I will also answer this question, since I had a huge break trying to do this job, and I finally did it. In addition, I am not a major expert in the fine details of HTTP, so I will give an unprofessional answer.
Firstly, this is how I checked if my GZipFilter is working or not. Launched Firefox, made sure that I have a Firebug addon, launched the Firebug addon, went to the "Net" tab. Then I turned to the URL, which should return a GZipped response. Here is what Firebug shows:

The Size column indicates the size of the response. If you hover over the Size label with the mouse, it will tell you that if the response is compressed, then the compressed size of the response will be displayed.
All this was done with the Jetty GZip enabled. Then I removed the GZip filter declaration from my web.xml, restarted Jetty and retried the test. This time around the answer was the same size as before, which clearly indicates that GZip compression is not working.
After some trial and error, I did this in Firebug under the “Request Headers” section to see the value for the “Accept” header. I noticed that there were such values as "application / xml" and "text / xml", but the way I set my original GZIp filter parameter "mimeTypes" contained only "text / xml" (and was missing) the application / XML "). It was configured like this:
<filter> <filter-name>GzipFilter</filter-name> <filter-class>org.eclipse.jetty.servlets.GzipFilter</filter-class> <init-param> <param-name>mimeTypes</param-name> <param-value>text/html,text/plain,text/xml,application/xhtml+xml,text/css,application/javascript,image/svg+xml,application/json,application/xml; charset=UTF-8</param-value> </init-param> </filter> <filter-mapping> <filter-name>GzipFilter</filter-name> <url-pattern>/*</url-pattern> </filter-mapping>
After adding the value "application / xml" to the list:
<filter> <filter-name>GzipFilter</filter-name> <filter-class>org.eclipse.jetty.servlets.GzipFilter</filter-class> <init-param> <param-name>mimeTypes</param-name> <param-value>text/html,text/plain,text/xml,application/xhtml+xml,application/xml,text/css,application/javascript,image/svg+xml,application/json,application/xml; charset=UTF-8</param-value> </init-param> </filter> <filter-mapping> <filter-name>GzipFilter</filter-name> <url-pattern>/*</url-pattern> </filter-mapping>
I reinstalled my previous test, and now I'm pretty sure that the response size is smaller:

Also note that now the response headers contain additional fields called "Content-Encoding" with a value of "gzip".
So, basically the idea is to check what values you send in the "Accept" request header, and make sure that all these values are configured in the initialization parameter of the "mimeTypes" GZip file.