HALT Testing and several parameters to test again were already menitoned by the other answers. Statistical considerations were described well and all relevant environmental parameters were accounted for.
However it is crucial to clearly define test goals and not to mix up different test methods. That's what I want to add here.
HALT (highly accelerated lifetime testing)
The original purpose of HALT is not to proove the fitness of your product for a certain lifetime under certain environment. In fact the purpose is somehow to disprove it. (Take that sentence with a grain of salt)
At the end of a successful HALT assessment the DUT will be dead. And if an iterative process has been chosen, a lot of DUT will be dead.
The iterative HALT process is a method to find weaknesses. Environmental parameters and working conditions of a device are worsened step by step until failure. The failure is investigated in detail. In most cases this failure is then fixed by ad hoc or preliminary measures just to make a continuation of the test possible. The next failure (hopefully at a different point of the device) is recorded and investigated. This is repeated until the device can't be further fixed for a next stress level. During this process a lot of potential weaknesses arise. These weaknesses can then be assessed and decisions can be made whether the weaknesses are problematic for normal use cases or not.
Type tests
Different from HALT tests are type tests. These focus on single parts or even raw materials like insulators. E.g. prepregs, cores, conformal coatings can be tested. The type test tries to prove the real lifetime. By applying adequate reasoning a tradeoff between stress level and test time is calculated to prove, that a material or part will survive the desired lifetime at expected conditions. For dielectrics a common formula used relates time and voltage like this:
$${U_{op}}^6 * T_{lifetime}={U_{test}}^6 * T_{test}$$
This way, the test time for the dielectrics can be reduced drastically. Be aware, that this test basically will effect the weardown during lifetime. Hence it also will damage the product. After testing with no dielectric breakdown the lifetime is proven. Nevertheless a microscopic analysis of the DUT is recommended.
Tests on products
Hi-Pot tests are often used to find defective parts during production process. IPC recommends voltage levels of 250V up to 500V. This is done typically to prove the electrical safety (to prevent harm to humans) of a product certified for certain insulation classes or other code. For reliability testing, this is however of limited use, as the tests last typically only milliseconds and the extended behaviour under stress can't be monitored.
Real lifetime tests
Are recommended. They may bring up things not related to mean environment conditions or even the other way round. Having a device operate for years under expected conditions may collect events from the outside you haven't thought of in advance. Even if you are only some month ahead of the sold devices operating in the field you still gain the opportunity to react to unforseen failures in some cases.
Updates I really find noteworthy
MTBF
Other answers mention the MTBF and give a proper explanation. However it is not easy to get a reasonable MTBF for your product. If your product consists of subcomponents (and that is to be assumed when it is some electronic device) which may or may not have a given MTBF for themselves it is a fairly difficult task to derive a total MTBF. That's because the failure rates change during lifetime and adhere different distributions. This was, as well explained by other answers.
However if you are interested in the calculation methods used for gaining a prediction of lifetime and failure rates you should read something about
FTA
This is an acronym for fault tree analysis which breaks down your device in a hierarchical way down to the single components. Each component or feature is then assigned a FIT (failure in time) value. The FIT values are then filled into a calculation scheme driven from the hierarchical view. An experienced engineer then can then take into account the cumulative effects of different failure mechanisms by setting parameters accordingly. Single points of failure become more visible and you will see which compoments will cause the device to fail most likely. You can then reselect components which will bring better FIT values. E.g. for capacitors you can select types with higher voltage or temperature rating.
FMEA
If you think you have set up a test strategy and have calculated some statistics, you still may want to assess risks for certain scenarios. The way to go is the Failure Mode and Effects Analysis. This is a method which comes in a lot of different flavours. It can be used during design phase but as well for examining potential problems during manufacturing.
Last but not least, it is very unlikely that you will rule out every failure in advance. You can try and do whatever you want, the most obscure error will strike, when you don't expect it. In most cases you will have limited time, to track the error down. When you are in that pinch, have a look at
8D
which is a superseding methodology to deal with real failures. 8D, properly executed until the 8th step, can help you to
- quickly get the really very cause of the problem (so called root cause)
- find a way to not let the error terminate your business immediately
- implement a solution which is not duct tape
- learn a lot, learn a lot and learn a lot.
I have another very personal message in terms of failure:
Failures are opportunities
Failures give us the chance to learn a lot. Not only about the mistakes made and the pitfalls of some tech thingies, but also about the greater link of different fields of working. Therefore I recommend a mind change. Most people live in fear of failures. I tend to live in fear, too. But what does fear do? In many cases it makes us avoid the perimeters of hardship and hazard unconciously.
I have been and I'm still working on devices with utmost complexity and our team tries to hail every problem or failure we encounter as a possibility to grow better. This way we manage to reduce the uneasy feeling when thinking about the million ways our products may break.
Try to build a culture with your team on how to deal with and think of failures.