This research article primarily focuses on the estimation of parameters of a linear regression model by the method of ordinary least squares and depicts Gauss-Mark off theorem for linear estimation which is useful to find the BLUE of a linear parametric function of the classical linear regression model. A proof of generalized Gauss-Mark off theorem for linear estimation has been presented in this memoir. Ordinary Least Squares (OLS) regression is one of the major techniques applied to analyse data and forms the basics of many other techniques, e.g. ANOVA and generalized linear models [1]. The use of this method can be extended with the use of dummy variable coding to include grouped explanatory variables [2] and data transformation models [3]. OLS regression is particularly powerful as it relatively easy to check the model assumption such as linearity, constant, variance and the effect of outliers using simple graphical methods [4]. J.T. Kilmer et.al [5] applied OLS method to evolutionary and studies of algometry.
The main objective of this research article is to propose test statistics for testing general linear hypothesis about parameters in stochastics linear regression model using studentized residuals, RLS estimates and unrestricted internally studentized residuals. In 1998, M. Celia Rodriguez -Campos et.al [1] introduced a new test statistics to test the hypothesis of a generalized linear model in a regression context with random design. Li Cai et.al [2] provide a new test statistic for testing linear hypothesis in an OLS regression model that not assume homoscedasticity. P. Balasiddamuni et.al [3] proposed some advanced tools for mathematical and stochastical modelling.
This research article uses Matrix Calculus techniques to study least squares application of nonlinear regression model, sampling distributions of nonlinear least squares estimators of regression parametric vector and error variance and testing of general nonlinear hypothesis on parameters of nonlinear regression model. Arthipova Irina et.al [1], in this paper, discussed some examples of different nonlinear models and the application of OLS (Ordinary Least Squares). MA Tabati et.al (2), proposed a robust alternative technique to OLS nonlinear regression method which provide accurate parameter estimates when outliers and/or influential observations are present. Xu Zheng et.al [3] presented new parametric tests for heteroscedasticity in nonlinear and nonparametric models.
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