Question

Suppose that Θ̂1, Θ̂2 and Θ̂3 are estimators of ?. We know that ?(Θ̂1) = ?(Θ̂2) = ?, ?(Θ̂3) ≠ ?, ?(Θ̂1) = 12, ?(Θ̂2) = 10 and ?(Θ̂3 − ?) 2 = 6 . Compare these three estimators. Which do you prefer? Why?

Answer 1

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 to estimate the population mean, μ. There are three desirable properties every good estimator should possess. These are:

1. Unbiasedness - An estimator is said to be unbiased if its expected value is identical with the population parameter being estimated. That is if θ is an unbiased estimate of θ, then we must have E (θ) = θ.
2. Efficiency - The concept of efficiency refers to the sampling variability of an estimator. If two competing estimators are both unbiased, the one with the smaller variance (for a given sample size) is said to be relatively more efficient.
3. Consistency - If an estimator, say θ, approaches the parameter θ closer and closer as the sample size n increases, θ is said to be a consistent estimator of θ. Stating somewhat more rigorously, the estimator θ is said is be a consistent estimator of θ if, as n approaches infinity, the probability approaches 1 that θ will differ from the parameter θ by no more than an arbitrary constant.