Estimates
Point vs Interval Estimates
| Type |
Definition |
Example |
| Point estimate |
Single-value estimate of a parameter |
\(\bar{x}\) as estimate of \(\mu\) |
| Interval estimate |
Range expressing uncertainty around estimate |
\([\bar{x} - z\frac{s}{\sqrt{n}},\; \bar{x} + z\frac{s}{\sqrt{n}}]\) (95% CI) |
Confidence Interval for a Mean (large sample)
| Symbol |
Meaning |
| \(\bar{x}\) |
Sample mean |
| \(s\) |
Sample standard deviation |
| \(n\) |
Sample size |
| \(z\) |
z-score for desired confidence (e.g., 1.96 for 95%) |
Formula
\[\bar{x} \pm z \cdot \frac{s}{\sqrt{n}}\]
Estimates — Common Methods
| Method |
When to use |
Pros |
Cons |
| Method of moments |
Simple parametric estimation |
Easy to compute |
Can be inefficient / biased |
| Maximum likelihood (MLE) |
Parametric models |
Asymptotically efficient |
Requires model; can be complex to compute |
| Bayesian estimation |
When prior knowledge exists |
Incorporates prior; yields full posterior |
Needs prior specification; may be computationally intensive |
| Nonparametric (bootstrap) |
For CI without parametric assumptions |
Flexible; few assumptions |
Computationally heavy |