Распределения, обладающие свойством почти отсутствия памяти (ALM) и их приложения

Данная статья обобщает фундаментальные исследования, посвященные классу вероятностных распределений, известных как распределения с почти отсутствием памяти (ALM). В то время как классическое экспоненциальное распределение однозначно характеризуется свойством отсутствия памяти в любом возрасте, распределения ALM проявляют это свойство выборочно, в возрастах, образующих дискретную последовательность {0, c, 2c, ...} для положительной константы c. Это обобщение предоставляет мощный и универсальный аппарат для моделирования времени жизни, времени до отказа и времени ожидания в периодических, нестационарных условиях. Мы представляем всестороннюю характеристику распределений ALM через несколько эквивалентных свойств: специфическую аналитическую форму функции распределения, периодическую интенсивность отказов и вероятностное представление, в котором случайная величина разлагается в сумму усеченной на [0, c) компоненты и геометрически распределенной компоненты, масштабированной на c. Это разложение имеет ключевое значение как для теоретического анализа, так и для практических приложений. Строго устанавливается внутренняя связь между распределениями ALM и периодическими неоднородными пуассоновскими процессами. Кроме того, в статье рассматриваются важные статистические аспекты, включая оценку параметров методами максимального правдоподобия и моментов, а также проверку гипотез для различения моделей. Применимость теории ALM демонстрируется в различных областях, включая надежность технических систем, работающих в условиях циклических нагрузок, страхование для моделирования убытков с сезонной структурой и науки об окружающей среде для изучения явлений, управляемых периодическими силами. Заполняя пробел между теоретической вероятностью и необходимостью моделирования реальной периодичности, распределения ALM предлагают надежный и математически элегантный инструмент для анализа в нестационарных условиях.

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