unbiasedness

This simple simulation-study shows the unbiasedness of the Sample Variance through a Monte-Carlo simulation in STATA.

This simulation is conducted using STATA since its programming capabilities facilitate the generation of multiple datasets and iterative analyses.

Introduction

Monte-Carlo methods are numerical techniques relying on random sampling to approximate results. This work performs a Monte-Carlo simulation in order to examine whether the sample variance is an unbiased estimator for the population variance.

Data generating

$X$ is a continuous random variable simulated from a uniform distribution such that $X∼U(a,b)$. It is known that, for $a=0$ and $b=1$

$$ \text{Var}(X)=\frac{(b-a)^2}{12}=0.8\overline{3} $$

gen x = runiform()

Method

Firstly, we generate a fixed number of independent repetitions of the rv X with a fixed amount of observations each. Consequently, we replace those repetitions with their sample variances.

clear
local N = 5000
local rep = 100
matrix stats = J(`rep', 1, .)
forv i = 1/`rep' {
	clear
	qui set obs `N'
	gen x = runiform()
	qui sum x
	matrix stats[`i', 1] = r(sd)
}	
clear
qui set obs `rep'
svmat stats, n(sd)
gen sample_var = sd1^2
drop sd1

Thus, we expect each sample variance to be approximately equal to $0.8\overline{3}$, as shown in the equation above. As a matter of fact, t-test result suggests a convergence of sample variance to our theoretical value, confirming that sample variance is an unbiased estimator of the population variance.

local var_unif = 1/12
ttest sample_var == `var_unif'

Output