Maximum-Likelihood Estimation of the Number-Flux Distribution of Radio Sources in the Presence of Noise and Confusion

The application of the method of maximum likelinood to the determination of the slope of the number- flux-density relationship is extended to include the presence of experimental errors in the flux-density measurements. It is shown that these experimental errors have a significant effect on the number counts at higher natios of flux density to error than is often recognized. The case of noise-limited fiux-density measurements is treated in some detail, and it is found that, provided the lower limit of a survey is chosen to be at least five times the rms noise, the enhancement in the source density as a function of fiux density can be readily calculated. For the case of significart confusion errors in the flux-density measurements the importance of a Monte Carlo approach is emphasized. Several methods that have been used previously are discussed and a number of shortcomings noted. (auth)