Properties of complex-valued power means of random variables and their applications

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.

[1]  B. Bercu.  On the Elephant Random Walk with Stops Playing Hide and Seek with the Mittag–Leffler Distribution , 2022, Journal of Statistical Physics.

[2]  Christof Schötz.  Strong laws of large numbers for generalizations of Fréchet mean sets , 2022, Statistics.

[3]  Mitsuhiro Itoh,et al.  Information geometry of the space of probability measures and barycenter maps , 2021, Sugaku Expositions.

[4]  M. Barczy,et al.  Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables , 2021, Aequationes mathematicae.

[5]  K. Okamura,et al.  Confidence disc and square for Cauchy distributions , 2021, Ukrains’kyi Matematychnyi Zhurnal.

[6]  K. Okamura,et al.  Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution , 2021, Brazilian Journal of Probability and Statistics.

[7]  K. Okamura,et al.  Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution , 2021, Annals of the Institute of Statistical Mathematics.

[8]  Y. Otobe,et al.  Characterizations of the Maximum Likelihood Estimator of the Cauchy Distribution , 2021, Lobachevskii Journal of Mathematics.

[9]  K. Okamura.  Characterizations of the Cauchy distribution associated with integral transforms , 2020, 2007.09707.

[10]  Frank Nielsen,et al.  On Voronoi Diagrams on the Information-Geometric Cauchy Manifolds , 2020, Entropy.

[11]  Jun Xu,et al.  A novel fractional moments-based maximum entropy method for high-dimensional reliability analysis , 2019, Applied Mathematical Modelling.

[12]  Jochen Einbeck,et al.  Quantile-Based Estimation of the Finite Cauchy Mixture Model , 2019, Symmetry.

[13]  Amlan Jyoti Das,et al.  Does the generalized mean have the potential to control outliers? , 2019, Communications in Statistics - Theory and Methods.

[14]  Mahesh D. Pandey,et al.  An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments , 2017 .

[15]  Miguel de Carvalho,et al.  Mean, What do You Mean? , 2016 .

[16]  J. Shabbir,et al.  Using extreme values and fractional raw moments for mean estimation in stratified random sampling , 2016 .

[17]  Tomonori Fukunaga,et al.  On convexity of simple closed frontals , 2015 .

[18]  Mitsuhiro Itoh,et al.  Geometry of Fisher Information Metric and the Barycenter Map , 2015, Entropy.

[19]  Zawar Hussain,et al.  A Class of Estimators for Finite Population Mean in Double Sampling under Nonresponse Using Fractional Raw Moments , 2014, J. Appl. Math..

[20]  Henry P. Wynn,et al.  Empirical geodesic graphs and CAT(k) metrics for data analysis , 2014, Statistics and Computing.

[21]  Zbynek Pawlas,et al.  Fractional absolute moments of heavy tailed distributions , 2013, 1301.4804.

[22]  M. Henmi,et al.  Geometry of Pre‐contrast Functions and Non‐conservative Estimating Functions , 2011 .

[23]  H. Le,et al.  Limit theorems for empirical Fr\'echet means of independent and non-identically distributed manifold-valued random variables , 2011, 1102.0228.

[24]  Henryk Gzyl,et al.  Maxentropic solution of fractional moment problems , 2006, Appl. Math. Comput..

[25]  Aldo Tagliani,et al.  Stieltjes moment problem via fractional moments , 2005, Appl. Math. Comput..

[26]  Aldo Tagliani,et al.  On the proximity of distributions in terms of coinciding fractional moments , 2003, Appl. Math. Comput..

[27]  R. Bhattacharya,et al.  LARGE SAMPLE THEORY OF INTRINSIC AND EXTRINSIC SAMPLE MEANS ON MANIFOLDS—II , 2003, math/0507423.

[28]  P. L. Novi Inverardi,et al.  Maximum Entropy Density Estimation from Fractional Moments , 2003 .

[29]  R. Bhattacharya,et al.  Nonparametic estimation of location and dispersion on Riemannian manifolds , 2002 .

[30]  Aldo Tagliani,et al.  Hausdorff moment problem via fractional moments , 2002, Appl. Math. Comput..

[31]  Tomasz J. Kozubowski,et al.  Fractional moment estimation of linnik and mittag-leffler parameters , 2001 .

[32]  H. Le,et al.  Locating Fréchet means with application to shape spaces , 2001, Advances in Applied Probability.

[33]  T. Porter.  SHAPE AND SHAPE THEORY (Wiley Series in Probability and Statistics) , 2000 .

[34]  T. K. Carne,et al.  Shape and Shape Theory , 1999 .

[35]  A. Pakes.  On the convergence of moments of geometric and harmonic means , 1999 .

[36]  E. Lehmann.  Elements of large-sample theory , 1998 .

[37]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[38]  H. Le,et al.  ON THE CONSISTENCY OF PROCRUSTEAN MEAN SHAPES , 1998 .

[39]  P. McCullagh.  Möbius transformation and Cauchy parameter estimation , 1996 .

[40]  G. Casella,et al.  Deriving Generalized Means as Least Squares and Maximum Likelihood Estimates , 1992 .

[41]  W. Kendall.  Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence , 1990 .

[42]  K. L. Saxena,et al.  Estimating parameters from mixed samples using sample fractional moments , 1989 .

[43]  P. Bullen.  Handbook of means and their inequalities , 1987 .

[44]  S. P. Mukherjee,et al.  Estimation of Weibull Parameters Using Fractional Moments , 1984 .

[45]  J. Ghosh,et al.  ON THE VALIDITY OF THE FORMAL EDGEWORTH EXPANSION , 1978 .

[46]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[47]  G. Letac.  Which functions preserve Cauchy laws , 1977 .

[48]  R. Askey.  Orthogonal Polynomials and Special Functions , 1975 .

[49]  G. M. Tallis,et al.  The Use of Fractional Moments for Estimating the Parameters of a Mixed Exponential Distribution , 1968 .

[50]  P. R. Rider,et al.  THE METHOD OF MOMENTS APPLIED TO A MIXTURE OF TWO EXPONENTIAL DISTRIBUTIONS , 1961 .

[51]  Tomislav Buric,et al.  Expectations of large data means , 2023, Journal of Mathematical Inequalities.

[52]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[53]  Henryk Gzyl,et al.  Hausdorff moment problem and fractional moments , 2010, Appl. Math. Comput..

[54]  Henryk Gzyl,et al.  Stieltjes moment problem and fractional moments , 2010, Appl. Math. Comput..

[55]  A. Tagliani,et al.  Optimal predictive densities and fractional moments , 2009 .

[56]  H. Ziezold,et al.  Mean Figures and Mean Shapes Applied to Biological Figure and Shape Distributions in the Plane , 1994 .

[57]  Arak M. Mathai,et al.  On fractional moments of quadratic expressions in normal variables1 , 1991 .

[58]  H. Ziezold.  On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces , 1977 .

[59]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[60]  M. Fréchet.  Les éléments aléatoires de nature quelconque dans un espace distancié , 1948 .

[61]  Mitio Nagumo.  Über eine Klasse der Mittelwerte , 1930 .

[62]  E. Hayes.  Mean Values. , 2022, Science.