@ARTICLE{26543120_137780228_2014, 
	author = {Vasiliy Vel'dyaksov and Alexey Shvedov}, 
	keywords = {, simple regression, fuzzy random variables, hypothesis testing, confidence intervalsbootstrap percentile intervals},
	title = {Hypothesis Testing in Regression Models with Fuzzy Data},
	journal = {HSE Economic Journal },
	year = {2014},
	volume = {18},
	number = {3},
	pages = {508-523},
	url = {https://ej.hse.ru/en/2014-18-3/137780228.html},
	publisher = {},
	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 yi = A + bxi + εi, i = 1,…,n, where A, x1,…,xn - fuzzy numbers; b - real number; ε1,…,εn, y1,…,yn - 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.},
	annote = {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 yi = A + bxi + εi, i = 1,…,n, where A, x1,…,xn - fuzzy numbers; b - real number; ε1,…,εn, y1,…,yn - 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.}
}