Pareto Distribution — $ \mathrm{Pareto}(x_m,\alpha) $
The Pareto distribution is the canonical heavy-tailed distribution. It is used to illustrate power-law behavior, tail dominance, failure of moments, and limits of classical theorems.
Definition
Parameters:
- Scale (minimum) $ x_m > 0 $
- Shape (tail index) $ \alpha > 0 $
Support:
\(x \ge x_m\)
PDF:
\(f(x)
=
\frac{\alpha x_m^\alpha}{x^{\alpha+1}},
\quad x \ge x_m\)
CDF:
\(F(x)
=
1-\left(\frac{x_m}{x}\right)^\alpha\)
Survival Function:
\(\mathbb{P}(X>x)
=
\left(\frac{x_m}{x}\right)^\alpha\)
Interpretation
- Models power-law tails
- Large values are not exponentially suppressed
- Extremes dominate sums and averages
Classic examples:
- Wealth distributions
- File sizes
- Insurance losses
- Network degrees
Moments and Tail Behavior
Existence of Moments
For $ k>0 $, \(\mathbb{E}[X^k] < \infty \quad \Longleftrightarrow \quad \alpha > k\)
Mean
\(\mathbb{E}[X] = \begin{cases} \frac{\alpha x_m}{\alpha-1}, & \alpha>1 \\ \infty, & \alpha \le 1 \end{cases}\)
Variance
\(\operatorname{Var}(X) = \begin{cases} \frac{\alpha x_m^2}{(\alpha-1)^2(\alpha-2)}, & \alpha>2 \\ \infty, & \alpha \le 2 \end{cases}\)
Heavy-Tail Classification
-
Regularly varying tail: \(\mathbb{P}(X>x) = x^{-\alpha} L(x)\) with slowly varying $ L(x) $
-
Pareto is the prototype heavy-tailed distribution
Moment Generating Function
- MGF does not exist for any $ t>0 $
This is a key contrast with:
- Exponential
- Gamma
- Normal
Relationship to Other Distributions
Pareto vs Exponential
| Feature | Pareto | Exponential |
|---|---|---|
| Tail | Power law | Exponential |
| MGF | Does not exist | Exists |
| Memoryless | No | Yes |
| Extreme events | Dominant | Rare |
Connection to Uniform
If \(U \sim \mathrm{Unif}(0,1),\) then \(X = x_m U^{-1/\alpha} \sim \mathrm{Pareto}(x_m,\alpha)\)
(Used in simulation.)
Pareto Type I and II
- Type I: standard Pareto (this page)
- Type II (Lomax): shifted version
Transformations
Log Transformation
If $ X \sim \mathrm{Pareto}(x_m,\alpha) $, then \(\log X \sim \mathrm{Exp}(\alpha) \quad \text{(shifted by } \log x_m \text{)}\)
This explains why Pareto appears in log-scale arguments.
Sums and Limits
- Sums of Pareto variables:
- Often dominated by the maximum
- Classical CLT may fail
- Stable limits may replace Normal limits when $ \alpha<2 $
Likelihood and Estimation
Likelihood
\(\ell(\alpha) = n\log\alpha + \alpha n\log x_m - (\alpha+1)\sum\log x_i\)
MLE (known $ x_m $)
\(\hat{\alpha} = \frac{n}{\sum \log(x_i/x_m)}\)
Bayesian Notes
- Conjugate priors do not have a clean closed form
- Often treated with improper or hierarchical priors
Why Professors Love Pareto
Pareto is used to show:
- Failure of LLN assumptions
- Failure of variance-based arguments
- Importance of tail conditions
- Sharp contrast with exponential families
It is the go-to counterexample distribution.
Key Theorems and Facts (Prelim-Relevant)
- Power-law tail
- Finite moments depend on $ \alpha $
- MGF does not exist
- Extremes dominate sums
- Log transform gives Exponential
Exam Takeaways
- Heavy tail ⇒ think Pareto
- Moments exist only above thresholds
- CLT may fail
- Compare against Exponential
- Used as a counterexample, not a model
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