Computer Algebra application to the distribution of sample correlation coefficient

Shigekazu Nakagawa, Naoto Niki, Hiroki Hashiguchi

Date: July 18th (Thursday)
Time: 16:00-16:25
Abstract Let $r$ be the correlation coefficient based on a sample of size $n$ from a population $F$. The value of $r$ remains unchanged however observations may be exchanged or permuted. The statistic $r$ is one of symmetric statistics.

Our goal is divided into two parts:

obtaining the asymptotic cumulants of $r$,
as a consequence of 1., deriving the asymptotic expansions for probability integrals and percentiles of $r$, where $F$ has population cumulants of requisite order.

In accomplishing 1, the computational difficulty arises there. If we want the more precise approximation, the order of requisite approximate cumulants of $r$ increases. Raising the order may cost a large amount of algebraic computation often beyond human power. Here computer algebra induces an increasing interest.

The authors have taken a new look at symmetric statistics and developed the symbolic and algebraic algorithm for changing of bases of symmetric polynomials. Both the asymptotic cumulants and the approximate distributions are given.

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