Especially if M = 1, the random variables are called negatively upper orthant dependent (NUOD), negatively lower orthant dependent (NLOD), and negatively orthant dependent (NOD), respectively (see, for example, [10, 13, 14]).
negatively upper orthant dependent (NUOD) if P(X_{i}>x_{i},i=1,2,ldots,n leprod _{i=1}^{n}P(X_{i}>x_{i}) (forall x_{1},ldots,x_{n}in_{n}in R), negatively lower orthant dependent (NLOD) if P(X_{i}le x_{i},i=1,2,ldots,n leprod _{i=1}^{n}P(X_{i}le x_{i}) (forall x_{1},ldots,x_{n}in_{n}in R), negatively orthant dependent (NOD) if they are both NUOD and NLOD.
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Say that the random variables {X i : i ≥ 1} are widely upper orthant dependent (WUOD), if there exists a finite real number sequence {g U (n): n ≥ 1}, such that for each n ≥ 1 and for all x i ∈, 1 ≤ i ≤ n, P ⋂ i = 1 n { X i > x i } ≤ g U ( n ) ∏ i = 1 n P ( X i > x i ).
Definition 2.1 We say that the r.v.s { η n, n ≥ 1 } are widely upper orthant dependent (WUOD) if there exists a finite real sequence { g U ( n ), n ≥ 1 } satisfying for each n ≥ 1 and for all x i ∈ , 1 ≤ i ≤ n, P r ( ⋂ i = 1 n { η i > x i } ) ≤ g U ( n ) ∏ i = 1 n P r ( η i > x i ) ; (2.5).
If there exists a finite sequence of real numbers ({g_{u}(n),ngeq1}) such that, for each (ngeq1) and for all (z_{i}in -infty, infty)), (1leq ileq n), P Biggl(bigcap_{i=1}^{n}(z_{i}in -inftyinftyleq g_{u}(n)prod_{i=1leqn}P(Z_{ileq_{i}), then we say that the random variables ({Z_{n},ngeq1}) are widely uPper orthant dependent (WUOD).
A finite collection of random variables X1, X2,..., X n is said to be negatively upper orthant dependent (NUOD) if for all real numbers x1, x2,..., x n, P ( X i > x i, i = 1, 2, …, n ) ≤ ∏ i = 1 n P ( X i > x i ), (1.1).
A finite collection of random variables X1, X2,..., X n is said to be negatively upper orthant dependent (NUOD) if for all real numbers x1, x2,..., x n, P ( X i > x i, i = 1, 2, …, n ) ≤ ∏ i = 1 n P ( X i > x i ), (1.2). and negatively lower orthant dependent (NLOD) if for all real numbers x1, x2,..., x n, P ( X i ≤ x i, i = 1, 2, …, n ) ≤ ∏ i = 1 n P ( X i ≤ x i ).
Definition 1.3 A finite collection of random variables X1, X2,...,X n is said to be negatively upper orthant dependent (NUOD) if for all real numbers x1,......,x n, P X i > x i, i = 1, 2,..., n ≤ ∏ i = 1 n P X i > x i, and negatively lower orthant dependent (NLOD) if for all real numbers x1,......,x n, P X i ≤ x i, i = 1, 2,..., n ≤ ∏ i = 1 n P X i ≤ x i.
The exact biomechanical mechanism and its contribution to the increased risk of shoulder pain development in this upper extremity dependent population are largely unknown.
