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Ridge estimation (RE) is an alternative method to ordinary least squares when there exists a collinearity problem in a linear regression model. The variance inflator factor (VIF) is applied to test if the problem exists in the original model and is also necessary after applying the ridge estimate to check if the chosen value for parameter k has mitigated the collinearity problem. This paper shows that the application of the original data when working with the ridge estimate leads to non-monotone VIF values. García et al. (2014) showed some problems with the traditional VIF used in RE.We propose an augmented VIF, VIF R .j; k/, associated with RE, which is obtained by standardizing the data before augmenting the model. The VIF R .j; k/ will coincide with the VIF associated with the ordinary least squares estimator when k D 0. The augmented VIF has the very desirable properties of being continuous, monotone in the ridge parameter and higher than one.

The utilization of structural high-performance concrete reinforced with polypropylene fibers in applications demanding long exposure times to high temperatures, such as in case of thermal energy storage systems, is analyzed in this work. Different temperatures: 100, 300, 500 and 700°C (hot), as well as cooled-down states (cold) and different exposure times (6, 24 and 48 h) have been analyzed. The thermogravimetric analysis, fracture behavior, compressive strength, the Young's modulus and tensile strength of concrete were also experimentally determined and subsequently, a comprehensive analysis of the thermal and mechanical behavior of highperformance concrete under different thermal conditions has been carried out, broadening with longer exposure times the available results about the behavior of high-performance fiber reinforced concrete subjected to high-temperatures. When the temperature is increased, the mechanical properties of this concrete diminishes. The results shown that once thermal and moisture equilibriums are reached there is no influence of the exposure time. They provide useful information about the influence of different parameters of fiber-reinforced concrete subjected to high temperatures.

The raise regression has been proposed as an alternative to ordinary least squares estimation when a model presents collinearity. In order to analyze whether the problem has been mitigated, it is necessary to develop measures to detect collinearity after the application of the raise regression. This paper extends the concept of the variance inflation factor to be applied in a raise regression. The relevance of this extension is that it can be applied to determine the raising factor which allows an optimal application of this technique. The mean square error is also calculated since the raise regression provides a biased estimator. The results are illustrated by two empirical examples where the application of the raise estimator is compared to the application of the ridge and Lasso estimators that are commonly applied to estimate models with multicollinearity as an alternative to ordinary least squares.

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