Orthogonal Spacings

Abstract

It is well known that if X₁, X₂ are independent exponential random variables then the two corresponding spacings X_{1: 2} and X_{2: 2} − X_{1: 2} are independent random variables. Under mild continuity regularity conditions, such independence characterizes the possibly shifted exponential distribution. If instead, it is only assumed that the two spacings are orthogonal, a variety of non-exponential distributions can be expected to be encountered. Certain classes of distributions which do not have orthogonal spacings are described. Negatively correlated spacings appear to be most common. Several strategies are investigated for identifying distributions for which the two spacings are orthogonal. Extensions to samples of sizes greater than two are discussed.

Keywords:

• Characterization
• Covariance
• Uncorrelated
• Independent spacings
• Order statistics

Mathematics Subject Classification:

• Primary: 60E05
• Secondary: 62G30