  The battle of Kursk between Soviet and German is known to be the biggest tank
battle in the history. The present paper uses the tank and artillery data from
the Kursk database for fitting both forms of homogeneous and heterogeneous
Lanchester model. Under homogeneous form the Soviet (or German) tank casualty
is attributed to only the German(or Soviet) tank engagement. For heterogeneous
form the tank casualty is attributed to both tank and artillery engagements. A
set of differential equations using both forms have been developed, and the
commonly used least square estimation is compared with maximum likelihood
estimation for attrition rates and exponent coefficients. For validating the
models, different goodness-of-fit measures like R2, sum-of-square-residuals
(SSR), root-mean-square error (RMSE), Kolmogorov-Smirnov (KS) and chi-square
statistics are used for comparison. Numerical results suggest the model is
statistically more accurate when each day of the battle is considered as a
mini-battle. The distribution patterns of the SSR and likelihood values with
varying parameters are represented using contour plots and 3D surfaces.
