Asymptotic Theory of Dual Generalized Order Statistics from Heterogeneous Population

Abstract

The outcomes of several real-life experiments arise in descending order. Dual generalized order statistics (DGOS) have been introduced as a unification of several models of descendingly ordered random variables like reversed ordered order statistics, lower k-records and lower Pfeifer records. The asymptotic theory (AT) proceeds by assuming that it is possible (in principle) to keep collecting additional data, so that the sample size grows infinitely. Under this assumption, many results can be obtained that are unavailable for samples of finite size. The AT is widely used in various statistical approaches, such as ordered random variables, time series models, estimation, testing hypotheses and so on. While the AT of DGOS from homogeneous population, i.e., all of the data points come from the same distribution, has been soundly investigated, no research has been devoted to this problem for heterogeneous population, i.e., the data points come from more than one distribution. This paper gives a closer look at the AT of DGOS based on data from a finite mixture of distributions normalized by the same continuous strictly monotonic sequence or a mixture of continuous strictly monotonic sequences.