This dissertation propose two new kernel functions to estimate a density unction with small biases and errors .The errors of the estimats are measured using mean squared error (MSEf (f(x,X ))), mean integrated squared error (MISE(f) and asymptotic mean integrated squared error (AMISE(f) ), and The estimates of these error measures are also given .The AMISE0(f) of the density estimates are compared to the kernel density estimates for the uniform, Epanechnikov and Gaussian kernel functions. One kernel function is derived by minimizing the sum of ...