15)
for all x, r = 1, 2 . . . , n and t (r ) = t (t − 1) . . . (t − r + 1) is the descending factorial expression. The proof being similar to that of Theorem 2.2, we give only the main steps. Assume that (2.14) holds. Then for n = 1, a1 (x1 ) = k−pk−x , 1 and
S 1 ( x1 ) =
x −1 t =0
1−
k k−p−t
=
k + p − x1 p − x1
k+p p
so that the result is true for n = 1. We notice that p − x1 − x2 (p − x1 − x2 )(p − x1 − x2 − 1) (k)(2) = 1−2 + . (k − p − x1 − x2 )(2) k − p − x1 − x2 (k + p − x1 − x2 )(k + p − x1 − x2 − 1) By induction
(k)(b) p − x1 − x2 − · · · − xb = 1−b + ··· (k − p − x1 − x2 − · · · − xb )(b) k − p − x1 − x2 − · · · − xb (p − x1 − · · · − xb )(b) . + (−1)b−1 (k − p − x1 − · · · − xb )(b) Using this
x (k)(b) S (k − p − x1 − · · · − xb )(b) b t1 = x 1 tb = x b k + p − x1 − · · · − xb − 1 p − x1 − · · · − xb − 1 = . k+p
S (xb , 1) = Sb (xb ) −
p
···
p
p
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
45
Now calculate S (xb , 2), S (xb , 3), . . . successively
S (xb+1 ) =
k + p − x1 − · · · − xb+1 p − x1 − · · · − xb+1
k+p p
for all positive integers b. This proves the ‘if’ part. The ‘only if’ part is obtained by direct calculation. 3. If X1 , X2 . . . , Xn are independent lifetimes, then an (x) = h1 (x1 )h2 (x2 ) . . . hn (xn )
(2.16)
and conversely, where hi (xi ) is the hazard rate of Xi . Proof. The proof of the first part is obvious and that of the second part is obtained by the method of Theorem 2.1, by starting from S2 (x1 , 1) = S1 (x1 ) −
∞
h1 (t )h2 (0)S1 (t1 )
t =x 1
= S1 (x1 ) − h2 (0)
∞
f1 (t1 )
t =x 1
= S1 (x1 )[1 − h2 (0)] = S1 (x1 )P [X2 ≥ 1] and then arriving at S2 (x2 ) = P [X1 ≥ x1 ]P [X2 ≥ x2 ]. The case of n variables now follows from (2.8) and (2.9).
4. In modelling multivariate data, the dependent relation between the constituent variables play a crucial role. The knowledge of whether the association is positive or negative may limit the choice of models considered for the data. We prove a result in this connection. Theorem 2.4. When E (Xi Xj ) < ∞, i ̸= j, Cov(Xi , Xj ) = E
1 a2 (Xi , Xj )
−E
1 a1 ( X i )
E
1
a1 (Xj )
.
(2.17)
Proof.
E
1 a2 (Xi , Xj )
=
∞ ∞
S2 (xi , xj )
x i =0 x j =0
= S2 (0, 0) +
∞ x i =1
Si (xi ) +
∞
Sj (xj ) +
x j =1
∞ ∞
S2 (xi , xj )
i =1 j =1
= 1 + E (Xi ) + E (Xj ) + E (Xi Xj ). Since E
1 ai (Xi )
=
∞
xi =0
Si (xi ) = 1 + E (Xi ), the conclusion follows.
5. In the bivariate case, the vector hazard rate (1.2) is related to a2 (x1 , x2 ) by a2 (x1 , x2 ) = b1 (x1 , x2 ) − b1 (x1 , x2 + 1) + b1 (x1 , x2 + 1)b2 (x1 , x2 )
= b2 (x1 , x2 ) − b2 (x1 + 1, x2 ) + b2 (x1 + 1, x2 )b1 (x1 , x2 ).
