Parametric Estimation

Quantitative Risk Management in Python

Jamsheed Shorish

Computational Economist

A class of distributions

  • Loss distribution: not known with certainty
  • Class of possible distributions?
    • Suppose class of distributions $f(x; \theta)$
    • $x$ is loss (random variable)
    • $\theta$ is vector of unknown parameters
  • Example: Normal distribution
    • Parameters: $\theta = (\mu, \sigma)$, mean $\mu$ and standard deviation $\sigma$
  • Parametric estimation: find 'best' $\theta^\star$ given data
  • Loss distribution: $f(x,\theta^\star)$
Quantitative Risk Management in Python

Fitting a distribution

  • Fit distribution according to error-minimizing criteria
    • Example: scipy.stats.norm.fit(), fitting Normal distribution to data
      • Result: optimally fitted mean and standard deviation
  • Advantages:
    • Can visualize difference between data and estimate using histogram
    • Can provide goodness-of-fit tests
Quantitative Risk Management in Python

Goodness of fit

  • How well does an estimated distribution fit the data?
  • Visualize: plot histogram of portfolio losses

histogram image of losses

Quantitative Risk Management in Python

Goodness of fit

  • How well does an estimated distribution fit the data?
  • Visualize: plot histogram of portfolio losses
  • Normal distribution with norm.fit()

histogram image of losses with normal distribution

Quantitative Risk Management in Python

Goodness of fit

  • How well does an estimated distribution fit the data?
  • Visualize: plot histogram of portfolio losses
  • Example:
    • Normal distribution with norm.fit()
    • Student's t-distribution with t.fit()
    • Asymmetrical histogram?

histogram image of losses with normal distribution

Quantitative Risk Management in Python

Anderson-Darling test

  • Statistical test of goodness of fit
    • Test null hypothesis: data are Normally distributed
    • Test statistic rejects Normal distribution if larger than critical_values
  • Import scipy.stats.anderson
  • Compute test result using loss data
from scipy.stats import anderson

anderson(loss)

AndersonResult(statistic=11.048641503898523,
critical_values=array([0.57 , 0.649, 0.779, 0.909, 1.081]),
significance_level=array([15. , 10. ,  5. ,  2.5,  1. ]))
Quantitative Risk Management in Python

Skewness

  • Skewness: degree to which data is non-symmetrically distributed
    • Normal distribution: symmetric

normal distribution

Quantitative Risk Management in Python

Skewness

  • Skewness: degree to which data is non-symmetrically distributed
    • Normal distribution: symmetric
    • Student's t-distribution: symmetric

normal and t distributions

Quantitative Risk Management in Python

Skewness

  • Skewness: degree to which data is non-symmetrically distributed
    • Normal distribution: symmetric
    • Student's t-distribution: symmetric
  • Skewed Normal distribution: asymmetric
    • Contains Normal as special case
    • Useful for portfolio data, where e.g. losses more frequent than gains
    • Available in scipy.stats as skewnorm

normal, t and skewed normal distributions

Quantitative Risk Management in Python

Testing for skewness

  • Test how far data is from symmetric distribution: scipy.stats.skewtest
  • Null hypothesis: no skewness
  • Import skewtest from scipy.stats
  • Compute test result on loss data
    • Statistically significant => use distribution class with skewness
from scipy.stats import skewtest

skewtest(loss)

SkewtestResult(statistic=-7.786120875514511,
pvalue=6.90978472959861e-15)
Quantitative Risk Management in Python

Let's practice!

Quantitative Risk Management in Python

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