Introduction

In statistical estimation, the concept of consistency plays a pivotal role in determining the reliability and accuracy of an estimator. Consistency refers to the property that an estimator becomes increasingly accurate as the sample size grows, converging to the true parameter value as the sample size approaches infinity. This chapter delves into the intricacies of estimator consistency, explores its significance in statistical analysis, and provides insights into why this concept is indispensable for robust inference.

Estimator Consistency

Consistency is a desirable property for any estimator. It ensures that as the sample size increases, the estimate gets closer to the true value of the parameter being estimated. Formally, an estimator $\hat{\theta}_n$ is consistent for a parameter $\theta$ if for any small positive value $\epsilon$, the probability that $|\hat{\theta}_n - \theta| > \epsilon$ approaches zero as the sample size $n$ grows. Mathematically, this can be expressed as:

$$\lim _{n \rightarrow \infty} P\left(\mid \hat{\theta}_n-\theta \mid >\epsilon\right)=0$$

Example: Consistency of Sample Mean

Consider estimating the mean $\mu$ of a population. The sample mean $\bar{x}$ is a consistent estimator for $\mu$. As the sample size $n$ increases, the sample mean’s accuracy improves, ensuring it converges to the true population mean.

Example: Consistency of Maximum Likelihood Estimator

Maximum Likelihood Estimators (MLE) are often consistent. For instance, the MLE of the mean of a normal distribution is the sample mean. As the sample size grows, the MLE becomes a more accurate estimate of the population mean.

Usefulness of Consistency

Consistency is a fundamental property that guarantees that the estimator’s performance improves with larger sample sizes. This concept is invaluable for several reasons:

• Reliable Inference: A consistent estimator ensures that as you collect more data, your estimate gets closer to the true parameter value. This reliability is crucial for making accurate inferences about the population.

• Decision Making: In decision-making scenarios, a consistent estimator provides more accurate information about the underlying population, enabling better choices.

• Model Assessment: In model selection and comparison, consistency allows for meaningful assessment of model performance as the data size increases.