Point Estimation

Overview

Point estimation uses sample data to calculate a single value (point estimate) as the best guess for an unknown population parameter.

Key Terms

Term	Definition
Parameter	Unknown population characteristic ($\mu$, $\sigma$, $p$)
Estimator	Rule or formula used to calculate estimate
Estimate	Specific value calculated from data

Common Point Estimators

Parameter	Point Estimator
Population mean ($\mu$)	Sample mean ($\bar{x}$)
Population variance ($\sigma^2$)	Sample variance ($s^2$)
Population proportion ($p$)	Sample proportion ($\hat{p}$)
Population SD ($\sigma$)	Sample SD ($s$)

Properties of Good Estimators

1. Unbiasedness

An estimator is unbiased if:

$$E(\hat{\theta}) = \theta$$

The expected value of the estimator equals the parameter.

• $\bar{x}$ is unbiased for $\mu$
• $s^2$ is unbiased for $\sigma^2$ (with $n-1$ in denominator)
• $\hat{p}$ is unbiased for $p$

2. Efficiency

Among unbiased estimators, the most efficient has the smallest variance.

3. Consistency

As $n \to \infty$, the estimator converges to the true parameter:

$$\hat{\theta} \xrightarrow{p} \theta$$

4. Sufficiency

Uses all relevant information in the sample.

Bias and Variance

$$\text{Mean Squared Error (MSE)} = \text{Variance} + \text{Bias}^2$$ $$\text{MSE}(\hat{\theta}) = \text{Var}(\hat{\theta}) + [E(\hat{\theta}) - \theta]^2$$

For unbiased estimators: MSE = Variance

Methods of Estimation

Method of Moments

Match sample moments to population moments:

$$\text{Sample mean} = \text{Population mean}$$ $$\bar{x} = \mu$$

Maximum Likelihood Estimation (MLE)

Find $\theta$ that maximizes the likelihood of observing the data:

$$L(\theta) = P(\text{data} \mid \theta)$$ $$\hat{\theta}_{MLE} = \arg\max L(\theta)$$

Examples

Example 1: Estimating Mean

Sample data: 12, 15, 18, 21, 24

$$\bar{x} = \frac{12 + 15 + 18 + 21 + 24}{5} = \frac{90}{5} = 18$$

Point estimate of $\mu$ is 18.

Example 2: Estimating Proportion

In 200 voters, 120 support a candidate:

$$\hat{p} = \frac{120}{200} = 0.60$$

Point estimate of $p$ is 0.60 (60%).

Example 3: Unbiased Variance

Why divide by $(n-1)$?

If we used $n$:

$$E\left[\frac{\sum(x_i - \bar{x})^2}{n}\right] = \frac{n-1}{n} \times \sigma^2 \neq \sigma^2 \quad \text{(biased)}$$

Using $n-1$:

$$E\left[\frac{\sum(x_i - \bar{x})^2}{n-1}\right] = \sigma^2 \quad \text{(unbiased)}$$

Standard Error of Estimators

Estimator	Standard Error
$\bar{x}$	$\sigma/\sqrt{n}$ or $s/\sqrt{n}$
$\hat{p}$	$\sqrt{p(1-p)/n}$ or $\sqrt{\hat{p}(1-\hat{p})/n}$

Limitations

Point estimates:

• Give a single value, not a range
• Don't indicate uncertainty or precision
• May be exactly wrong (just one value)

→ This motivates interval estimation (confidence intervals)