There are some facts which could be proven by basic Physics and common sense.
1. IR detection works on line of sight principle by virtue of basic fact that electromagnetic Radiation travels in straight line. At 12,000m of altitude ( max stated cruising Altitude of F-16), An Aircraft could see 391 Km into horizon.
Calculation: d2 = h2 + 2Rh
For utilizing full 700 km capacity, an Aircraft has to be 38 Km up in sky, more than maximum altitude of Eurofighter and one that could be achieved only in no-weapon, minimum fuel configuration.
2. Detection of IR emissions depend on strength of emission. A ballistic missile plum is large and could theoretically be detected from great distance. In addition to that, a BM is gaining altitude thus bringing it within the range of both Airborne and Ground based detectors ( a cornerstone of BMD ). On the other hand, an AAM has very small heat signature and detecting it from large distance is next to impossible, and this cannot be made possible by designing better detectors as this limitation is imposed by signal detection theory. In case of a small emitter, if you try to improve true positive, you could do that only by increasing false positives at same time. This is what makes an F-22 stealthy, not it's shape or coating; if this restriction is not there, enemy would simply change software codes of it's radar to pick small RCS object.All Radars pick up low RCS objects; they just assign them to be background noise. Limitation imposed by SDT ensures that if a conventional radar is set to detect F-22, it would be buzzing all the time.
3. IR emissions depend highly on weather. Cloudy sky would reduce maximum effective distance an AAM launch could be detected.
These are conclusions that could be drawn from simple physics.