METHODS OF CHARACTERISTIC FUNCTIONS IN PROBLEMS OF STATISTICAL ESTIMATION BY CENSORED SAMPLES

摘要

Let X be a random variable with an unknown distribution function F(x) = P(X < x), x ∈ R, and with the characteristic function C(t)= ∫ from -∞ to ∞ of e~(itx) dF(x), t∈R.(1) In the case where there exists a total sample X~((n)) = (X_1,X_2,...,X_n) from independent observations over the random variable X, in the capacity of an estimation of a characteristic function C(t) an empirical characteristic function C_n~(Emp)=∫ from -∞ to ∞ of e~(itx) dF_n~(Emp)(x), t∈R.(2) is used, where F_n~(Emp)=1 Σ from i=1 to n of I(X_i < x) (3) is an empirical distribution function and I(A) is an indicator of the event A. The empirical distribution function has a lot of very good properties such as unbiasedness, a uniformly strong convergence, a convergence to the Gauss process on every finite interval for t, etc. (see). It is successfully used in estimation tasks of unknown variables of the distribution F, especially in such cases where the famous moment method and the maximum plausibility method are not easily realized or are not suitable. This work is devoted to the consideration of similar tasks, when the random variable X is randomly censored on two sides. At the same time, for the characteristic function (1) we constructed an estimation that is used further for composing estimation equations for unknown parameters. As an example, a method of estimation of a scale parameter of the centralized Cauchy distribution was considered.