Statistics Fundamentals

Statistics Fundamentals

Core Concepts

Conventions and Decision Rules

Sample variance: use n-1

When estimating variance or standard deviation from a sample of returns, divide by n - 1 (Bessel's correction), not n. Dividing by n systematically underestimates dispersion. Standard deviation of returns is "volatility"; annualize with sigma_annual = sigma_period * sqrt(periods_per_year) (e.g., * sqrt(12) for monthly, * sqrt(252) for daily).

Normality testing: Jarque-Bera and its limits

JB = (n/6) * (skew^2 + excess_kurtosis^2 / 4), distributed chi-squared with 2 df under the null of normality (5% critical value: 5.99).

Low-power caveat: with small samples (n below roughly 50), JB rarely rejects even for clearly non-normal data — failing to reject is weak evidence of normality, not confirmation. With large samples, financial return series almost always reject due to fat tails and (for equities) negative skewness. Treat the test as a screen, and pair it with a look at the actual skew/kurtosis magnitudes and extreme observations.

Covariance estimation and Ledoit-Wolf shrinkage

The sample covariance matrix Sigma_hat = (1/(n-1)) (X - X_bar)^T (X - X_bar) becomes poorly conditioned or singular when the number of assets p approaches the number of observations n. Plugging it into a mean-variance optimizer then produces extreme, unstable weights that flip with small data changes.

Shrinkage blends the sample matrix toward a structured target:

$$\hat{\Sigma}_{shrunk} = \delta \cdot F + (1 - \delta) \cdot \hat{\Sigma}$$

where F is the target (e.g., scaled identity) and delta is the shrinkage intensity. Ledoit-Wolf (2004) derives the delta that minimizes expected squared Frobenius distance to the true covariance matrix, trading a little bias for a large variance reduction — yielding better-conditioned, invertible matrices and stable portfolio weights.

Note: the bundled script's shrunk_covariance implements a simplified shrinkage-intensity estimate, not the full Ledoit-Wolf estimator. For production work use sklearn.covariance.LedoitWolf.

Regression diagnostics (CAPM and factor models)

For the single-factor CAPM regression R_i - R_f = alpha + beta * (R_m - R_f) + epsilon:

• beta = rho * sigma_i / sigma_m (market sensitivity); alpha is the risk-adjusted excess return.
• In a single-factor regression, R^2 = rho^2.
• Judge coefficients by t-statistics (t = coefficient / SE); with n - 2 df, |t| above roughly 2 indicates 5% significance. A positive alpha point estimate with |t| < 2 is not evidence of skill.
• Adding regressors always raises R-squared; use adjusted R-squared, AIC/BIC, or cross-validation to guard against overfitting.

Bootstrap procedure

Non-parametric resampling for the sampling distribution of a statistic when analytical standard errors are unavailable (Sharpe ratio, alpha), the distribution is non-normal, or samples are small:

1. From the original n observations, draw B resamples of size n with replacement (B = 1,000-10,000).
2. Compute the statistic on each resample.
3. Percentile method: the (1 - alpha) confidence interval is the alpha/2 and 1 - alpha/2 percentiles of the bootstrap distribution; the bootstrap standard error is the std of the B statistics.

Caveat: the i.i.d. bootstrap ignores autocorrelation and volatility clustering; use block bootstrap for serially dependent return series.

Standard Analysis Workflow

Given a return series, run this sequence:

1. Descriptive stats — mean, volatility (n-1), skewness, excess kurtosis; annualize for reporting.
2. Distribution checks — Jarque-Bera (mind the low-power caveat), inspect skew/kurtosis magnitudes and largest outliers; decide whether normal-based methods (parametric VaR, t-tests) are defensible.
3. Covariance/correlation (multi-asset) — sample covariance and correlation matrices; if p is large relative to n, apply shrinkage before any optimization.
4. Regression diagnostics — CAPM or factor regression; report alpha/beta with t-stats and R-squared; check residuals for structure.
5. Bootstrap CIs — for statistics without clean analytical standard errors (Sharpe, alpha, drawdown), bootstrap confidence intervals rather than reporting bare point estimates.

Worked Examples