Characterizations of the exponential distribution by order statistics

Let X 1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX 1,n is identically distributed as X 1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

The adult respiratory distress syndrome. Clinical features, factors influencing prognosis and principles of management

Low-dose IL-2 is efficacious in steroid-refractory cGVHD, with objective responses in >50% of patients, and durable disease control. IL-2 initiation earlier after cGVHD onset, prior to severe impairment of Treg:Tcon ratios, improves likelihood of clinical response.

Determining linkage and mode of inheritance: mode scores and other methods

The lod score method remains a popular approach for detecting linkage and estimating the recombination fraction theta between a marker locus and a trait locus. However, its implementation requires knowledge of all parameters of the genetic mechanism, including the number of loci involved and the genotype specific penetrance, which could depend on factors such as age. When some of the penetrance parameters phi are unknown, several methods are available, and have been reviewed by Hodge and Elston [(1994) Genet Epidemiol 11:329-342]. These include the "wrod score" (lod score maximized over theta under a wrong value of phi) and "mod score" (lod score maximized over both theta and phi) methods for inference on theta. It has further been proposed that the mod score also be used for estimating phi. In this paper, we review and assess the adequacy of these two methods for inferences on both phi and theta. In particular, all of the methods can be seen as variations on likelihood inference, using the information in the conditional likelihood for the marker data, given the trait data. The loss of efficiency of the mod is compared to that of the full likelihood, which utilizes all information available in the trait data. We also propose an alternative, based on the pseudo-likelihood, where phi is estimated via the trait information and plugged into the conditional likelihood. This method is compared to the mod score method, and the advantages and disadvantages of each are elucidated. In particular, it is seen that the pseudo-likelihood method can be more efficient than the mod score method if the ascertainment scheme can be modeled. As examples, both a random sample and a multiplex ascertainment scheme are considered. In addition, the pseudo-likelihood method leads to likelihood ratio tests for detecting linkage with a simple, known asymptotic reference distribution, a feature not shared by the mod score. Finally, we discuss the advantages of using the pseudo-likelihood method over the full likelihood method, both of which are valid when the ascertainment scheme is known.