This problem based on the calculation of Mean Square Error gives a detailed solution to ISI M.Stat 2019 PSB Problem 5, with a tinge of simulation and code.
Suppose are independent random variables such that
where are all distinct and unknown. Consider
and another random variable
which is distributed as Binomial
where
Between
and
which is a better estimator of
in terms of their respective mean squared errors?
.
~ Binomial
.
If is unbiased for
, then MSE(
.
Observe that .
This results in the fact that .
Therefore, is a better estimate thatn
w.r.t Mean Square Error.
Let's verify this as usual by simulation.
library(statip)
library(Metrics)
N = 10
p = runif(10, 0, 1)
X = rep(0,N)
vX = NULL
vY = NULL
for (j in 1:1000)
{
for (i in 1:N)
{
X[i] = rbern(1,p[i])
}
Z = sum(X) #sum of Xi random variables
Y = rbinom(1,N,mean(p)) #Y random variable
vX = c(vX, Z)
vY = c(vY, Y)
}
k = rep(sum(p), 1000)
mse(k, vX) #MSE of Sum Xi #1.57966
mse(k, vY) #MSE of Y #2.272519
Hence, the theory is verified by this simulation. I hope it helps.
This problem based on the calculation of Mean Square Error gives a detailed solution to ISI M.Stat 2019 PSB Problem 5, with a tinge of simulation and code.
Suppose are independent random variables such that
where are all distinct and unknown. Consider
and another random variable
which is distributed as Binomial
where
Between
and
which is a better estimator of
in terms of their respective mean squared errors?
.
~ Binomial
.
If is unbiased for
, then MSE(
.
Observe that .
This results in the fact that .
Therefore, is a better estimate thatn
w.r.t Mean Square Error.
Let's verify this as usual by simulation.
library(statip)
library(Metrics)
N = 10
p = runif(10, 0, 1)
X = rep(0,N)
vX = NULL
vY = NULL
for (j in 1:1000)
{
for (i in 1:N)
{
X[i] = rbern(1,p[i])
}
Z = sum(X) #sum of Xi random variables
Y = rbinom(1,N,mean(p)) #Y random variable
vX = c(vX, Z)
vY = c(vY, Y)
}
k = rep(sum(p), 1000)
mse(k, vX) #MSE of Sum Xi #1.57966
mse(k, vY) #MSE of Y #2.272519
Hence, the theory is verified by this simulation. I hope it helps.