Parameter estimation based on grouped or continuous data for truncated exponential distributions

Parameter estimation based on truncated data is dealt with; the data are assumed to obey truncated exponential distributions with a variety of truncation time-a, data are obtained by truncation time b(1), a(2) data are obtained by truncation time b(2) and so on, whereas the underlying distribution is the same exponential one. The purpose of the present paper is to give existence conditions of the maximum likelihood estimators (MLEs) and to show some properties of the MLEs in two cases: 1) the grouped and truncated data are given (that is, the data each express the number of the data value falling in a corresponding subinterval), 2) the continuous and truncated data are given.