변화시점(變化時點) 문제(問題)에 대한 통계적(統計的) 검정(檢定)과 사례 분석(事例 分析) = Statistical Testing and Analysis of Case Studies for the Change point Problem

국문 초록 (Abstract)

時間이 흐름에 따라서 順次的으로 추출한 標本들에서 變化가 발생하는 경우 그 變化時點을 알고 있다면 단순한 두 標本集團의 問題가 되어 이는 두 標本 比較問題로 귀착된다. 그러나 變化時點을 모르고 있다는 假定은 새로운 檢定問題를 제기하게 되는데 이러한 問題를 變化時點 問題(change point problem)라 지칭한다. 기존의 많은 分析들은 단순한 數植上의 差異 등으로써 時間의 흐름에 따르는 變化를 서술하는데 그쳐왔다. 따라서 먼저 分析對象 資料들의 變化 有無를 檢定해야 하며 만약 變化가 발생했다면 그 變化時點을 推定할 필요가 있다. 본 논문에서는 이에 대한 統計的 方法을 서술하고 變化時點 問題에 대한 事例들을 分析하였다.

다국어 초록 (Multilingual Abstract)

Consider a specific problem where independent observations are generated sequentially over time and the distribution of this random sequence is subjected to change at several possible points du-ring data collection. Then it is of interest to elicit information from the observations concerning the possibility of such change points in this random sequence. In most of these cases, the numeric di-fferences among observations have been used to describe the change over time. Accordingly, we need to test whether the observations change over time, and if so, to estimate the change poi-nts. There may be several change points, but, here we are concerned with at most one. If the possible change point is known, it becomes a tra-ditional two sample comparison problem. Otherwise, it introduces a new testing problem, which is called the change point problem.
Let X ＜sub＞1＜/sub＞, X ＜sub＞2</sub＞, …, X ＜sub＞N＜/sub＞ be independent random varia-bles that are successively drawn from a conti-nuous population F(·) such that X ＜sub＞1＜/sub＞, X ＜sub＞2</sub＞, "`, X ＜sub＞ r ＜/sub＞∼ F(x) , X＜sub＞ r+l ＜/sub＞, X＜sub＞ r+2＜/sub＞, …, X＜sub＞N＜/sub＞ ∼ F(x- △), -∞<△<∞， where r is the change point, and △ is the loca-tion-shift parameter which is the magnitude of the postulated change after the change point. Two statistical testing methods for this change point problem have been introduced. We are then able to use the parametric testing procedure proposed by Sen and Srivastava if we can assume that F (X) is a normal distribution function. For many cases when we can not make such an assumption, we use the nonparametric testing procedure by Pettitt.
Three case examples for the changes point problem have been studied. With the parametric testing method by Sen and Srivastava, it can be concluded that there was a change in the out-patient average / day between 1969 and 1988, and the change point is estimated to be 1981. Whether there was a change in the death rate from diabe-tes mellitus between 1981 and 1990 was also tes-ted, as was the death rate from malignant neop-lasm of the uterus between 1983 and 1990. Using the nonparametric testing method by Pettitt, it is concluded that there was an increase in the death rate from diabetes mellitus in 1985, and there was no change for the death rate from malignant neoplasm of the uterus.