Question

Show that if θ is an unbiased estimator of θ and var (θ) ≠ 0, then θ² is not an unbiased estimator of θ²

Answer 1

An unbiased estimator θ with non-zero variance cannot produce an unbiased estimator θ² with variance greater than zero.

Step-by-step explanation:

In statistics, an estimator is considered unbiased if its expected value is equal to the parameter it is estimating. In this case, θ is an unbiased estimator of θ. However, if var(θ) ≠ 0, it means that the estimator has non-zero variance, indicating that it is not a perfect estimator.

Now, if we square θ to get θ², the variance of θ² will be larger than zero since var(θ²) = E(θ⁴) - (E(θ²))². This implies that θ² cannot be an unbiased estimator of θ².