Did you try this? Jest Test
We had created this for [SOLVED] Problem with testing aurelia components using “jest”
I checked it again for you and works perfectly with TS and Jest. See the result
We had a problem with downloading npm_modules If you want to have a working sample (included npm_modules) without a problem you should download this. (I uploaded for you 
)
Extract that and execute npm test. It should work like above picture
I believe you can learn about Aurelia testing, Typescript and Jest integration with it.