Method of moments estimator: what to do when first moment is zero?

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I'm trying to solve a problem in which a function has a pdf with only one unknown parameter. The expected value came out to be zero when I computed it over the range for which the function is defined. Does this mean the estimator for the unknown parameter is equal to zero? Since the first moment is zero, do I need to find the second moment and use that as the estimate instead?

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Yes, if the first moment doesn't depend on the parameter, using the method of moments on the first moment doesn't make sense (and it generally indicates that the sample mean won't contain much useful information about the parameter). Using the second moment seems like a reasonable alternative.