Unbiasedness of the Estimator (2013)

The unbiasedness of the estimator is probably the most important property that a good estimator should possess. In statistics, the bias (or bias function) of an estimator is the difference between this estimator’s expected value and the true value of the parameter being estimated. An estimator is said to be unbiased if its expected value equals the corresponding population parameter; otherwise, it is said to be biased. Let us discuss in detail the unbiasedness of the estimator.

Introduction to Unbasedness of the Estimator

In the world of statistics and data analysis, estimators play a crucial role in drawing conclusions from data. One of the most important properties of an estimator is unbiasedness. Understanding this concept helps statisticians and data scientists ensure that their estimates are as accurate and representative as possible. Let us explore the definition of unbiasedness, why it matters, and how it applies to real-world data analysis.

Unbiasedness of the Estimator

Suppose in the realization of a random variable $X$ taking values in probability space i.e. ($\chi, \mathfrak{F}, P_\theta$), such that $\theta \varepsilon \Theta$, a function $f:\Theta \rightarrow \Omega $ has to be estimated, mapping the parameter set $\Theta$ into a certain set $\Omega$, and that as an estimator of $f(\theta)$ a statistic $T=T(X)$ is chosen. if $T$ is such that
\[E_\theta[T]=\int_\chi T(x) dP_\theta(x)=f(\theta)\]
holds for $\theta\varepsilon \Theta$ then $T$ is called an unbiased estimator of $f(\theta)$. The unbiased estimator is frequently called free of systematic errors.

Unbiased Estimator

Suppose $\hat{\theta}$ be an estimator of a parameter $\theta$, then $\hat{\theta}$ is said to be unbiased estimator if $E(\hat{\theta})=0$.

• If $E(\hat{\theta})=\theta$ then $\hat{\theta}$ is an unbiased estimator of a parameter $\theta$.
• If $E(\hat{\theta})<\theta$ then $\hat{\theta}$ is a negatively biased estimator of a parameter $\theta$.
• If $E(\hat{\theta})>\theta$ then $\hat{\theta}$ is a positively biased estimator of a parameter $\theta$.

Bias of an estimator $\theta$ can be found by $$[E(\hat{\theta})-\theta]$$

• $\overline{X}$ is an unbiased estimator of the mean of a population (whose mean exists).
• $\overline{X}$ is an unbiased estimator of $\mu$ in a Normal distribution i.e. $N(\mu, \sigma^2)$.
• $\overline{X}$ is an unbiased estimator of the parameter $p$ of the Bernoulli distribution.
• $\overline{X}$ is an unbiased estimator of the parameter $\lambda$ of the Poisson distribution.

In each of these cases, the parameter $\mu, p$ or $\lambda$ is the mean of the respective population being sampled.

However, sample variance $S^2$ is not an unbiased estimator of population variance $\sigma^2$, but consistent.

It is possible to have more than one unbiased estimator for an unknown parameter. The sample mean and the sample median are unbiased estimators of the population mean $\mu$ if the population distribution is symmetrical.

Why is Unbiasedness Important?

• Accuracy of Estimates: Unbiased estimators provide estimates that are correct on average, which reduces the chances of consistently overestimating or underestimating the true parameter.
• Reliability in Statistical Inference: When conducting hypothesis tests or constructing confidence intervals, unbiased estimators ensure that statistical conclusions are valid and trustworthy.
• Foundation for Further Statistical Properties: Many other desirable properties, such as consistency and efficiency, build upon the unbiasedness of an estimator.

Limitations of Unbiased Estimators

While unbiasedness is a desirable property, it is not the only criterion for a good estimator. Some unbiased estimators may have a high variance, making them unreliable for small samples. In such cases, biased but low-variance estimators (e.g., regularized estimators) might be preferred.

Summary

Unbiased estimators are a fundamental concept in statistics, ensuring that estimates are accurate on average. However, they must be used carefully, considering other factors like variance and sample size. Understanding unbiasedness helps in making informed statistical decisions and improving data-driven analysis.

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