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6. The hazard rate (2.1) can be related to the multivariate mean residual life function in the following way. In discrete set up, the multivariate mean residual life function of X is defined as the vector mn (x) = (m1 (x), m2 (x), . . . , mn (x)),
(2.18)
where, mi (x) = E (Xi − xi |X > x),
=
=
i = 1, 2 . . . , n; xi = −1, 0, 1 . . .
∞
1
Sn (x + en ) t =x +1 1 1 ∞
1
Sn (x + en ) t =x +1 i i
...
∞
(ti − xi )f (t )
tn = x n + 1
S (xi + 1, . . . , xi−1 + 1, ti , xi+1 + 1, . . . , xn + 1)
with en = (1, 1, . . . , 1). In the bivariate case, 1 − a2 (x1 , x2 ) =
m 1 ( x1 , x2 ) − 1 S2 (x1 + 1, x2 + 1)
+
+
m 2 ( x1 , x2 ) − 1 m2 (x1 , x2 + 1)
m1 (x1 , x2 ) − 1 m2 (x1 , x2 + 1) − 1 m1 (x1 + 1, x2 ) m2 (x1 + 1, x2 + 1)
.
(2.19)
This result can be extended to the n-variate case, but being lengthy, it is not given here. 7. By virtue of Theorem 2.1, all the functional forms of a2 (x1 , x2 ) in Table 1 along with the distribution of X1 (obtained by setting x2 = 0 in S2 (x1 , x2 )) characterize the respective bivariate models. 8. In reliability and survival analysis, the lifetime distribution of a device or organism after it has survived a specific age is of considerable interest. Instead of considering the usual covariance in such situations, the residual covariance, which is a time-dependent measure of association is more desirable. Therefore, we define the covariance of residual life as a measure in this context by C (x1 , x2 ) = M (x1 , x2 ) − m1 (x1 , x2 )m2 (x1 , x2 )
(2.20)
where M (x1 , x2 ) = E [(X1 − x1 )(X2 − x2 )|X1 > x1 , X2 > x2 ] is the product moment of residual life. We see that M (x1 , x2 ) =
=
1
∞
∞
(t1 − x1 )(t2 − x2 )f (t1 , t2 )
S (x1 + 1, x2 + 1) t =x +1 t =x +1 1 1 2 2 1
∞
∞
S (x1 + 1, x2 + 1) t =x +1 t =x +1 1 1 2 2
S (t1 , t2 ).
(2.21)
It is easy to see that M (x1 , x2 ) = E
1 a2 (X1 , X2 )
|X1 > x1 , X2 > x2 .
The constancy of the hazard rate is of general importance in the univariate and multivariate cases. From the definitions of a2 (x1 , x2 ) and M (x1 , x2 ), we have the following result. Theorem 2.5. The bivariate scalar hazard rate a2 (x1 , x2 ) = k, a constant, for all x1 , x2 in N if and only if M (x1 , x2 ) = 1k . Remark 2.1. The relationship a2 (x1 , x2 ).M (x1 , x2 ) = 1 is satisfied by several distributions as is evident from Example 2.1. The analogous results for continuous distributions were studied in Gupta and Sankaran [6], Navarro et al. [10] and Navarro and Sarabia [11].
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
47
9. The equilibrium distribution of a discrete random variable X with probability mass function f (x) and survival function S (x) is defined as f1,E (x) =
S1 (x + 1) E (X )
,
x = 0, 1, 2 . . . .
(2.22)
A direct generalization provides that for the random vector X, the equilibrium distribution is fn,E (x) =
Sn (x + en ) E (X1 X2 . . . Xn )
.
(2.23)
By summation, fn,E (x) Sn,E (x)
Sn (x + en )
=
∞ t1 = x 1 + 1
∞
...
