The Arctangent Regression and the Estimation of Parameters of the Cauchy Distribution
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The arctangent regression and the estimation of parameters of the Cauchy distribution Ivan H. Krykun (Presented by I. I. Skrypnik) Abstract. Some estimates of the parameters of a nonlinear regression between the variables of X and Y are constructed for the arctangent as a regression function. The obtained estimates are used to evaluate the unknown parameters of the Cauchy distribution. Computer simulations are performed, and the estimates are compared with another estimates such as the quantile ones, maximum liklyhood estimates, and some others. The confidence intervals for parameters of the Cauchy distribution are obtained. Keywords. Nonlinear regression, Cauchy distribution, estimation of parameters of the distribution, confidence interval.
Introduction In practice and in theory, we frequently deal with the interdependences between some variables X and Y which are described by the equations of a nonlinear regression, namely, polynomial, exponential, hyperbolic, and logarithmic ones. At the same time, the author did not meet the arctangent regression, though the properties of an arctangent such as the continuity, differentiability, monotonicity, boundedness, and definiteness on the whole axis are convenient for the usage in practice. So, it is natural to assume the expediency to consider the arctangent regression. The examples of its possible application will be given in Proposition 4. The standard technique of evaluation of parameters of the arctangent regression by the method of least squares gives no possibility to find those estimates explicitly, but this model after a nonlinear change usual for the nonlinear regression analysis can be reduced to a linear one. From whence, it is possible to get the estimates of parameters of the arctangent regression. We recall that a random variable satisfies the Cauchy distribution with the parameters (a, γ) (where γ > 0), if its probability density function has the form [1, p. 123] f (x) =
1 γ , −∞ < x < ∞, π (x − a)2 + γ 2
and cumulative distribution function, respectively, F (x) =
1 1 x−a + arctan . 2 π γ
The parameters a and γ are called, respectively, a location parameter and a scale one. It is known [2, p. 112] that the Cauchy distribution has neither mathematical expectation nor moments of high orders. This distribution belongs to the distributions with “heavy tails” for which the law of large numbers does not hold. In particular, the arithmetic mean of random variables obeying Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 17, No. 2, pp. 196–214 April–June, 2020. Original article submitted September 30, 2019 c 2020 Springer Science+Business Media, LLC 1072 – 3374/20/2495–0739 ⃝
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the Cauchy distribution is also a random variable satisfying the Cauchy distribution with the same parameters of location and scale [2, p. 114]. Several approaches to the determination of parameters of the Cauchy distribution are available in the literature. But, by virtue of the above-mentioned properties of the Cauchy distribution, those approaches give rather nonexact result
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