Spline estimate of the time series trend for a random number of data at measurement instants

In many problems of economy, science, and technology one deals with time series. The observed values of a random process $y(t)$ at instants $t_1, t_2, \dots, t_N, \dots$ form a time series. One of the main goals of time series analysis is the problem of separating the trend. It is a systematic component because a selected trend allows one:

• to predict the future based on the knowledge of the past;
• to manage the process generating the series;
• to describe characteristic features of the series.

In the classical theory of time series, the process is measured at regular intervals, exactly one observation at each time instant. However, there exists an organization of the measurement process when the number of measurements is random. Such situations arise especially often in economic systems, for example, in the stock market. This leads to the necessity of developing the theory of time series analysis for a situation where the number of measurements at each time instant is random.
Note that selecting a trend polynomial whose order exceeds four is inexpedient in view of a large error in the evaluation of polynomial coefficients. At the same time, if the number of observations is large, a low order polynomial can be inadequate to describe the true trend. The solution is in a spline estimate of the time series trend.
In this work, we construct a theory of selecting a time series trend by splines of the first, second, and third orders when the number of measurements in each time is random. The estimates for the spline coefficients are obtained in an explicit form. We have investigated statistical characteristics of the obtained estimates.