Testing and estimation of the index of stability of univariate and bivariate symmetric $\alpha-$stable distributions via modified Greenwood statistic

1. Problem Statement

The primary challenge addressed in this paper is the statistical testing and estimation of the stability index ($\alpha$) for symmetric $\alpha$-stable distributions, particularly in distinguishing between Gaussian distributions ($\alpha = 2$) and heavy-tailed $\alpha$-stable distributions ($\alpha < 2$).

• The Difficulty: As the stability index $\alpha$ approaches 2, $\alpha$-stable distributions become increasingly similar to Gaussian distributions. Classical tests for multivariate normality (e.g., Mardia's, Jarque-Bera, Henze-Zirkler) often lack sufficient power to detect deviations from normality when $\alpha$ is close to 2, especially with small sample sizes.
• The Gap: While the Greenwood statistic and its modified version have been used for univariate positive or symmetric data, there was a lack of methodology extending these statistics to bivariate (and multivariate) sub-Gaussian distributions for hypothesis testing and parameter estimation.

2. Methodology

The authors propose a testing and estimation framework based on a Modified Greenwood Statistic, originally designed for symmetric distributions. The methodology is divided into univariate and bivariate settings.

A. The Modified Greenwood Statistic

For a univariate sample $X = (X_1, \dots, X_n)$, the statistic is defined as:
$$S_X = \frac{\sum_{i=1}^n |X_i|^2}{(\sum_{i=1}^n |X_i|)^2}$$
Key Property: Within the class of symmetric $\alpha$-stable distributions, $S_X$ is stochastically decreasing with respect to $\alpha$. As $\alpha$ increases (approaching 2), the value of the statistic tends to decrease. This monotonicity allows it to serve as a reliable test statistic.

B. Bivariate Extensions (Sub-Gaussian Case)

The paper extends this concept to bivariate random vectors $\vec{X} = (X_1, X_2)$ drawn from a sub-Gaussian distribution (a special case of multivariate $\alpha$-stable distributions). Two specific test statistics are proposed:

1. Scenario 1 ($S^1_{\vec{X}}$): Summation Approach

   • Definition: $W_i = X^{(i)}_1 + X^{(i)}_2$. The statistic is calculated on the sum: $S^1_{\vec{X}} = S_W$.
   • Property: The sum of two components of a bivariate sub-Gaussian vector is itself a univariate symmetric $\alpha$-stable variable. Crucially, $S^1_{\vec{X}}$ is independent of the correlation ($\rho$) and the scale parameters of the individual components. It retains the stochastic monotonicity with respect to $\alpha$.

2. Scenario 2 ($S^2_{\vec{X}}$): Squared Norm Approach

   • Definition: $Y_i = (X^{(i)}_1)^2 + (X^{(i)}_2)^2$. The statistic is calculated on the squared Euclidean norm: $S^2_{\vec{X}} = S_Y$.
   • Property: The behavior of this statistic depends on a parameter $\beta$, which is a function of the correlation $\rho$ and the ratio of variances.
   • Stochastic Ordering: For a fixed $\beta$, $S^2_{\vec{X}}$ is stochastically decreasing with respect to $\alpha$. However, its distribution is sensitive to the dependence structure ($\beta$), requiring specific calibration (e.g., using $\beta=0$ for the most conservative critical region).

C. Testing and Estimation Procedures

• Hypothesis Testing: The authors formulate tests for $H_0: \alpha = 2$ (Gaussian) vs. $H_1: \alpha < 2$. Rejection regions are constructed using Monte Carlo simulations to estimate quantiles of the test statistics under the null hypothesis.
• Confidence Intervals: By inverting the hypothesis test, the authors construct confidence intervals for the stability index $\alpha$.
• Simulation: Extensive Monte Carlo simulations (10,000 samples for critical values, 1,000 for power analysis) are used to validate the theoretical properties and compare performance against classical tests.

3. Key Contributions

1. Extension to Bivariate/Sub-Gaussian: The paper successfully generalizes the modified Greenwood statistic from univariate to bivariate sub-Gaussian distributions, introducing two distinct extensions ($S^1$ and $S^2$).
2. Novel Testing Framework: It introduces a robust testing methodology that leverages the stochastic monotonicity of the Greenwood statistic to distinguish between Gaussian and heavy-tailed distributions.
3. Superior Performance in "Hard" Cases: The proposed methods are specifically optimized for the difficult case where $\alpha$ is close to 2 (near-Gaussian) and sample sizes are small ($n < 30$).
4. Parameter Estimation: The methodology provides a mechanism for constructing confidence intervals for the stability index $\alpha$ in both univariate and bivariate settings.

4. Results

• Simulation Studies:
  • Small Samples ($n=10, 30$): The $S^1$ test (summation approach) significantly outperforms classical tests (Mardia's Kurtosis/Skewness, Henze-Zirkler, Jarque-Bera, Royston) when $\alpha$ is close to 2.
  • Large Samples ($n=100$): Classical tests like Mardia's Kurtosis and Royston's test generally show higher power, though the proposed methods remain competitive.
  • Impact of Correlation: The $S^2$ test's power is highly dependent on the correlation parameter $\beta$. It performs exceptionally well when $\beta$ is small (high correlation), but the $S^1$ test is recommended when the correlation structure is unknown or when sample sizes are small, as it is invariant to correlation.
• Real-World Data Application:
  • Dataset: Financial data from a copper mining company (KGHM), analyzing USD/PLN exchange rates and Copper prices.
  • Analysis: The data was segmented into two regimes (2008 crisis and stable market) and modeled using a VAR(1) model with $\alpha$-stable innovations.
  • Findings:
    • The proposed tests ($S^1$ and $S^2$) rejected the null hypothesis of bivariate normality for both regimes, confirming the presence of heavy tails ($\alpha < 2$).
    • The estimated stability indices were $\hat{\alpha} \approx 1.72$ (crisis) and $\hat{\alpha} \approx 1.84$ (stable) for USD/PLN, and similar values for Copper prices.
    • The 95% confidence intervals for $\alpha$ consistently excluded 2, validating the heavy-tailed nature of the financial residuals.

5. Significance

• Financial Risk Management: The ability to accurately distinguish between Gaussian and heavy-tailed models is critical in finance, where extreme events (crashes) are often underestimated by Gaussian models. This paper provides a tool to detect these heavy tails even when the data appears "almost" Gaussian.
• Robustness: The $S^1$ statistic offers a correlation-invariant test, making it highly practical for real-world data where the dependence structure between variables is complex or unknown.
• Methodological Advancement: By extending the Greenwood statistic to multivariate settings, the authors provide a new, computationally efficient, and theoretically grounded approach for testing multivariate normality in the presence of heavy tails, filling a gap left by existing literature which struggles with the $\alpha \to 2$ limit.

In conclusion, the paper establishes the modified Greenwood statistic as a powerful, versatile tool for stability index estimation and hypothesis testing in both univariate and bivariate $\alpha$-stable contexts, demonstrating particular superiority in small-sample, near-Gaussian scenarios.