Any additional calls to scripts will slow down your site. However, Google Analytics instructs you to place it in a specific place so that it isn't loaded until the page has loaded. (It used to be before the </body> tag but I believe it's now supposed to be the last <script> in the <head> tag.) Don't worry about it too much; the benefits of analytics will far outweigh the extra call to a remote file.
Focus on other optimizations (database queries, CSS sprites, fewer HTTP requests). Analytics is necessary in today's site market and is indispensable; IMO it is not an option to forgo it.
As far as having your own "statistics app," I assume you're talking about building your own proprietary statistics codebase? I would discourage that, because it takes a lot of time and effort and in the end you will not have the same optimizations that Google has employed an entire project's worth of software engineers to make. Remember that while it's always great to create your own product, you don't have to reinvent the wheel, especially when it comes to things like this that have many sensible drop-in solutions that are widely available for free.
With respect to non-Google analytics solutions, one other of note is Clicky. I'm not as experienced with it as I am with GA, but I've heard many reviews that it is more precise and more informative than GA. However, just as an end-user browsing the web I've noticed a lot of times that its calls to Clicky's website do tend to slow down pages, and noticeably so; I cannot really say that I have seen the same effect with GA.
One last thing I would caution against is this: Do not employ more than one analytics solution unless you are trying to find the best one to suit your needs. It's just overkill to run two remotely-hosted analytics solutions on every single one of your pages, so what I would encourage you to do is try out a few for the first few weeks or so of your site (yes, pages will slow down during this trial phase) and then simply stick with the one that you like best. That will also give you the added benefit of being able to see first-hand what the speed implications are on your unique hosting environment for each script.
Here's some other analytics solutions that you might check out: