Comparing two estimators Suppose I have two estimators, one is unbiased and another one is biased. But the biased one has smaller MSE(Mean Squared Error) than the unbiased one.
Can we figure out the better one in this case? If yes, then which one is the better estimator and why?
 A: Suppose you have a random sample with $n = 5$ observations from a
normal distribution with unknown $\mu$ and $\sigma^2.$
In estimating $\sigma^2,$ the usual sample variance
$V_1 = \frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2$ is unbiased for $\sigma^2:$
$E(V_1) = \sigma^2.$
By contrast, the maximum likelihood estimator of $\sigma^2,$ which is
$V_0 = \frac{1}{n}\sum_{i=1}^n(X_i-\bar X)^2,$ is biased, but has smaller MSE.
[This is true for any $n,$ but I choose $n=5$ so that the bias
of $V_0$ (negligible for large and moderate $n)$ will be unmistakable in my simulation.]
set.seed(2020)
m = 10^6;  n = 5;  mu = 100;  sg = 10
v1 = replicate(m, var(rnorm(n,mu,sg)))
v0 = (n-1)*v1/n 
mean(v0);  mean(v1)
[1] 79.95946  # aprx E(V0) < 100
[1] 99.94932  # aprx E(V1) = 100
mean((v0-sg^2)^2)
[1] 3606.298  # aprx MSE(V0) < MSE(V1) 
mean((v1-sg^2)^2)
[1] 5007.307  # aprx MSE(V1) = 5000

For $\sigma^2 = 100,$ we have $E(V_0) = 80, E(V_1) = 100.$
Also, $MSE(V_0) = 3200 + 400 = 3600 < MSE(V_1) = Var(V_1) = 5000.$
Histograms of v1 and v0:

par(mfrow = c(2,1))
 hdr1="Unbiased Sample Variance"
 hist(v1, br=30, prob=T, xlim=c(0,800), col="skyblue2", main=hdr1)
  abline(v=100, col="red", lty="dotted")
 hdr2="MLE of Population Variance" 
 hist(v0, br=30, prob=T, xlim=c(0,800), col="skyblue2", main=hdr2)
  abline(v=100, col="red", lty="dotted")
par(mfrow = c(1,1))

Note: A few authors have advocated use of the MLE, bias notwithstanding. However, traditional methods of inference
for variances using the chi-squared distribution would
have to be altered to use the MLE, and many statisticians
believe underestimating $\sigma^2$ is a strong argument
against the MLE. (Another complication is that dividing by
$n+1$ results in an even greater decrease in MSE.)
