Estimation of the central moments of a random vector based on the  definition of the power of a vector

Katarzyna  Budny

doi:https://doi.org/10.59170/stattrans-2017-001

Estimation of the central moments of a random vector based on the definition of the power of a vector

Katarzyna Budny Cracow University of Economics. Statistics in Transition new series, vol. 18, 2017, 1, pages: 7-26 Published online: 1 March 2017 https://doi.org/10.59170/stattrans-2017-001

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ABSTRACT

The moments of a random vector based on the definition of the power of a vector, proposed by J. Tatar, are scalar and vector characteristics of a multivariate distribution. Analogously to the univariate case, we distinguish the uncorrected and the central moments of a random vector. Other characteristics of a multivariate distribution, i.e. an index of skewness and kurtosis, have been introduced by using the central moments of a random vector. For the application of the mentioned quantities for the analysis of multivariate empirical data, it appears desirable to construct their respective estimators. This paper presents the consistent estimators of the central moments of a random vector, for which essential characteristics have been found, such as a mean vector and a mean squared error. In these formulas, the relevant orders of approximation have been taken into account.

KEYWORDS

central moment of a random vector, estimator, multivariate distribution, power of a vector.

REFERENCES

BILODEAU, M., BRENNER, D., (1999). Theory of Multivariate Statistics. Springer-Verlag, New York.

BUDNY, K., (2009). Kurtoza wektora losowego. Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu. Ekonometria (26), Vol. 76, s. 44–54. [Kurtosis of a random vector. Research Papers of Wrocław University of Economics. Ekonometria (26), Vol. 76, pp. 44–54].

BUDNY, K., (2012). Kurtoza wektora losowego o wielowymiarowym rozkładzie normalnym. w: S. FORLICZ (red.). Zastosowanie metod ilościowych w finansach i ubezpieczeniach. CeDeWu, Warszawa, s. 41–54. [Kurtosis of normally distributed random vector. in: S. FORLICZ (ed.). The application of quantitative methods in finance and insurance. CeDeWu, Warsaw, pp. 41–54].

BUDNY, K., (2014). Estymacja momentów zwykłych wektora losowego opartych na definicji potęgi wektora. Folia Oeconomica Cracoviensia, Vol. 55, s. 81–96. [Estimation of the raw moments of a random vector based on the definition of the power of a vector. Folia Oeconomica Cracoviensia, Vol. 55, pp. 81–96].

BUDNY, K., TATAR, J., (2009). Kurtosis of a random vector – special types of distributions. Statistics in Transiton - new series, Vol. 10, No. 3, pp. 445–456.

BUDNY, K., TATAR, J., (2014). Charakterystyki wielowymiarowych wielkości finansowych oparte na definicji potęgi wektora. Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach, Vol. 189, s. 27–39. [Characteristics of multivariate financial items based on definition of the power of a vector. Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach, Vol. 189, pp. 27–39].

BUDNY, K., SZKLARSKA, M., TATAR, J., (2014). Wielowymiarowa analiza sytuacji społeczno-demograficznej Polski. Studia i Materiały "Miscellanea Oeconomicae", Rok 18, Nr 1/2014, s. 273–288. [Multivariate analysis of the socio-demographic situation of Poland, Studia i Materiały "Miscellanea Oeconomicae", Vol. 18, No. 1/2014, pp. 273–288].

CRÁMER, H., (1958). Metody matematyczne w statystyce. Wyd.1. PWN, Warszawa. [Mathematical Methods of Statistics. 1st ed. PWN, Warsaw].

FUJIKOSHI, Y., ULYANOV, V. V., SHIMIZU, R., (2010). Mutivariate Statistics: high-dimensional and large-sample approximations, John Wiley & Sons, Inc.

HOLMQUIST, B., (1988). Moments and cumulants of the multivariate normal distribution, Stochastic Analysis and Applications, 6, pp. 273–278.

ISSERLIS, L., (1918). On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables, Biometrika, Vol. 12, No. 1/2, Nov.

JAKUBOWSKI, J., SZTENCEL, R., (2004). Wstęp do rachunku prawdopodobieństwa. Wyd. 3. Script, Warszawa. [Introduction to Probability Theory. 3rd ed. Script, Warsaw].

JOHNSON, N. J., KOTZ, S., KEMP, A. W., (1992). Univariate discrete distributions:, 2nd ed. John Wiley & Sons, Inc.

OSIEWALSKI, J., TATAR, J., (1999). Multivariate Chebyshev inequality based on a new definition of moments of a random vector, Statistical Review, Vol. 46, No. 2, pp. 257–260.

SHAO, J., (2003). Mathematical statistics, 2nd ed. Springer.

TATAR, J., (1996). O niektórych miarach rozproszenia rozkładów prawdopodobieństwa. Przegląd Statystyczny, Vol. 43, No. 3–4, s. 267–274. [On certain measures of the diffusion of probability distribution. Statistical Review, Vol. 43, No. 3–4, pp. 267–274].

TATAR, J. (1999). Moments of a random variable in a Hilbert space. Statistical Review, Vol. 46, No. 2, pp. 261–271.

TATAR, J., (2000). Asymetria wielowymiarowych rozkładów prawdopodobieństwa. Materiały z XXXV Konferencji Statystyków, Ekonometryków i Matematyków Akademii Ekonomicznych Polski Południowej. [Asymmetry of multivariate probability distributions. Conference Proceedings, Cracow University of Economics].

TATAR, J., (2002). Nierówność Lapunowa dla wielowymiarowych rozkładów prawdopodobieństwa. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Krakowie, Vol. 549, s. 5–10. [Lapunav's inequality for multivariate probability distribution. Cracow Review of Economics and Management, Vol. 549, pp.5–10].

TATAR, J., (2008). Miary zależności wektorów losowych o różnych wymiarach. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Krakowie, Vol. 780, s. 53–60 [Measures of dependence of random vectors of different sizes. Cracow Review of Economics and Management, Vol. 780, pp. 53–60