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The present paper examines models for the expected contribution to the X-ray background (XRB) from clusters of galaxies in a flat universe with a spectrum of fluctuations corresponding to the standard cold dark matter picture and for a power-law spectrum P(k) varies as k sup n with n = -2. A self-similar scaling model for clusters proposed by Kaiser (1986) is presented, along with several variants. Photon statistics from synthesized maps of deep Einstein fields show a granularity similar to that deduced by Hamilton and Helfand (1987) from the Einstein data but overproduces by a large factor the number of discrete sources observed as clusters, and the fluctuation limit derived from the recent correlation analyses of the background. Two alternative models are designed to be consistent with the local luminosity function data; for these, expected deep images for the Rosat satellite are generated. Clusters of galaxies should be a significant source of background at energies around 2 keV, and it is proposed that clusters of galaxies contribute up to 30 percent of the unresolved component.

Methods of estimation of parameter values and confidence regions by maximum likelihood and Fisher efficient scores starting from Poisson probabilities are developed for the nonlinear spectral functions commonly encountered in X-ray astronomy. It is argued that these methods offer significant advantages over the commonly used alternatives called minimum chi-squared because they rely on less pervasive statistical approximations and so may be expected to remain valid for data of poorer quality. Extensive numerical simulations of the maximum likelihood method are reported which verify that the best-fit parameter value and confidence region calculations are correct over a wide range of input spectra.