This thesis investigates diagnostic methods in parametric regression models with some alternatives to ordinary least squares estimator. Of which, the Liu estimator has been developed as an alternative to the ordinary least squares estimator in the presence of collinearity among the regressors in linear regression models. Firstly, we presented the DFFITS and different versions of the Cook distance analogous to the ones given for the ordinary linear regression models of each individual observation on the Liu estimates. We suggested a version of the Cook distance based on one-step approximation. The mean shift outlier model for the Liu regression has also been investigated. Moreover, using the Sherman-Morrison-Woodbury theorem, we proposed approximate versions of the DFFITS and the Cook distance. Next, the pseudo likelihood function is given for estimating the regression coefficients and shape parameter as well as to establish local influence diagnostics. The normal curvatures of local influence are deduced under arbitrary perturbation schemes to detect influential observations. Then, we discussed the assessment of local influence under the modified ridge regression with normal error distribution. Using a pseudo-likelihood function, we expressed the normal curvatures of local influence for useful perturbation schemes in interpretable forms.Finally, local influence diagnostic methods in the modified ridge regression are deduced under heavy-tailed error distribution. The methods of analysis we presented have considerable significance for the detection of influential observations in the Liu and the modified ridge regression models. Examples using real-life data sets are used to illustrate the proposed methodology.