## FANDOM

32.681 Seiten

 Typus Verschleierung Bearbeiter Hindemith Gesichtet
Untersuchte Arbeit:
Seite: 97, Zeilen: 10-33
Quelle: Daley VereJones 2003
Seite(n): 231, 248, 249, Zeilen: 231: 12-17; 248: 38; 249: 1-3, 6-25
The main aim of the Proposition 4.3.14 was to make conditional the distribution of the current marks as time progresses, on marks and time points of all preceding events, i.e., the full set of the time points (0, T), irrespective of the marks and their relative positions in time.

Another view to look at is through the hazard functions. Instead of specifying the conditional densities pn(∙∣∙) as in (4.3.10) we express them in terms of their hazard functions

[GLEICHUNGEN, identisch zur Quelle, siehe link ]

Using the conditioning in the hazard functions may now include the values of the preceeding marks as well as the length of the current and preceeding intervals. Thus, all the information is summarized in the internal history H ≡ {Ht : t ≥ 0} of the process and of this form the amalgam of hazard function functions [sic] and mark densities can be represented as a single composite function for the MPP as follows

[GLEICHUNG, identisch zur Quelle, siehe link ]

where h1(t) is the hazard function for the location of the initial point, h2(t∣(t1, x1)) the hazard function for the location of the second point conditioned by the location of the first point and the value of the first mark, and so on, while f1(x∣t) is the density for the first mark given its location, and so on.

Hence a regular MPP N on R+ × X can be represented piecewise by the function λ*(t, x) named the conditional intensity function with respect to its internal history H.

Predictability, as it was defined in (4.3.5), is important in it that the hazard functions refer to the risk at the end of a time interval, not at the beginning of the next interval.

[Seite 248: 38]

In the equations above we condition the distribution of the current mark, as

[Seite 249: 1-3]

time progresses, on both marks and time points of all preceding events; in the proposition, we condition on the full set of time points in (0, T), irrespective of the marks and of their relative positions in time.

[Seite 231: 12-17]

We now make a seemingly innocuous but critical shift of view. Instead of specifying the conditional densities pn(∙∣∙) directly, we express them in terms of their hazard functions

[GLEICHUNGEN, siehe (7.2.2) ]

[Seite 249: 6-25]

The conditioning in the hazard functions may now include the values of the preceding marks as well as the length of the current and preceding intervals. All this information is collected into the internal history H ≡ {Ht : t ≥ 0} of the process so that the amalgam of hazard functions and mark densities can be represented as a single composite function for the MPP, namely

[GLEICHUNG, siehe (7.3.2) ]

where h1(t) is the hazard function for the location of the initial point, h2(t∣(t1, κ1)) the hazard function for the location of the second point conditioned by the location of the first point and the value of the first mark, and so on, while f1(κ∣t) is the density for the first mark given its location, and so on.

Definition 7.3.II. Let N be a regular MPP on R+ × K. The conditional intensity function for N, with respect to its internal history H, is the representative function λ*(t, κ) defined piecewise by (7.3.2).

Predictability is again important in that the hazard functions refer to the risk at the end of a time interval, not at the beginning of the next time interval, [...]

 Anmerkungen Ein Quellenverweis fehlt. Sichter (Hindemith), WiseWoman