When dealing with modeling processes, everything can happen since you cannot predict the performance of a model before you test it on a data set. But you can confidently say that since the models are trained to fit the calibration data it is natural that the calibration efficiency (precision) is higher than validation.

Furthermore, I don't think you would find such a paper. I mean, to be honest, the subject it is not important to try to find a way to analytically prove whether the validation precision of a modeling process will tend to be lower. What will be the gain of such a result? Take also into account that it is not even certain that such proof even exists.

Moreover, to make a robust statistical claim we need to analyze a very huge set of models with a very much more enormous set of data. It is virtually impossible to study every model using every kind of data for every purpose possible.

I think if you stated more precisely, what you exactly you are after you definitely can find your answer here.

Cheers!

Sina Masoumzadeh I understand your points. I remember reading some papers a few years ago which stated that the validation efficiency of their model was lower than the calibration efficiency "which was expected". I can't seem to find them or similar papers now though.

Hello,

Since the dataset of the calibration and validation periods are completely independent, you may get the results with more or less accuracy.

What matters is that your results fall within the acceptable range of accuracy according to model evaluation and the purpose of modelling.

Simply, report as it is.