Origin of Terms in Likelihood Equation (Curious)

[canadaboy]

The following link shows the form of the complete likelihood equation (for complete data, censored data, and interval data). http://www.weibull.com/LifeDataWeb/analysis_parameter_methods.htm#likelihood_function

The terms for interval data refer to probabilities of failing in an interval: [F(I(i+1)) - F(Ii)].

The terms for censored data refer to probabilities of surviving [1-F(j)]

However, the terms for complete data use the pdf function f(t) instead of the cdf function F(t).

Is there a reason for not using the CDF as the basis for the terms in the complete data portion? Does it make the resulting math easier to solve?

Just curious.

[Pantelis]

In the case of interval (as well as right censored) data the probability is over a range of times, not an exact time, thus you look at the area under the pdf (integrate f(t) yielding F(T)) over the appropriate time range. When dealing with the exact time you are looking at a single point, thus no integration is needed. I guess one could cast the entire equation in terms of f(t) - using integrals for the censored items -, but F(t) serves the same purpose.

Hope this helps

[canadaboy]

Yes, but I'm not sure how you find the time to answer so many of the posts on this site, given your position, but I indeed thank you nonetheless. . .

Rewording what you said and asking for confirmation. . .

Would the whole MLE approach still work if F(t) were used for complete data instead of f(t)?

[Pantelis]

Thanks :-)
With regards to your last question.
In short No. F(t) is cumulative and you do not want that, but rather the probability at that exact time. Now, one may argue that since time is continuous you could use an interval that bounds your “exact” time (i.e. t(+/-)Dt) and use the interval formulation using F(t), which then reverts back to f(t) as Dt approaches zero.

[canadaboy]

Nice explanation. Pointwise for complete data makes sense. Thank you. Slowly but surely I'll get this stuff.