# Estimation of the parameters of the mixture k ≥ 2 of logarithmic-normal distributions.

• Volume: 3, Issue: 2, page 167-175
• ISSN: 0213-8190

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## Abstract

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In the mixture k ≥ 2 of logarithmic-normal distributions, with density function (1), the parameters μ1, ..., μk satisfying conditions (2) and the parameters p1, ..., pk satisfying conditions (3) are unknown. Using moments of orders r = -k, -k+1, ..., 0, 1, ..., k-1 we get a system of 2k equations (8), an equivalent of matrix equation (10). The equation (13) has exactly one solution with regard to A. If in the equation (13) we substitute the unbiased and consistent estimators D'r for the coefficients Dr, we can get the matrix A with the estimators a'i of the coefficients ai in the equation (11) and the estimators of the roots of the above equations C1 ≤ ... ≤ Ck. Consequently on the basis of (6) we get the estimators μi, i = 1, ..., k. Similarly on the basis of equation (16) and the condition (3) we get the estimators of the remaining parameters. The author does not know any other papers dealing with the estimation of the mixture parameters of finite number of identical distributions where moments of negative order are used.

## How to cite

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Wasilewski, Mariusz J.. "Estimation of the parameters of the mixture k ≥ 2 of logarithmic-normal distributions.." Trabajos de Estadística 3.2 (1988): 167-175. <http://eudml.org/doc/40519>.

@article{Wasilewski1988,
abstract = {In the mixture k ≥ 2 of logarithmic-normal distributions, with density function (1), the parameters μ1, ..., μk satisfying conditions (2) and the parameters p1, ..., pk satisfying conditions (3) are unknown. Using moments of orders r = -k, -k+1, ..., 0, 1, ..., k-1 we get a system of 2k equations (8), an equivalent of matrix equation (10). The equation (13) has exactly one solution with regard to A. If in the equation (13) we substitute the unbiased and consistent estimators D'r for the coefficients Dr, we can get the matrix A with the estimators a'i of the coefficients ai in the equation (11) and the estimators of the roots of the above equations C1 ≤ ... ≤ Ck. Consequently on the basis of (6) we get the estimators μi, i = 1, ..., k. Similarly on the basis of equation (16) and the condition (3) we get the estimators of the remaining parameters. The author does not know any other papers dealing with the estimation of the mixture parameters of finite number of identical distributions where moments of negative order are used.},
author = {Wasilewski, Mariusz J.},
keywords = {Estimación paramétrica; Distribución logarítmica; Distribución normal; Mezclas; Momentos; mixture of logarithmic-normal distributions; moments of negative order; unbiased estimator; consistent estimator; Vandermonde determinant},
language = {eng},
number = {2},
pages = {167-175},
title = {Estimation of the parameters of the mixture k ≥ 2 of logarithmic-normal distributions.},
url = {http://eudml.org/doc/40519},
volume = {3},
year = {1988},
}

TY - JOUR
AU - Wasilewski, Mariusz J.
TI - Estimation of the parameters of the mixture k ≥ 2 of logarithmic-normal distributions.
PY - 1988
VL - 3
IS - 2
SP - 167
EP - 175
AB - In the mixture k ≥ 2 of logarithmic-normal distributions, with density function (1), the parameters μ1, ..., μk satisfying conditions (2) and the parameters p1, ..., pk satisfying conditions (3) are unknown. Using moments of orders r = -k, -k+1, ..., 0, 1, ..., k-1 we get a system of 2k equations (8), an equivalent of matrix equation (10). The equation (13) has exactly one solution with regard to A. If in the equation (13) we substitute the unbiased and consistent estimators D'r for the coefficients Dr, we can get the matrix A with the estimators a'i of the coefficients ai in the equation (11) and the estimators of the roots of the above equations C1 ≤ ... ≤ Ck. Consequently on the basis of (6) we get the estimators μi, i = 1, ..., k. Similarly on the basis of equation (16) and the condition (3) we get the estimators of the remaining parameters. The author does not know any other papers dealing with the estimation of the mixture parameters of finite number of identical distributions where moments of negative order are used.
LA - eng
KW - Estimación paramétrica; Distribución logarítmica; Distribución normal; Mezclas; Momentos; mixture of logarithmic-normal distributions; moments of negative order; unbiased estimator; consistent estimator; Vandermonde determinant
UR - http://eudml.org/doc/40519
ER -

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