soxumis saxelmwifo universitetis S r o m e b i VII

studying the smoothness properties of empirical functions. For this reasonwe consider a slightly modified likelihood functionLnx FtFti(0)n1i1 i1ti1FtiF tFtFini ii FtFt f x; . (4)Lemma. Let the following conditions be fulfilled:(a) the distribution function F x, is continuous with respectto both variables and has the continuous derivativeF x, f x, ;x(b) the function L nx, has the absolute maximum n.Then nis an asymptotically consistent and asymptotically effectiveestimator of the true value of the parameter 0.i(1)The proof follows from the respective theorems of [6], [7].i1inj1il 3. Estimation of a mean for a normal distributionwith incomplete observation.t1 222Let X be a normally distributed random value with densityp t e , where is the unknown mean and is known..2Let the interval [ a , b]be inaccessible ( a and b may be infinite, too) forthe observer, and also the quantity of individual observations in this intervalbe unknown. However we have observations outside this interval:X1, X2,...,X n. It is required to estimate by these observations. For thiswe use maximum pseudo-likelihood estimators.To construct the likelihood function, note that if we denote by kthe number of terms of the total sampling which have occurred in [ a , b],kthen will be the frequency of occurrences in the interval [ a , b].k nTherefore, by the Bernoulli-Kholmogorov theorem,k pk na.s.,36