ON CHARACTERIZATION OF THE GAMMA DISTRIBUTION.
STANFORD UNIV CALIF DEPT OF STATISTICS
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Let X sub 1, X sub 2,... be a sequence of i.i.d. random variables and S sub n summation, j 1 to j n, of X sub j. If X sub 1 has a gamma distribution, Z sub n is identically equal to Ssubscript n, superscript rES subscript n, superscript r n or 1 is a reverse martingale sequence for any positive r. These reverse martingales find applications in sequential analysis. In this paper the converse is proved for any integer r 1, and this provides a characterization of the gamma distribution in fact, it is sufficient that the reverse martingale sequence have finite length r. Another characterization is also proved, extending the case r 2 to non-identically distributed r.v.s. Author
- Statistics and Probability