Laplace Transform driven Stein-type Goodness-of-fit Tests for Pareto Distribution

Abstract: The Pareto distribution plays a crucial role in various disciplines, necessitating robust goodness-of-fit tests for its validation. This article introduces a novel tests based on Stein's characterization and the Laplace transform, offering a fresh perspective on model assessment. We establish the asymptotic properties of the proposed test and evaluate its empirical performance against existing methods in terms of size and power. Our findings demonstrate that the new test often outperforms or performs comparably to established tests. In addition, real data applications illustrate its practical utility.

• The paper introduces a hybrid framework combining Stein’s method and the Laplace transform to design goodness-of-fit tests for the Pareto distribution.
• The proposed tests, including integral, Kolmogorov-Smirnov, and Cramér-von Mises types, are computationally efficient with robust asymptotic properties.
• Empirical simulations and real data analyses confirm the tests' strong power in detecting heavy-tailed and non-Pareto structures.

Laplace Transform Driven Stein-type Goodness-of-fit Tests for Pareto Distribution

Introduction

This paper addresses the challenge of constructing robust statistical goodness-of-fit (GoF) tests for the Pareto distribution, exploiting a novel synergy between Stein’s method and the Laplace transform. The Pareto distribution is instrumental in modeling heavy-tailed phenomena across economics, finance, insurance, and environmental sciences. Despite the wide availability of classical GoF procedures, leveraging unique distributional characterizations for test construction has proven advantageous. The integration of Stein-type differential identities with the analytic flexibility of the Laplace transform yields new families of GoF statistics, capable of discriminating Pareto-like structure against general alternatives.

Characterization: Stein’s Method and Laplace Transform Fusion

The methodology builds upon a fixed-point Stein characterization for Pareto distributions, specifically adapting the framework of Betsch and Ebner (2021). The classical Pareto law is characterized by:

$$f_X(s) = \mathbb{E}\left(\frac{\alpha+1}{X} \mathbb{I}(X > s)\right),\quad s > 1,$$

where $f_X$ is the density function for $X$. Substitution into the Laplace transform representation,

$\mathscr{L}_X(t) = \mathbb{E}\left(e^{-Xt}\right),$

leads to the identity:

$$\mathscr{L}_X(t) = \mathbb{E}\left(\frac{\alpha+1}{X} \cdot \frac{e^{-t}-e^{-tX}}{t}\right).$$

This establishes the foundation for a test statistic based on deviations from the above characterization.

Proposed Test Statistics

Three classes of test statistics are introduced:

• Integral Type ($\mathscr{DS}_1$):

$$\mathscr{DS}_1 = \frac{\hat{\alpha} + 1}{n} \sum_{i=1}^n \frac{\log X_i}{X_i} - \frac{1}{n} \sum_{i=1}^n \frac{1}{X_i}$$

• Kolmogorov-Smirnov Type ($\mathscr{DS}_2$):

$$\mathscr{DS}_2 = \sup_{s \in [0, 1]} \left| \frac{1}{n} \sum_{i=1}^n \frac{\hat{\alpha}+1}{X_i} \left(\frac{s - s^{X_i}}{-\log s}\right) - \frac{1}{n} \sum_{i=1}^n s^{X_i} \right|$$

• Cramér-von Mises Type ($\mathscr{DS}_3$):

Formulated via quadratic functional over the space of Laplace-transformed deviations.

All statistics depend on the MLE of the shape parameter $f_X$0, with explicit representations that allow efficient computation.

Asymptotic Properties

Detailed asymptotic analysis establishes limiting distributions under the null hypothesis. For $f_X$1, under the Pareto null, the statistic converges in distribution to a normal law $f_X$2, with closed-form expressions for $f_X$3 as a function of $f_X$4. For $f_X$5 and $f_X$6, weak convergence is shown to Gaussian processes and related supremum or quadratic functionals, which includes a correction for parameter estimation.

Empirical Evaluation and Numerical Results

Extensive Monte Carlo studies benchmark the proposed statistics against established GoF procedures, including EDF-based tests (Kolmogorov-Smirnov, Cramér-von Mises, Anderson-Darling), entropy-based, Mellin/Fourier transform-based, and characterization-derived tests. Simulation results reveal:

• The proposed Laplace-Stein tests exhibit type I error rates at nominal levels across all tested $f_X$7 for null Pareto samples.
• For diverse alternatives (Gamma, Inverse-Beta, Tilted Pareto, Benini, Log-Gamma, Log-Normal, Weibull, Levy, Burr, Inverse-Gaussian, Log-Weibull, Frechet, Half-Normal, Chi-Square, Dhillon, Log-Logistic, Linear Failure Rate), the proposed tests achieve power competitive with or superior to existing methods.
• The empirical power is especially pronounced for moderate to large sample sizes and alternatives with deviations in tail behavior or shape.

For $f_X$8, the power values for $f_X$9, $X$0, $X$1 often exceed those of competitors, particularly in heavy-tailed and skewed alternatives.

Application to Real Data

The paper demonstrates practical utility on two datasets:

• LIV Golf Earnings (2022): For large normalized earnings, all test $X$2-values are well above the conventional threshold. The proposed statistics validate the Pareto hypothesis, matching prior analysis.
• Airplane Failure Times: For transformed failure times, nearly all test $X$3-values fall below 0.05, correctly rejecting the null. This highlights high sensitivity of the Laplace-Stein statistics to non-Pareto structure.

Implications and Future Directions

The hybrid Stein-Laplace framework yields GoF tests with both theoretical guarantees and practical discrimination. The tests are computationally tractable, amenable to analytic and bootstrap-based inference, and applicable in classical, heavy-tailed data settings.

Potential future extensions include:

• Adapting tests for unknown scale parameters via joint estimation strategies.
• Generalizing to right-censored and truncated data, relevant in survival and reliability contexts.
• Exploring other transform domains (e.g., Fourier, Mellin) for test construction with Stein-type characterizations.
• Applying the framework to broader distribution families with established Stein identities (e.g., exponential, gamma, log-normal) for unified GoF theory.

Conclusion

The paper advances GoF methodologies for the Pareto distribution by integrating Stein’s method with Laplace transform theory. The resulting statistics are rigorously characterized, exhibit robust empirical power, and demonstrate practical effectiveness across simulated and real-world datasets. The analytic and computational tractability, alongside strong performance, positions these tests as versatile tools for distributional validation, especially in heavy-tailed environments (2604.22486).