结果并不令人惊讶.对于具有大峰度的分布,样本方差的预期方差大致为mu4 / N,其中mu4是分布的第4个时刻.对于对数正态,mu4指数地取决于参数sigma ^ 2,这意味着对于足够大的sigma值,样本方差将相对于真实方差在整个地方.这正是你所观察到的.在你的例子中,mu4 / N~(coeffOfVar ^ 8)/ N~50 ^ 8 / 1e6~4e7.

用于推导样本变量的预期方差.请参阅http://mathworld.wolfram.com/SampleVarianceDistribution.html.以下是一些代码,以更精确的方式说明这些想法.注意样本方差的方差和理论预期值的大值,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