Exponential Parameter Estimation for Generalized Gaussian Probability Distribution Function

The generalized Gaussian probability distribution function is a flexible distribution model capable of adapting to different tail behaviors, commonly used in practical applications such as signal processing and image analysis. One of its core parameters is the distribution's shape parameter (also known as the exponent parameter), which determines the distribution's kurtosis and tail characteristics. Estimation of the exponent parameter typically employs two methods: moment estimation and maximum likelihood estimation.

Moment estimation is based on matching sample moments with theoretical moments. Properties of the generalized Gaussian distribution show that its higher-order moments have analytical relationships with the shape parameter, allowing the shape parameter to be derived by calculating higher-order sample moments (such as fourth-order moments or absolute moments). Moment estimation is computationally simple and suitable for large sample sizes, but due to the sensitivity of higher-order moments to outliers, the estimation results may lack robustness. In MATLAB implementation, this typically involves calculating sample moments using functions like moment() or custom moment calculations, then solving for the shape parameter through algebraic equations.

Maximum likelihood estimation finds the parameter values most likely to generate the observed data by optimizing the likelihood function. Compared to moment estimation, maximum likelihood estimation has superior statistical properties, especially with moderate or small sample sizes, typically yielding higher estimation accuracy. However, the logarithmic likelihood function of the generalized Gaussian distribution may involve complex higher-order exponential operations, requiring numerical optimization methods (such as Newton-Raphson iteration or gradient descent) for solution, resulting in higher computational complexity. Code implementation often uses optimization functions like fminsearch() or fminunc() in MATLAB, where the likelihood function calculation requires careful handling of numerical stability issues.

Comparison of the two estimation methods: Computational efficiency: Moment estimation is computationally simpler and suitable for scenarios requiring high real-time performance; maximum likelihood estimation has higher computational cost but offers better accuracy. Applicability: Maximum likelihood estimation is more reliable with small sample sizes; moment estimation can serve as a fast alternative when sample sizes are large and computational resources are limited. Robustness: Moment estimation is susceptible to extreme values, while maximum likelihood estimation performs stably when distribution assumptions hold.

In practical applications, the advantages of both methods can be combined, for example using moment estimation to provide initial values to accelerate the convergence of maximum likelihood estimation. This hybrid approach can be implemented in code by first computing moment estimates as starting points for optimization algorithms, significantly reducing the number of iterations required for convergence.