Properties of an Estimator

A point estimator (PE) is a sample statistic used to estimate an unknown population parameter. It is a random variable and therefore varies from sample to sample. A good example of an estimator is the sample mean, \(x\), which helps statisticians estimate the population mean, μ. There are three desirable properties every good estimator should possess. These are:

1. Unbiasedness.
2. Efficiency.
3. Consistency.

Let us now look at each property in detail.

Unbiasedness

We say that a PE β’_j is an unbiased estimator of the true population parameter β_j if the expected value of β’_j is equal to the true β_j. Putting this in standard mathematical notation, an estimator is unbiased if:

E(β’_j) = β_j­   as long as the sample size n is finite.

Bias is the difference between the expected value of the estimator and the true value of the parameter. Therefore, this difference is and should be zero if an estimator is unbiased. Given the foregoing  statement, a non-zero difference indicates bias. A biased estimator can be less or more than the true parameter, giving rise to positive and negative biases.

Efficiency

Suppose we have two unbiased estimators – β’_j1 and β’_j2 – of the population parameter β_j:

E(β’_j1) = β_j­  and E(β’_j2) = β_j­

We say that β’_j1 is more efficient relative to β’_j2  if the variance of the sample distribution of β’_j1 is less than that of β’_j2  for all finite sample sizes.

In short, if we have two unbiased estimators, we prefer the estimator with a smaller variance because this means it is more precise in statistical terms. It is also important to note that the property of efficiency only applies in the presence of unbiasedness. This is because we only consider the variances of unbiased estimators.

Consistency of an Estimator

Let β’_j(N) denote an estimator of β_j­ where n represents the sample size. We would consider β’_j(N) a consistent point estimator of β_j­ if its sampling distribution converges to the true value of the population parameter β_j­ as n tends to infinity.

This intuitively means that if a PE is consistent, its distribution becomes more concentrated around the real value of the population parameter involved. As such, we could say that as n increases, the probability that the estimator ‘closes in’ on the actual value of the parameter approaches 1.