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Abstract
The thesis ties up three inter-related papers on the Leslie Matrix (LM). The three papers deal with various aspects of the approximate LM, which is used for population projections. The first paper provides an insight into a widely used first row formula for Leslie Matrix. It develops a new derivation that we argue is better in terms of its generality and clarity than the existing derivations and investigates the pattern of age-group level inaccuracies that underlie the formula's overall accuracy for populations with stationary age distribution at the start of the projection period. We find that for human populations, the errors under-estimate the contribution of younger mothers and over-estimate that of older mothers.
The second paper derives a new method of deriving intrinsic rate of growth of a stable population using LM. The method developed departs from the tradition of identifying the roots of Lotka equation. It adopts an approach that recovers true r by finding the dominant eigen value of the exact Leslie Matrix (LM) from that of the usual approximate LM. It proves to be more accurate than existing methods in almost all of the 896 cases considered in the paper. The average absolute error is 0.00000036 and for all the cases, the error falls within a very narrow band around 'true r' (the best estimate if lx values for individual age-groups is available) indicating its reliability as an accurate measure of r.
The third paper develops an 'exact' LM for a non-stable population and assesses the level of inaccuracy implied by the use of approximate LM. We do this by deriving exact expressions corresponding to the two approximate formulae used in the approximate LM. We compare the results of a 100-year projection for a set of real world population, mortality and fertility schedules obtained using both the exact LM and the approximate LM. We find the difference in total population projected to be around 3%, which is more than the 1.8% amount by which the 2002 UN population projections (medium variant) for the year 2050 were revised in 2004.
