Estimating Complete Life Tables for Populations with Limited Size: From Graduation to Equivalent Construction

Reliable complete life tables cannot be directly calculated for small populations, but they are necessary to improve population projections that are the basis of various programs such as social security. Estimating complete life tables for small populations is a solution and is related to mortality graduation. In the rich literature of mortality graduation, various life-table functions are graduated using different smoothing models, but a definition of the true age patterns of life-table functions can hardly be found. The vagueness about the true age patterns of life-table functions is the essential difficulty in mortality graduation. This article removes the essential difficulty by analyzing the computational structure of a life table and by constructing a complete life table that is equivalent to the input abridged life table. Inputting an abridged life table to equivalently construct a complete life table refers to that, using the constructed complete life table to compute an abridged life table, the result is identical to the input life table. For a constructed complete life table, the true age pattern between the abridged age groups is defined as the observed age pattern of the input abridged life table. Within the abridged age groups, the true age pattern is defined as the smoothest pattern using the variation of functionals. The basis of these definitions is the law of large numbers. Utilizing these definitions, equivalent constructions and construction-based graduations are proposed and applied successfully to the data of women in Iceland and Estonia.