Hypothesis Testing in Regression Models with Fuzzy Data

Abstract

Regression analysis is in wide use in scientific investigation. Fuzzy linear regression is an actively developing area of research since in many real-life situations dependent or independent variables are not given as real numbers. The regression problem with fuzzy data is treated in the literature with different kinds of input-output data.

We consider the model y_i = A + bx_i + ε_i, i = 1,…,n, where A, x₁,…,x_n – fuzzy numbers; b – real number; ε₁,…,ε_n, y₁,…,y_n – fuzzy random variables.

In [Veldyaksov, Shvedov, 2014] A,b estimates were proposed, using ordinary least squares approach. The estimates rely on calculus of variations, and on previous research conducted for the case when A is a crisp (real) number.

Estimate for b is also proposed in [Veldyaksov, Shvedov, 2014]. In first part of this paper, we prove that this estimate is unbiased. We use new fuzzy random variables definition from [Shvedov, 2013].

In second part of this paper we refer to a number of numerical tests to compare confidence intervals for b coefficient, calculated both using traditional approach, and bootstrap approach. We show that the intervals become closer, as number of observations grows. We also propose a procedure for hypothesis testing for b coefficient in regression models with fuzzy data.