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2007.05.29

[統計]相関係数の不偏推定量

 大学が停電(定期的な)で、復旧作業で出動したら、電気製品の中に復旧しないものがあり、対応で翌日までかかってしまった、という新たなトラブル発生で、”トラブルは降る雨のごとく”という状態となる中で、ふいに母集団が二次元正規分布のときの相関係数(ここでは、いわゆる普通の相関係数である、Pearsonのrのこと)の不偏推定量について調べざるをえなくなった。よくある状況なので、統計の入門的なテキストでもたいていは扱われている状況だが、不偏推定量については書いてない。
 サンプルのデータで計算した積和を(n-1で割ったもの)は母集団の共分散の不偏推定量だが、相関係数(標本相関係数)は母相関係数の不偏推定量ではなく、arcsin(標本相関係数)がarcsin(母相関係数)の不偏推定量といったところまでは、確かKendall&Stuar本などで見た記憶があった。他の仕事の合間に電子ジャーナルでごそごそ調べているうち、ややこしい関数で表される不偏推定量が見つかり(Olkin&Pratt、Ann.Math.Statist.29:201-211,1958による、一様最小分散不偏推定量)、その近似式(r{1+(1-r^2)/(2n-2)})に到達して一応の目的を達することができた。
(追記、5/31)
 上の近似式も誤差は小さいのですが、Olkin&PrattとLehmann&Casellaの本(2版)に基づき補足しておきます。Olkin&Prattは超幾何関数で表現される不偏推定量を得て、その級数展開による近似も与えています。近似式はnが1次の形で出てくるものを使うなら、(r{1+(1-r^2)/(2n-6)})がよく、8以上のnなら1%以下の誤差とのことです。
 なお、nが20だと、母相関係数が0.4から0.6くらいなら、標本相関係数は2%くらいバイアスしていることになります。

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投稿: Catalina | 2020.07.14 22:41

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