IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties

X1,...,Xn is a random sample from a normal distribution with variance σ². Estimator S² is the sample variance with divisor n-1, with variance 2σ⁴/(n-1). Estimator V uses divisor n. For n = 5, what is the bias of V as a multiple of σ²?

The bias of V is -0.20σ². Since V equals (n-1)/n times the unbiased S², its expectation is 0.8σ² when n is 5, which falls short of σ² by 0.2σ².

Explanation

V = (n-1)S²/n, so E[V] = (n-1)σ²/n = 0.8σ² for n=5. Bias = 0.8σ² - σ² = -0.2σ². The value -0.25 wrongly uses 1/(n-1).

One question tells you little. A timed set on Estimators and their properties shows your real accuracy, how long you take and where you lose marks.