Statistical Models for Intercepted Data

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Abstract
Consider a population of replaceable items of different ages, and suppose that for each item there is a certain measurable attribute that is a monotone function (perhaps random) of the item's age. To fix ideas, we refer to this attribute as deterioration and we assume that as items age their level of deterioration increases. We further assume that as the level of deterioration hits a certain threshold value (perhaps unknown), the item dies and disappears from the population, so it is not available for sampling any more. Since items that deteriorate at a slower rate live longer, any sample obtained over a short period of time would overrepresent slowly deteriorating items. Consequently, under such circumstances longitudinal studies are often needed to statistically study deterioration as a function of age. This article presents a statistical model that enables us to study the deterioration function, from a single cross-section “snapshot” of the population or some random parts of it. Specifically, this is done by deriving the joint sampling distribution of the item's age and deterioration level at the time of sampling (taking into account the model's assumptions and the bias introduced by the sampling mechanism) and deriving parametric and nonparametric likelihood functions based on this distribution. Potential applications from a variety of fields are discussed. The value of the methodology goes beyond the formulas for bias correction in the specific models. It is well known that statistical estimates based on longitudinal studies often differ from estimates based on cross-section surveys, and this article provides quantitative insight into this discrepancy. Specifically, even in situations where the models do not apply exactly, they could often provide the quantitative (and qualitative) guidelines to the form of bias correction needed for inferences about longitudinal behavior, based on cross-section surveys or surveys conducted over relatively short periods of time.

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