I want to use a Weibull analysis for demonstrating the reliability of a piece of equipment that will be set to cycle. How do I select the proper scale and shape parameters for the distribution if I have no previous reliability data for the equipment.
Weibull analysis is often used to model failure data as part of reliability engineering. With failure-time data available you can use nonlinear fitting or graphical methods to estimate the parameters of the distribution. You don't yet, however, have failure information on this equipment.
One advantage of the Weibull distribution is its reproductive property, which is useful when considering failure times under the weakest-link principle: that is, the failure time of a device composed of multiple components will be the time that any component fails. Then if you have information about the Weibull distributions of the component failure times you can (in some circumstances) combine those distributions into an overall Weibull distribution for the device as a whole. That's shown for example in this document.
So if the 5 parts for which you have reliability information are the least reliable of your equipment's parts, you might be able to use that reproductive property to estimate a Weibull distribution for your equipment's failure time. Otherwise you might have to wait until you get multiple equipment failures on your own. There do seem to be Bayesian approaches to Weibull failure modeling that would allow you to update estimates as you proceed.