Applying the maximum likelihood method for valuation

We consider various applications of the maximum likelihood method to problems of mathematical statistics and asset valuation. Most of the known valuation methods are based on the assumption of the normal distribution of prices of identical assets. However, this assumption is difficult to substantiate, especially with a small sample size or in the presence of "outliers". We proceed from a less severe assumption about the logarithmic convexity of the price distribution density and present a method for constructing the most plausible log-convex distribution density of a random variable from its independent realizations. In the valuation tasks, the mode of a corresponding price distribution of identical assets allows a natural interpretation as an asset value, since, according to valuation standards, the market value of an asset is its most likely (most probable) price in a transaction made under certain conditions. We show that comparing the modal and mean values of the assets prices allows the appraiser to characterize the state of the market of such assets. In a market approach to the valuation of assets, the value of an asset is determined from the data on the prices of its analogues. In such cases, appraisers build several options for the parametric regression of the price of an asset on its characteristics, which differ in the type of regression function. Usually, at the same time, appraisers use the criterion of minimum variance of deviations, oriented to their normal distribution. In our opinion, the maximum likelihood criterion is more appropriate here. We use it to simultaneously estimate the calibration parameters of the regression dependence and the log-convex density of the distribution of deviations from this dependence, as well as to construct convex nonparametric regression dependencies and modal regression dependencies (which is especially important for valuation).