  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.
