We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.
The second and third authors were supported by JSPS KAKENHI 19K14549 and 16K05196 respectively.
The authors wish to give their gratitude to an anonymous referee for his or her comments. They also wish to give their gratitude to Prof. Ken-iti Sato for references and to Prof. Matyas Barczy for comments.
"Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution." Braz. J. Probab. Stat. 36 (2) 385 - 407, June 2022. https://doi.org/10.1214/22-BJPS531