. Sn (t )
tn = x n + 1
This leads to the following result. Theorem 2.6. The scalar hazard rate of the multivariate equilibrium distribution is an,E (x) =
1 Mn (x + en )
(2.24)
where Mn (x) = E [(X1 − x1 )(X2 − x2 ) . . . (Xn − xn )|X > x] . The converse of this result is also true. 3. Ageing properties In this section, we discuss various ageing properties based on the multivariate scalar hazard rate. 3.1. No-ageing property We say that the lifetime X has multivariate no-ageing property if and only if it satisfies P [Xr = xr |Xr ≥ xr ] = P [Xr = 0r |X ≥ 0r ]
(3.1)
for all xr in N and for r = 1, 2, . . . , n and 0r = (0, 0 . . . , 0). This means that the conditional probability of failure of a n-component system at age xr upon survival of age xr , r = 1, 2 . . . , n is the same as the failure of the system at the starting point of its operation. Equivalently one can also express it as the probability of failure is the same at all ages. In terms of MSHR, (3.1) is equivalent to the statement r
ar (xr ) = kr ,
a constant for all xr , 0 < kr < 1, for r = 1, 2, . . . , n.
(3.2)
Theorem 3.1. A lifetime X satisfies the multivariate no-ageing property (3.2) if and only if Sn (x) = (1 − k1 )x1 (1 − k2 )x2 . . . (1 − kn )xn ,
(3.3)
a multivariate geometric distribution with independent geometric marginals. The proof of the above theorem is easily obtained by applying Theorem 2.1. It may also be observed that in this case, ar (xr ) is the product of the marginal hazard rates of X1 , X2 , . . . , Xr . Further, the condition an (x) = k for some constant k, 0 < k < 1 neither guarantees the independence of the Xi′ s nor geometric marginals. Another result of interest in which the bivariate hazard rate is constant is given by Asha and Nair [1].
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N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
Theorem 3.2. A bivariate lifetime (X1 , X2 ) on N 2 has bivariate geometric distribution S (x1 , x2 ) =
x −x
px2 p11 2 : x1 ≥ x2 x −x px1 p22 1 : x1 ≤ x2
(3.4)
0 < p < pi < 1, i = 1, 2; 1 + p ≥ p1 + p2 if and only if the form of the bivariate hazard rate c1 : x1 > x2 c2 : x1 < x2 c3 : x1 = x2
a2 (x1 , x2 ) =
(3.5)
1 −1 c1 = (p1 − p)(1 − p1 )p− 1 , c2 = (p2 − p)(1 − p2 )p2 and c3 = 1 + p − p1 − p2 .
3.2. Positive and negative ageing Definition 3.1. We say that X has increasing (decreasing) multivariate scalar hazard rate, MIHR (MDHR) if and only if ar (xr ) is increasing (decreasing) in x1 , x2 , . . . , xr for r = 1, 2, . . . , n. In this definition, we mean increasing (decreasing) to be non-decreasing (non-increasing). Example 3.1. (a) The multivariate Waring distribution (2.10) is MIHR and the multivariate hypergeometric distribution in Theorem 2.3 is MDHR. (b) S (x1 , x2 ) =
(m)x1 (m+n)x1
a2 (x1 , x2 ) =
k + p − x1 p − x1
k+p p
, x1 = 0, 1, 2 . . . ; x2 = 0, 1, 2 . . . n for which nk
(m + n + x1 )(k + p − x2 )
is neither MIHR nor MDHR. Definition 3.2. A random vector X on N p is multivariate increasing (decreasing) scalar hazard rate average, MIHRA (MDHRA) if and only if Hr (xr ) =
1 x1 x2 . . . xr
x 1 −1
x r −1
...
t1 = 0
ar (tr )
(3.6)
tr = 0
is increasing (decreasing) in x1 , x2 , . . . , xr ; r = 1, 2 . . . , n. We call (x1 x2 . . . xr )Hr (xr ), the cumulative hazard rate of Xr . Theorem 3.3. X is MIHR (MDHR) ⇒ X is MIHRA (MDHRA). Proof. Since the proof of the n-variate case is apparent from that of the bivariate case, we give proof of the latter case only. Since (X1 , X2 ) is MIHR, a1 (x1 ) is increasing in x1 and a2 (x1 , x2 ) is increasing in x1 , x2 . x 1 −1 Using H1 (x1 ) = x1 t1 =0 a1 (t ), 1
H1 (x1 + 1) ≥ H1 (x1 ). Hence, H1 (x1 + 1) − H1 (x1 ) ⇐⇒
1
x1
x 1 + 1 t =0 1
a1 (t ) − H1 (x1 )
⇐⇒ a1 (x1 ) ≥ H1 (x1 ).
