엔

$N$

$N$특급[ μ + 1 / 2 σ2]

$\exp [\mu +1/2{\sigma }^2]$

$\exp[\mu+1/2\sigma^2]$

엑스¯→피이자형( X나는)

$\overline{X}{\to }_{p}E(X_i)$

$\bar X\to_pE(X_i)$

$$\exp[\hat\mu+1/2\hat\sigma^2]\to_p\exp[\mu+1/2\sigma^2],$$ μ^→피μ

$\hat{\mu }{\to }_p\mu$

$\hat\mu\to_p\mu$σ^2→피σ2

${\hat{\sigma }}^2{\to }_p{\sigma }^2$

$\hat\sigma^2\to_p\sigma^2$

그러나 MLE은 편견이 아닙니다.

엔

$N$

$N$μ^

$\hat{\mu }$

$\hat\mu$σ^2

${\hat{\sigma }}^2$

$\hat\sigma^2$엔= 100

$N=100$

$N=100$엔− 1

$N-1$

$N-1$μ

$\mu$

$\mu$σ2

${\sigma }^2$

$\sigma^2$

이자형( μ^+ 1 / 2 σ^2) ≈ μ + 1 / 2 σ2

$E(\hat{\mu }+1/2{\hat{\sigma }}^2)\approx \mu +1/2{\sigma }^2$

$E(\hat\mu+1/2\hat\sigma^2)\approx\mu+1/2\sigma^2$

$$E[\exp(\hat\mu+1/2\hat\sigma^2)]>\exp[E(\hat\mu+1/2\hat\sigma^2)]\approx \exp[\mu+1/2\sigma^2]$$

엔= 100

$N=100$

$N=100$

엔= 1000

$N=1000$

$N=1000$

로 만든 :

N <- 1000
reps <- 10000

mu <- 3
sigma <- 1.5
mm <- mle <- rep(NA,reps)

for (i in 1:reps){
  X <- rlnorm(N, meanlog = mu, sdlog = sigma)
  mm[i] <- mean(X)

  normmean <- mean(log(X))
  normvar <- (N-1)/N*var(log(X))
  mle[i] <- exp(normmean+normvar/2)
}
plot(density(mm),col="green",lwd=2)
truemean <- exp(mu+1/2*sigma^2)
abline(v=truemean,lty=2)
lines(density(mle),col="red",lwd=2,lty=2)

> truemean
[1] 61.86781

> mean(mm)
[1] 61.97504

> mean(mle)
[1] 61.98256

특급( μ + σ2/ 2)

$\exp (\mu +{\sigma }^2/2)$

$\exp(\mu+\sigma^2/2)$

$$V_t = (\sigma^2 + \sigma^4/2)\cdot \exp\left\{2(\mu + \frac 12\sigma^2)\right\},$$

$$\exp\left\{2(\mu + \frac 12\sigma^2)\right\}(\exp\{\sigma^2\}-1)$$

$$\exp\{\sigma^2\}>1+\sigma^2 + \sigma^4/2,$$ 특급( x ) = ∑∞나는 = 0엑스나는/ 나!

$\exp (x)=\sum _{i=0}^{\mathrm{\infty }}{x}^i/i!$

$\exp(x)=\sum_{i=0}^\infty x^i/i!$σ2> 0

${\sigma }^2>0$

$\sigma^2>0$

엔

$N$

$N$N <- c(50,100,200,500,1000,2000,3000,5000)

엔

$N$

$N$엔

$N$

$N$엔= 50

$N=50$

$N=50$

> tail(sort(mm))
[1] 336.7619 356.6176 369.3869 385.8879 413.1249 784.6867
> tail(sort(mle))
[1] 187.7215 205.1379 216.0167 222.8078 229.6142 259.8727

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