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Statistics · 300 Level
This is the complete lecture note for STA 321 Statistical Inference III, prepared for Statistics students at 300 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Criteria of Good Estimators Series 1 outline • Part 1.1: Introduction to Estimators • Definition of an estimator • Role of estimators in statistical inference • Part 1.2: Consistency of Estimators • Definition of consistency • Conditions for consistency • Examples of consistent and inconsistent estimators • Part 1.3: Unbiasedness and Efficiency • Definition of unbiasedness • Examples of unbiased and biased estimators • Efficiency and comparison of estimators • Part 1.4: Minimum Variance and Sufficiency • Minimum variance unbiased estimators (MVUE) • Concept of sufficiency • Factorisation theorem and examples • Part 1.5: Summary and Applications • Recap of properties of good estimators • Practical applications in statistical inference Introduction In statistical inference, the quality of an estimator determines the reliability of conclusions drawn from data. When researchers or analysts attempt to estimate unknown parameters, they rely on certain desirable properties to judge whether their chosen estimator is suitable. These properties ensure that the estimator not only provides accurate results but also performs well across repeated samples. Understanding these criteria is fundamental, as they form the backbone of sound
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS statistical practice and connect directly to later topics such as estimation methods, confidence intervals, and hypothesis testing. The aim of this Series is to equip you with the ability to identify and explain the essential properties that make an estimator “good”, and to evaluate estimators against these criteria in practical situations. We shall commence this Series by introducing the concept of estimators and their role in statistical inference. Next, we will examine the property of consistency, followed by a detailed discussion of unbiasedness and efficiency. Subsequently, we will explore minimum variance and sufficiency, including the factorisation theorem. Finally, we will wrap up by summarising the properties of good estimators and considering their applications in practice. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define an estimator and explain its role in statistical inference, • SLO 1.2 describe the property of consistency and evaluate whether an estimator is consistent, • SLO 1.3 define unbiasedness and efficiency, and compare estimators using these properties, • SLO 1.4 explain the concepts of minimum variance and sufficiency, and apply the factorisation theorem to identify sufficient statistics, and • SLO 1.5 summarise the properties of good estimators and illustrate their applications in statistical inference 1.1 Introduction to Estimators An estimator is a rule, formula, or statistic used to approximate an unknown population parameter based on sample data. For example, the sample mean is often used as an estimator of the population mean. Estimators are central to statistical inference because they allow us to make conclusions about populations using limited information from samples. Fill in the blank: An estimator is a __________________ used to approximate an unknown population parameter.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.1.1 Definition of an estimator An estimator is formally defined as a function of observed sample data that provides an estimate of a population parameter. If is the parameter of interest, then denotes the estimator. Estimators can be expressed as mathematical formulas, such as the sample mean Xˉ=1ni=1nXi. Illustration of how sample data is used to construct an estimator of a population parameter. 1.1.2 Role of estimators in statistical inference Estimators provide the foundation for statistical inference. They are used to make decisions, test hypotheses, and construct confidence intervals. A good estimator should capture the true parameter as closely as possible, while also being reliable across repeated samples. Fill in the blank: Estimators are essential in statistical inference because they allow us to make __________________ about populations using sample data. 1.1.3 Examples of common estimators
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • Sample mean (Xˉ): Estimates the population mean . • Sample variance (S2): Estimates the population variance 2. • Sample proportion (p): Estimates the population proportion p. These examples highlight how estimators are practical tools for summarising data and making inferences. Examples of common estimators and their corresponding parameters. Pilot Questions 1.1 1. An estimator is defined as: A. A function of sample data used to approximate a population parameter B. A population parameter itself C. A random experiment D. None of the above 2. The symbol is commonly used to denote: A. An estimator of a parameter B. The true parameter C. A random variable D. None of the above 3. The sample mean is an estimator of: A. The population mean B. The population variance C. The population proportion D. None of the above 4. The sample variance is an estimator of: A. The population variance B. The population mean C. The population proportion D. None of the above
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 5. Estimators are essential in statistical inference because they allow us to make: A. Conclusions about populations using sample data B. Predictions without data C. Decisions without parameters D. None of the above 1.2 Consistency of Estimators An important property of a good estimator is consistency. An estimator is said to be consistent if, as the sample size increases, it converges in probability to the true value of the parameter being estimated. In other words, with larger samples, the estimator becomes increasingly reliable and accurate. Fill in the blank: An estimator is consistent if it converges in __________________ to the true parameter as the sample size increases. 1.2.1 Definition of consistency Formally, an estimator nof a parameter is consistent if: limn→P(∣n-θ∣<ϵ)=1for all ϵ>0 This definition means that the probability of the estimator being close to the true parameter approaches one as the sample size grows. Illustration of consistency showing how the distribution of an estimator becomes concentrated around the true parameter as sample size increases.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.2.2 Conditions for consistency Consistency often requires two conditions: • The estimator must be unbiased or asymptotically unbiased. • The variance of the estimator must shrink to zero as the sample size increases. For example, the sample mean is a consistent estimator of the population mean because its variance decreases as the sample size grows. Fill in the blank: For an estimator to be consistent, its variance must __________________ as the sample size increases. 1.2.3 Examples of consistent and inconsistent estimators • Consistent estimator: The sample mean Xˉis consistent for the population mean . • Inconsistent estimator: The sample maximum is not a consistent estimator of the population mean, since it does not converge to as sample size increases. Examples of consistent and inconsistent estimators. Estimator Type What It Estimates Consistency Explanation Sample Mean (x̄) Estimator Population Mean (μ) ✅ Consistent As sample size (n) increases, x̄ gets closer to μ. Its variance shrinks (σ²/n), so estimates concentrate around the true mean. Sample Maximum (max) Estimator Population Mean (μ) (incorrect target) ❌ Inconsistent The maximum value does not stabilize around μ. As n grows, it tends to increase (especially in unbounded distributions), moving away from μ instead of converging to it.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot Questions 1.2 1. An estimator is consistent if: A. It converges in probability to the true parameter as sample size increases B. It always equals the true parameter C. It has zero variance in small samples D. None of the above 2. The formal definition of consistency requires: A. Probability of estimator being close to the true parameter approaches one as sample size grows B. Estimator always equals the parameter in finite samples C. Estimator has maximum variance D. None of the above 3. For an estimator to be consistent, its variance must: A. Shrink to zero as sample size increases B. Remain constant C. Increase with sample size D. None of the above 4. The sample mean is: A. A consistent estimator of the population mean B. An inconsistent estimator of the population mean C. Always biased D. None of the above 5. The sample maximum is: A. An inconsistent estimator of the population mean B. A consistent estimator of the population mean C. Always unbiased D. None of the above 1.3 Unbiasedness and Efficiency Two further properties that help us judge the quality of an estimator are unbiasedness and efficiency. These properties ensure that an estimator not only targets the correct parameter but also does so with minimal error compared to other possible estimators. Fill in the blank: An estimator is unbiased if its expected value is equal to the __________________ it is estimating. 1.3.1 Definition of unbiasedness An estimator of a parameter is said to be unbiased if: E[]=θ This means that, on average, the estimator hits the true parameter value. For example, the sample mean Xˉis an unbiased estimator of the population mean .
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