Advertisement

A note on equal distributions

  • Gwo Dong Lin
Article

Summary

It is known that the set
$$\left\{ {E\left( {X_{k_n ,n} } \right)\left| {n = 1,2, \cdots } \right.} \right\}, where 1 \leqq k_n \leqq n,$$
of expectations of order statistics of samples from a distributionF which has a finite expectation determinesF. In this note, we show that each of the sets
$$\begin{gathered} \left\{ {E\left( {X_{k_j ,n_j } } \right)\left| {j = 1,2, \cdots } \right.} \right\}, \hfill \\ where \left\{ {\left( {{{k_j } \mathord{\left/ {\vphantom {{k_j } {n_j }}} \right. \kern-\nulldelimiterspace} {n_j }}} \right)\left| {j = 1,2, \ldots } \right.} \right\} is dense in \left[ {0,1} \right], \hfill \\ \end{gathered} $$
$$\begin{gathered} \left\{ {E\left( {X_{1,1} } \right)} \right\} \cup \left\{ {E\left( {X_{k_j ,2j + 1} } \right)\left| {j = 1,2, \cdots } \right.} \right\} \cup \hfill \\ \left\{ {E\left( {X_{k'j,2j + 1} } \right)\left| {j = 1,2, \cdots } \right.} \right\},where 1 \leqq k_j< k'_j \leqq 2j + 1, \hfill \\ \end{gathered} $$
also determinesF.

AMS 1980 subject classifications

Primary 62E10 Secondary 30B60 

Key words and phrases

Distribution determine dense completeness and order statistics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Arnold, B. C. and Meeden, G. (1975). Characterization of distribution by sets of moments of order statistics,Ann. Statist.,3, 754–758.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Hoeffding, W. (1953). On the distribution of the expected values of the order statistics,Ann. Math. Statist.,24, 93–100.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Huang, J. S. and Hwang, J. S. (1975).L 1-completeness of a class of Beta distributions,Statistical Distribution in Scientific Work,3, 137–141.Google Scholar
  4. [4]
    Hwang, J. S. (1978). A note on Bernstein and Müntz-Szász Theorems with applications to the order statistics,Ann. Inst. Statist. Math.,30, A, 167–176.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Pollak, M. (1973). On equal distributions,Ann. Statist.,1, 180–182.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1984

Authors and Affiliations

  • Gwo Dong Lin
    • 1
  1. 1.Institute of StatisticsAcademia SinicaTaiwan

Personalised recommendations