In this study, generalized k-order Fibonacci hybrid quaternion is defined. We give recurrence relation, generating function, the summation formula and some properties for these quaternions. Furthermore, matrix representation quaternions determined. The Q_k defined numbers given By means of another matrices, several identities are also obtained.

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex of two distributions from case consecutive to general spacings. The sufficient conditions given in terms expected proportional shortfall order. As an application, compare deprivation within some parametric families income distributions.