By Georgy L. Shevlyakov

The sequence is dedicated to the book of high-level monographs and surveys which conceal the total spectrum of likelihood and records. The books of the sequence are addressed to either specialists and complex students.

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It is natural to require that, with the rotation of axes by some angle (the choice of other field bench-marks), the coordinates of a ‘typical’ representative in new axes must be rotated by the same angle. Obviously, such a requirement imposes sufficiently strict restrictions on the structure of a contrast function. 3. Orthogonal equivariancy 35 • if the measurements are made in a plane, then the contrast function depends only on two measurement characteristics: – the Euclidean distance between the measured object and the ‘typical’ representative; – the angle determining the direction at the ‘typical’ representative from the measured object; • if the measured object is of dimension greater than two, then the contrast function depends only on the Euclidean distance between the measured object and the ‘typical’ representative.

To guarantee uniqueness, we may add the requirement of concavity of ρ (u) to the above conditions. 1. 2. 1. 4) with respect to λ , we obtain n i=1 ∂ ϕ (xi − λ , m − λ ) ∂ (xi − λ ) dxi ∂ ϕ (xi − λ , m − λ ) −1 + dλ ∂ (m − λ ) dm −1 dλ = 0, and taking into account the independence of xi and m of λ , we have n i=1 ∂ ϕ (xi − λ , m − λ ) ∂ ϕ (xi − λ , m − λ ) + ∂ (xi − λ ) ∂ (m − λ ) = 0. 12) should hold for each {xi }, so we choose the data in such a way that xi = x, i = 1, …, n. 12) takes the form ∂ ϕ (u, v) ∂ ϕ (u, v) + ∂u ∂v n = 0, where u = xi − λ and v = m − λ .

If the following relation holds for each collection of the data: m(f(x1 ), …, f(xn )) = f(m(x1 , …, xn )). Now we describe some general requirements on the contrast function and the score function. These requirements mainly follow from the intuitively obvious assumptions about the dependence between the ‘typical’ representative and data. 1. Let x1 = x2 = … = xn = x be n equal observations. Then it is natural to assume that m = x: the ‘typical’ representative coincides with the observed value.