# Inference for three-parameter M-Wright distributions with applications

## TL;DR

This paper develops point and interval estimators for three-parameter M-Wright distributions, including methods for uncertainty quantification, and demonstrates their effectiveness on synthetic and real data.

## Contribution

It introduces novel estimation procedures for the three-parameter M-Wright family, including uncertainty quantification and asymptotic covariance analysis.

## Key findings

- Interval estimator for scale outperforms previous methods when location is zero.
- Asymptotic covariance structure enables correlation estimation between parameters.
- Proposed methods are validated on synthetic, age, and height data.

## Abstract

We propose point estimators for the three-parameter (location, scale, and the fractional parameter) variant distributions generated by a Wright function. We also provide uncertainty quantification procedures for the proposed point estimators under certain conditions. The class of densities includes the three-parameter one-sided and the three-parameter symmetric bimodal $M$-Wright family of distributions. The one-sided family naturally generalizes the Airy and half-normal models. The symmetric class includes the symmetric Airy and normal or Gaussian densities. The proposed interval estimator for the scale parameter outperformed the estimator derived in \cite{cah12} when the location parameter is zero. We obtain the asymptotic covariance structure for the scale and fractional parameter estimators, which allows estimation of the correlation. The coverage probabilities of the interval estimators slightly depend on the proposed location parameter estimators. For the symmetric case, the sample mean (or median) is favored than the median (or mean) when the fractional parameter is greater (or lesser) than 0.39106 in terms of their asymptotic relative efficiency. The estimation algorithms were tested using synthetic data and were compared with their bootstrap counterparts. The proposed inference procedures were demonstrated on age and height data.