Which Statistic Best Estimates the Parameter? (2026 Guide to Optimal Estimators)
Discover the top estimators like Maximum Likelihood Estimators (MLE), Minimum Variance Unbiased Estimators (MVUE), and robust methods such as trimmed means and shrinkage estimators. This guide compares them using key metrics like Mean Squared Error (MSE), bias-variance tradeoffs, and real-world performance, drawing from 2026 research insights.
Get step-by-step criteria for choosing the best estimator, backed by Cramér-Rao Lower Bounds (CRLB), bootstrap validation, and emerging theories on super-efficient estimators.
Quick Answer
No single "best" statistic exists universally--the optimal estimator depends on the bias-variance tradeoff, data contamination levels, and estimation goals. MSE-minimizing choices like MLE (asymptotically efficient per CRLB) or shrinkage estimators often outperform the sample mean in finite samples. MVUE achieves minimum variance among unbiased estimators, as per STAT 205B notes.
Key Takeaways: Best Estimators at a Glance
- Clean, IID data (normal dist.): Sample mean or MLE--lowest MSE asymptotically.
- Contaminated data (10-20%): PIC criterion beats competitors (PMC7516763); 20% trimmed mean yields 54.92 in skewed examples (garstats).
- Covariance estimation: Correlation shrinkage PRIAL 16-70% gains (PMC2748251).
- Asymptotic efficiency: MLE √n(θ̂ - θ) → N(0, I⁻¹(θ)); Method of Moments (MoM) worse variance (MathOverflow).
| Estimator | MSE | Bias | Variance | Best For |
|---|---|---|---|---|
| Sample Mean | Low (Gaussian) | 0 | σ²/n | IID normal |
| Median | Higher | Low | ~πσ²/2n | Skewed/outliers |
| MLE | Asymp. min | ~0 | CRLB | Large n, known dist. |
| 20% Trimmed Mean | 54.92 (ex.) | Low | Reduced | Contaminated (20%) |
| Shrinkage | 16-70% PRIAL | Biased | Low | High-dim covar. |
Understanding Point Estimators and What Makes One "Best"
Point estimators provide a single value g(X₁, ..., Xₙ) to approximate a population parameter θ from IID random samples (Analytics Vidhya). Unlike interval estimation (e.g., 95% confidence intervals, LIS Academy), they focus on the "best guess."
Bias: Bias_θ(W) = E_θ[W] - θ (STAT 205B). Unbiased if zero.
MSE: MSE_θ(W) = Bias² + Var_θ(W)--the gold standard for "best" (TDS: penalizes large errors quadratically under Gaussian noise).
An estimator is "best" if it minimizes MSE, satisfies CRLB for unbiased cases, or excels in robustness/consistency.
Bias-Variance Decomposition and Tradeoff Explained
MSE = Bias² + Variance + Irreducible Error (Medium/TDS). High bias underfits (e.g., linear model on quadratic data); high variance overfits (R for SL: 9th-degree polynomial EPE spikes).
Mini case: Polynomial degree on EPE--low degree: high bias; high degree: high variance (R for SL).
Tradeoff: Biased estimators like shrinkage reduce total MSE by trading bias for lower variance.
MSE Comparison: Which Statistic Minimizes Mean Squared Error?
MSE formulas: MSE(W) = E[(W - θ)²] = Var(W) + [E(W) - θ]² (STAT 205B). Minimize via CRLB: Var(W) ≥ [dE(W)/dθ]² / E[(∂/∂θ log f)²].
Bootstrap estimates MSE accuracy (StackExchange). Pitman closeness: P(|Ŵ - θ| < |W - θ|) > 0.5.
Example: Sample mean vs. 20% trimmed mean in skewed data--trimmed MSE lower at 54.92 (garstats).
Mean vs. Median vs. Trimmed Mean: Robustness Performance
In normal data, mean ≈ median ≈ 20% trimmed mean. Contaminated: mean pulled by outliers.
