In the recent past, we considered some parameter estimation problems, arising from a one dimensional heat diffusion system. In such problems one normally uses finite order models to approximate the infinite order heat diffusion system, and estimate the parameters involved. Therefore, all the parameter estimates are bound to be biased, even if one considers the availability of infinite data sequences. In this paper we perform an analysis of the bias as well as the variance of the parameter estimates, obtained by using certain approximations. We derive some expressions, which we support with simulations.