A method of maximum likelihood fit to data with non-uniform efficiencies

Estimations of physical parameters using data usually involve non-uniform experimental efficiencies. In this article, a method of maximum likelihood fit is introduced using the efficiency as a weight, while the probability distribution function is kept unaffected by the efficiency. A brief proof and pseudo-experiment studies suggest that this method gives unbiased estimation of parameters. For cases where the probability distribution function can be normalized analytically, this method significant reduces the usage of computing resources.