Empirical and Mixture Distributions
A mixture distribution blends several distributions, such as calm and volatile regimes. Learn how mixtures create fat tails, how they are fitted and their uses.
Markets do not behave the same way all the time. There are long stretches of calm trading and shorter bursts of turmoil. A mixture distribution models this by combining two or more simpler distributions, each describing a different state, with weights for how often each state occurs. Even if each component is normal, the mixture can have fat tails and skewness, which helps explain why market returns look so different from a single bell curve.
The idea#
A two component normal mixture:
f(x) = w × Normal(μ1, σ1) + (1 - w) × Normal(μ2, σ2)- w: probability of being in state 1 (for example, calm).
- 1 minus w: probability of being in state 2 (for example, turbulent).
Worked example#
Why mixtures produce fat tails#
Mixing a narrow distribution with a wide one creates a shape with a tall peak (many quiet days) and heavy tails (occasional extreme days). This is exactly what high kurtosis describes. Volatility clustering, where calm and turbulent periods alternate, can be seen as a dynamic version of this mixing. See Skewness and Kurtosis and GARCH.
Fitting mixtures#
- Expectation maximisation (EM) algorithm: a standard method to estimate the weights, means and standard deviations, based on maximum likelihood. See Maximum Likelihood.
- Number of components: chosen using criteria such as AIC or BIC, or by economic reasoning (for example, two regimes: normal and stress).
- Gaussian mixture models are also widely used in machine learning for clustering. See Machine Learning in Trading.
From static mixtures to regime switching#
A simple mixture assumes each day independently comes from one state. In reality, regimes persist: a stressed day is usually followed by more stressed days. Regime switching models, such as the Markov switching model introduced to economics by James Hamilton in 1989, let the state evolve over time with transition probabilities. They estimate the probability of being in each regime on each day. See Structural Breaks and Regime Changes.
| Model | Regime persistence | Use |
|---|---|---|
| Static mixture | None | Describing the overall distribution |
| Markov switching | Yes | Detecting and forecasting regimes |
| GARCH | Continuous volatility changes | Volatility forecasting |
Trading uses#
| Use | Example |
|---|---|
| Risk models | More realistic value at risk and stress scenarios. See Value at Risk (VaR) |
| Regime filters | Reduce exposure when the model signals a high volatility regime |
| Option pricing | Mixture models can reproduce volatility smiles. See Volatility Smile and Skew |
| Strategy evaluation | Test how strategies perform in each regime |
| Simulation | Generate realistic return paths for Monte Carlo. See Monte Carlo Simulation |
Limits#
- Regimes are identified after the fact more clearly than in real time.
- Parameters can be unstable with limited data on rare regimes.
- Too many components lead to overfitting. See Overfitting and Curve Fitting.
Frequently asked questions#
What is a mixture distribution?#
A distribution formed by combining two or more distributions, each representing a different state, weighted by how often each state occurs.
How do mixture distributions create fat tails?#
By combining a narrow distribution for calm periods with a wide one for turbulent periods, producing many quiet values and occasional extreme ones.
What is a regime switching model?#
A model in which the market moves between states, such as calm and volatile, with probabilities that make regimes persist over time.
Next, learn how to analyse data over time in Time Series Basics.
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Mentioned in
- Skewness and KurtosisMath and Statistics