Unbiasedness vs consistency of estimators - an example

Understanding the Difference Between Unbiased and Consistent Estimators

Introduction to Estimation Concepts

• The video discusses the difference between an estimator being unbiased versus consistent, using a specific example related to population parameters.

• The focus is on estimating a population mean (denoted as μ), which could represent various metrics like average height or GDP.

Defining the Sample and Estimator

• A sample from the population is denoted as tildeX , which will be used to estimate the population parameter μ.

• For this example, tildeX is defined mathematically as 1/n - 1 sum_i=1^n X_i , where X_i represents individual data points in the sample.

Analyzing Bias in Estimation

• To determine if tildeX is biased, we consider that each individual's height can be expressed as the mean height plus an unpredictable error with a mean of zero.

• By applying the expectations operator, it’s shown that E[tildeX] = nmu/n - 1 , indicating that this estimator does not equal μ for finite samples, thus confirming bias.

Understanding Consistency of Estimators

• The discussion shifts to whether this estimator is consistent. As sample size n approaches infinity, we analyze how bias behaves.