結果并不令人驚訝 . 對于具有大峰度的分布，樣本方差的預期方差大致為mu4 / N，其中mu4是分布的第4個時刻 . 對于對數正態，mu4指數地取決于參數sigma ^ 2，這意味著對于足夠大的sigma值，樣本方差將相對于真實方差在整個地方 . 這正是你所觀察到的 . 在你的例子中，mu4 / N~(coeffOfVar ^ 8)/ N~50 ^ 8 / 1e6~4e7 .

用于推導樣本變量的預期方差 . 見http://mathworld.wolfram.com/SampleVarianceDistribution.html . 下面是一些代碼，以更精確的方式說明這些想法 . 注意樣本方差的方差和理論預期值的大值，即使coeffOfVar = 5 .

exp.var.of.samp.var

(n-1)*((n-1)*mu4-(n-3)*mu2^2)/n^3

}

mu2.lnorm

(exp(sigma^2)-1)*exp(2*mu+sigma^2)

}

mu4.lnorm

mu2.lnorm(mu,sigma)^2*(exp(4*sigma^2)+2*exp(3*sigma^2)+3*exp(2*sigma^2)-3)

}

exp.var.lnorm.var

exp.var.of.samp.var(n,mu2.lnorm(mu,sigma),mu4.lnorm(mu,sigma))

}

exp.var.norm.var

exp.var.of.samp.var(n,sigma^2,3*sigma^4)

}

coeffOfVar

mean

sigma

mu

n

m

## Get variance of sample variance for lognormal distribution:

var.trial

cat("samp. variance (mean of",m,"trials):",mean(var.trial),"\n")

cat("theor. variance:",mu2.lnorm(mu,sigma),"\n")

cat("variance of the sample var:",var(var.trial),"\n")

cat("expected variance of the sample var:",exp.var.lnorm.var(n,mu,sigma),"\n")

> samp. variance (mean of 10000 trials): 105.7192

> theor. variance: 100

> variance of the sample var: 350997.7

> expected variance of the sample var: 494053.2

## Do this with normal distribution:

var.trial

cat("samp. variance (mean of",m,"trials):",mean(var.trial),"\n")

cat("theor. variance:",sigma^2,"\n")

cat("variance of the sample var:",var(var.trial),"\n")

cat("expected variance of the sample var:",exp.var.norm.var(n,mu,sigma),"\n")

> samp. variance (mean of 10000 trials): 3.257944

> theor. variance: 3.258097

> variance of the sample var: 0.002166131

> expected variance of the sample var: 0.002122826

總結

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