THE ACCURACY BY \(L_1\) METRIC OF KERNEL ESTIMATES FOR DENSITY CONSTRUCTED BY OBSERVATIONS WITH A CHAIN DEPENDENCE

A kernel estimate of the Rosenblatt-Parzen type of unknown density of a random variable X is constructed given the observations with a chain dependence \(X_i\) $X_i$ , \(i\ge 1\) $i\ge 1$ . The \(L_1\) $L_1$ -accuracy of the estimate is established. The specific form of the suggested estimate and the accuracy of its approximation are obtained using the Bartlett kernel. The derived result is refined in the case of the smoothing coefficient \(a_n=\sqrt{n}\) $a_n=\sqrt{n}$ . Finally, one example of a sequence with a chain dependence is constructed.