(3.7)
N.U. Nair, P.G. Sankaran / Statistical Methodology 27 (2015) 39–50
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Also, H1 (x1 ) =
x 1 −1 1
x 1 t =0
a1 (t ) ≤
x 1 −1 1
x t =0
a( t )
(3.8)
= a1 (x1 ),
(3.9)
since a1 (x1 ) ≥ a1 (t ) for all t = 0, 1, 2 . . . x − 1. From (3.7) and (3.8), H1 (x1 ) is increasing if a1 (x1 ) is increasing. In the case of two variables, assume that a2 (x1 , x2 ) is increasing in both x1 and x2 . H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) =
1
x1 x2
x1 x2 (x1 + 1)(x2 + 1)
x1 x2
x 1 −1 x 2 −1
− (x1 + 1)(x2 + 1)
a2 (t1 , t2 )
t1 =0 t2 =0
a2 (t1 , t2 ) .
t1 = 0 t2 = 0
Write x1 x2
x 1 −1 x 2 −1
a2 (t1 , t2 ) =
t1 = 0 t2 = 0
x 1 −1
a2 (t1 , t2 ) −
t1 = 0 t2 = 0
x2 −1
a2 (t1 , x1 ) −
t1 =0
a2 (x1 , t2 ) + a2 (x1 , x2 ).
t2 = 0
We have,
H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) = x1 x2
a2 (t1 , x2 ) −
+ x2 x1
x 2 −1
a2 (t1 , t2 )
t1 = 0 t2 = 0
t1 = 0
x1 −1 x2 −1
x 1 −1
x1 −1 x2 −1
a2 (x1 , t2 ) −
t2 = 0
+ x 1 x 2 a2 ( x 1 , x 2 ) −
a2 (t1 , t2 )
t1 = 0 t2 = 0 x 1 −1 x 2 −1
a2 (t1 , t2 ) .
(3.10)
t1 = 0 t2 = 0
Since a2 (x1 , x2 ) is increasing in x1 and x2 , a2 (x1 , t2 ) ≥ a2 (t1 , t2 ) for all t1 in [0, x1 − 1] and a2 (t1 , x2 ) ≥ a2 (t1 , t2 ) for all t2 in [0, x2 − 1]. Hence all the terms in the square braces in (3.10) remain non-negative. This means that H2 (x1 + 1, x2 + 1) − H2 (x1 , x2 ) ≥ 0 for all x1 and x2 and hence (X1 , X2 ) is MIHRA. Definition 3.3. We say that X is multivariate new better than used in hazard rate, MNBUHR (MNWUHR) if ar (xr ) ≥ ar (0r ) for all xr and r = 1, 2, . . . , n. From the definition, it follows that MIHR⇒ MNBUHR. Thus we see that the notions of ageing in the univariate case using hazard rate can be extended to the multivariate case employing the scalar hazard rate. 4. Conclusion In this paper, we have studied the properties of a discrete multivariate hazard rate. It is seen through the various results that it can carry out all the functions prescribed for a hazard rate. The properties obtained here compare favourably with those of the competing alternative definitions of multivariate hazard rates. The major distinction of the scalar form from the others is that while the former is indicative of the joint variation of the lifetimes, the latter does the same thing about the mean of the observations. We have provided only an illustrative account of the role that the scalar hazard can carry out in reliability modelling. The modelling aspects based on the above results like estimation of the hazard rate, selection of models for real data, etc. need to be studied further, hopefully in a future work.
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