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The phrase "a first estimate" is correct and usable in written English.
It can be used when referring to an initial approximation or calculation of a value or quantity before further analysis or refinement.
Example: "After reviewing the project requirements, we can provide a first estimate of the costs involved."
Alternatives: "an initial estimate" or "a preliminary estimate."
Exact(60)
The least square estimation of the circle enables us to get a first estimate of the sphere radius, R l.
Business investment was faster than first thought, at +2.5% (up from a first estimate of 0.8%).
Ratcliffe, J. H. Geocoding crime and a first estimate of a minimum acceptable hit rate.
According to a first estimate Spain also fell back into recession, which was less surprising.
See articleGermany's GDP grew by 3% last year, according to a first estimate.
A first estimate had put growth at 3.5%, but failed to reckon for rising consumer spending.
This paper provides a first estimate of the potential for mitigation in the whole Spanish agriculture.
A first estimate of the required shielding was obtained by means of a simple analytical model.
"This is a first estimate, and the real number could be one in eight instead of one in four.
The data also shows that consumer spending rose by 3.6%, down from a first estimate of 4.0%.
A first estimate put growth in the last three months of 2010 at a 3.2% annualised pace.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com