Normal sample (garstats): Mean=55, Trimmed=54.92, Median~54--similar.
Skewed/contaminated: Trimmed mean resists 20% outliers better.
| Pros/Cons | Mean | Median | 20% Trimmed |
|---|---|---|---|
| Pros | Unbiased, min var (normal) | Robust to outliers | Balances robustness/efficiency |
| Cons | Sensitive to contamination | Higher var (πσ²/2n) | Discards data |
| MSE (contam.) | High | Medium | Low (54.92 ex.) |
Case study: 20% contamination--trimmed outperforms mean (PMC7516763 analogs).
MLE vs. Method of Moments: Asymptotic Efficiency Showdown
MLE: √n(Ŵ - θ) → N(0, I⁻¹(θ))--asymptotically efficient (CRLB). Invariant to reparameterization.
MoM: Matches sample moments (e.g., θ̃ = √(X̄²/4 + X̄²) - X̄/2). Consistent but higher variance; fails invariance (StackExchange/MathOverflow: transformations worsen asymp. var.).
Contradiction resolved: MLE superior asymptotically; MoM simpler but suboptimal.
Advanced Winners: MVUE, Shrinkage, Empirical Bayes, and Super-Efficient Estimators
MVUE: Min var among unbiased, attains CRLB if complete/sufficient (STAT 205B). Example: Sample mean for normal μ.
Shrinkage: PRIAL 16-70% for correlations (PMC2748251); reduces risk in high-dim.
Empirical Bayes: Superior in hierarchical models via shrinkage.
Super-efficient (2026 theory): Beat CRLB at √n rate in some subsequences.
Pitman Estimator: Closest to θ in probability.
Bootstrap and Asymptotic Efficiency for Estimator Selection
Bootstrap CIs: Pros--nonparametric, handles complex metrics; Cons--slow, assumes IID (StackExchange). Percentile-t better for small trim; percentile for 20%+ (garstats). Tests consistency/convergence rates (Wikipedia).
How to Choose the Best Estimator: Step-by-Step Checklist
- Define goals: Unbiased (MVUE)? Min MSE (shrinkage)? Robust?
- Check CRLB: Var ≥ 1/I(θ).
- Bias-variance plot: Simulate tradeoff.
- Bootstrap/simulate: MSE comparison.
- Test contamination: Trim/PIC for 10-20%.
Case: Uniform(a,b)--MoM: â = X̄ - √3 S, b̂ = X̄ + √3 S.
Robust Estimator Selection Criteria and Model Criteria (PIC, DIC)
PIC (γ=0.3) excels at 10-20% contamination, beats DIC/MDIC (PMC7516763 γ-divergences). Checklist: >10% outliers? Use trimmed/PIC over mean.
Pros & Cons of Top Estimators: Quick Comparison Table
| Estimator | Pros | Cons | When to Use |
|---|---|---|---|
| MLE | Asymp. efficient, invariant | Not robust, needs dist. | Large n, clean data |
| MVUE | Min var (unbiased) | May not exist, high var | Known complete family |
| Shrinkage | 16-70% PRIAL risk cut | Biased | High-dim, covar. |
| Trimmed Mean | Contamination-resistant | Data loss | Skewed/20% contam. |
| MoM | Simple, consistent | Poor efficiency | Quick, moments known |
Resolves: MoM consistent but not efficient; MLE wins asymp.
FAQ
Why minimize MSE for estimators? Penalizes large errors quadratically; optimal under Gaussian (TDS).
MLE vs. MoM: which better asymptotically? MLE--attains CRLB; MoM higher variance (MathOverflow).
Bias-variance tradeoff? MSE = Bias² + Var; balance via regularization (Medium).
Bootstrap pros/cons? Pros: Flexible CIs; Cons: Compute-heavy, IID assumption (StackExchange).
Sample mean vs. trimmed? Mean for clean normal; trimmed for contamination (garstats).
MVUE always exist? No--often none with min var (StackExchange).
