A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models

Abstract

Linear regression is a simple but powerful tool for prediction. However, it still suffers from some deficiencies, which are related to the assumptions made when using a model like normality of residuals, uncorrelated errors, where the mean of residuals should be zero. Sometimes these assumptions are violated or partially violated, thereby leading to uncertainties or unreliability in the predictions. This paper introduces a new method to account for uncertainty in the residuals of a linear regression model. First, the error in the estimation of the dependent variable is calculated and transformed to a fuzzy number, and this fuzzy error is then added to the original crisp prediction, thereby resulting in a fuzzy prediction. The results are compared to a fuzzy linear regression with crisp input and fuzzy output, in terms of their ability to represent uncertainty in prediction.

Pricing Life Contingencies Linked to Impaired Life Expectancies Using Intuitionistic Fuzzy Parameters

“…Of course, these approaches would yield fuzzy predictions for yield spread. Alternatively, the original econometric models of Braun and Xu (2020) and Kung et al (2021) produce predictions through probabilistic confidence intervals that can be used to fit an FN representation by means of a probability-possibility transformation (Adjenughwure and Papadopoulos 2020;Al-Kandari et al 2020), which may be the basis for inducing intuitionistic quantifications by Definition 7 and Remark 7.…”

“…The results of Couso et al (2001), Dubois et al (2004), and Sfiris and Papadopoulos (2014) facilitate the inference of fuzzy numbers using probabilistic confidence intervals. These findings were employed in a regression framework by Adjenughwure and Papadopoulos (2020) and Al-Kandari et al (2020), where the variables of interest were predicted by fuzzy numbers induced from probabilistic confidence interval estimates derived from statistical regression. Remark 6 shows that TIFN can be induced from the estimated TFN.…”

Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance

“…In [50] it is suggested that by placing those confidence intervals one on top of the other, a FN close to triangular-shaped is obtained. So, we point out two alternatives to apply that idea: [49,51] propose making fuzzy predictions from statistical linear regression models. In [49] it is stated that a 1 − a statistical confidence interval of coefficients adjusted with a linear regression may be interpreted as the a-cut of a FN for these coefficients.…”

“…A similar approach may be developed from the results in [51]. However, in this case, it must be taken into account that their approach to making fuzzy predictions from a statistical regression is built up from the interval predictions of residuals instead of using interval estimates of coefficients.…”

A Fuzzy-Random Extension of Jamshidian’s Bond Option Pricing Model and Compatible One-Factor Term Structure Models

“…On the one hand, we will take advantage of the fact that the α-cut of a fuzzy number can be interpreted as a confidence interval in which the probability that the variable of interest is 1 − α [20]. This allows us to use the approaches of [21][22][23], who propose quantifying estimates from conventional regression models through fuzzy numbers induced by statistical confidence intervals, and [24], who use an analogous procedure to model uncertainty of variance by using fuzzy numbers. On the other hand, those fuzzy estimates will be grounded in the standard parameter-calibration methodologies based on least-squares regression [6].…”