In reply to Jo.

My first suggestion is "don't take advice from random people on the internet." Work with a trusted statistician at your institution.

1. No, what you propose is not Winsorization. It is called "single imputation of missing values," where you are substituting the minimum observed value for the missing value.
2. There are many papers for single imputation. I do not recommend its use for the same reasons that I do not recommend mean-value imputation.