Theory and application of empirical distribution functions in the inter-satellite calibration problem

In any attempt to construct proper time series when an overlapping period between two data sets exists, one has to observe the behavior of these data sets during the overlap period. In this manuscript we consider the problem of inter-satellite calibration. We consider a pair of distributions of observations which correspond to atmospheric temperatures measured from different satellites. Given the measurements from one satellite and the measurements from the other satellite, we estimate the function that relates these pairs of observations. This function depends upon the differences between corresponding quantiles from the two satellites, and therefore the techniques that will be used for the estimation of this function are based on the theory of the asymptotic behavior of sample quantiles. Using the asymptotic normality of the difference between two sample quantiles we construct a confidence region for the unknown difference function at fixed quantiles. Moreover, we can estimate the sample size needed such that the difference between the true and the estimated difference function is smaller than a specified value. The exact confidence coefficient of such a confidence region is derived and compared to a theoretical one. The degree of normality and continuity of the real data will be used as a guide as to whether the normal theory is appropriate or whether we should look for alternative ways for constructing a confidence region. The alternative method shown in this paper is based on the behavior of the data. It provides a confidence region which has satisfactory performance when compared to the one derived by using normal theory.