ESTIMATION OF MAXIMUM LIKELIHOOD IN THE CONDITIONS OF INSPECTIONS OF THE CURRENT STATE

Abstract

The article discusses the algorithm and methods of statistical processing of one type of grouped data – interval-censored samples with overlapping observation intervals. The samples are censored by the observation time interval of the measured value. The samples consist entirely of such interval observations. The initial information was the data of one-time and multiple inspections of the structure in operation in order to detect damage in the form of cracks. The stage of appearance of the initial crack (macrocrack) in the zone of stress concentration is determined. A macrocrack, for objective reasons, cannot be directly detected during inspections. Examples of describing the time of formation of initial cracks in the form of interval-censored samples with intersecting observation intervals are presented. The main focus was on the development of methods for statistical processing of the studied samples. The maximum likelihood method was applied for parametric estimation of the time of appearance of macrocracks. Basic maximum likelihood equations are obtained for distributions with location and scale parameters. The computational procedure of the accumulation method with the use of Fisher’s information is applied. Expressions for calculating the variances of parameter estimates are obtained. An equation for checking the existence of a solution to the system of likelihood equations is presented. The appearance of macrocracks can have a different physical nature, for example, the accumulation of fatigue or instantaneous damage due to an overload exceeding the calculated one. Different physical nature implies different types of preventive work to prevent them and is described by different probabilistic models. A likelihood ratio criterion is proposed to test the statistical hypothesis about the situation of sudden appearance of macrocracks at a load exceeding the calculated one. The results obtained show the features of the structure of the studied samples and possible methods of statistical processing. Bibliography 11 titles Table 2.