[Assuming Weibull is appropriate] Johnson Kotz and Balakrishnan's book has a lot of ways to estimate Weibull parameters. Some of these do not depend on the data not including zeroes (e.g. using the mean and standard deviation, or using certain percentiles).

[Assuming Weibull is appropriate] Johnson Kotz and Balakrishnan's book has a lot of ways to estimate Weibull parameters. Some of these do not depend on the data not including zeroes (e.g. using the mean and standard deviation, or using certain percentiles).

Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions. New York: Wiley, roughly on page 632.