Equality of MNC and Aitken Estimations of Linear Regression Model Paper in the Case of Heteroscedastic Deviations

• Date: 2019
• Main Author: Савкіна, Марта Юріївна
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2019
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/174206
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

At the paper in the case of heteroscedastic independent deviations a linear regression model whose function has the form  where  and  unknown parameters, is studied. Approximate values (observations) of functions  are registered at equidistant points of the segment  We formulate Theorem 1, which gives conditions on the variances of deviations, in which the Aitken estimation of parameter  coincides with its estimation of MNCs. Under these conditions, the Aitken and MNC estimations of the parameter  will not coincide. We also formulate Theorem 2, which gives the conditions for the coincidence of the Aitken estimation and the MNC estimation of parameter  Based on Theorems 1 and 2, in this paper the properties of variances of deviations that give equality with these estimations separately for parameter  and for parameter  are investigated. It is shown that for equality estimations of Aitken and MNC of the parameter the deviations will have the largest and smallest variance in two adjacent observation points located in the middle of the segment [0, 1], for the equality estimations of the parameter  — in the neighborhood of the point 2/3. The asymptotic values of the variances of all deviations are found, if the ratio of the largest to the smallest variance goes to infinity. It is proved that in this case, the variances of all deviations will be no more the smallest variance than 3 times for parameter  and not more than 5 times for parameter