Biased Estimator

Neil J.Salkind

doi:10.4135/9781412961288

Entry
Reader's guide
Entries A-Z
Subject index

Return to Entries

Biased Estimator

Edited by:
Neil J. Salkind
In:Encyclopedia of Research Design
Chapter DOI:https://doi.org/10.4135/9781412961288.n30
Subject:Research Design
Keywords:debt; estimators; parameters; statistical models

Request Permissions

Show page numbers Hide page numbers

In many scientific research fields, statistical models are used to describe a system or a population, to interpret a phenomenon, or to investigate the [Page 85]relationship among various measurements. These statistical models often contain one or multiple components, called parameters, that are unknown and thus need to be estimated from the data (sometimes also called the sample). An estimator, which is essentially a function of the observable data, is biased if its expectation does not equal the parameter to be estimated.

To formalize this concept, suppose θ is the parameter of interest in a statistical model. Let

be its estimator based on an observed sample. Then

is a biased estimator if E (

) ≠ θ where E denotes the expectation operator. Similarly, one may say that

is an unbiased estimator if E (

) = θ. Some examples follow.

Example 1

Suppose an investigator wants to know the average amount of credit card debt of undergraduate students from a certain university. Then the population would be all undergraduate students currently enrolled in this university, and the population mean of the amount of credit card debt of these undergraduate students, denoted by θ, is the parameter of interest. To estimate θ, a random sample is collected from the university, and the sample mean of the amount of credit card debt is calculated. Denote this sample mean by

1. Then E(

)1) = θ; that is,

1 is an unbiased estimator. If the largest amount of credit card debt from the sample, call it

2, is used to estimate θ, then obviously

2 is biased. In other words, E(

2) ≠ θ.

Example 2

In this example a more abstract scenario is examined. Consider a statistical model in which a random variable X follows a normal distribution with mean μ and variance σ2, and suppose a random sample X1, …, Xn is observed. Let the parameter θ be μ. It is seen in Example 1 that

the sample mean of X1,…, Xn, is an unbiased estimator for θ. But

2 is a biased estimator for μ2 (or θ2). This is because X follows a normal distribution with mean μ and variance

. Therefore,

Example 2 indicates that one should be careful about determining whether an estimator is biased. Specifically, although

is an unbiased estimator for θ, g(

) may be a biased estimator for g(θ) if g is a nonlinear function. In Example 2, g(θ) = θ2 is such a function. However, when g is a linear function, that is, g(θ) = aθ + b where a and b are two constants, then g(

) is always an unbiased estimator for g(θ).

Example 3

Let X1,…, Xn be an observed sample from some distribution (not necessarily normal) with mean μ and variance σ2. The sample variance S2, which is defined as