Variation diminishing property of densities of uniform generalized order statistics

Let f(*,r), r >= 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0, 1) of any linear combination Sigma(r)(j)(=1) a(j)f(*,j) does not exceed the number of sign changes in the sequence (a(1), center dot center dot center dot, a(r)). This result is applied to study monotonicity and convexity properties of f(*,r).