Question

3. Classifying Life Distributions. Suppose a continuous lifetime T has survival function S(O), hazard function h(i), cumulative hazard function (1), and mean residual life m(t). Consider the following properties that I might have: I. h(t) is nondecreasing for 120, called increasing failure rate (IFR). II. HIV/1 is nondefreasing for >0, called increasing failure rate on the average (IFRA). II. ml) Sm(0) for all / 20, called new better than used (NBU). IV. m(1) decreases in 1, called decreasing mean residual life. (a) Prove that I II III. (b) Prove that I = IV III.

Answer 1

Survival for S(+).. Hazred fn-h(E) Cumulative hazard fr. H(E). Mean residual life. – m(t) PER - n(t) is non decreasing for t o . IFRA - H() is non- devedasing for to NBU - m(t) = m(o) & to. DMIRL- m(t) decreases in b. .

Le SE IFR S (t+x) It S (E) 2 - I log S(t) TE s(at) > 5* (t). = n(N) olere - I log S(L) sto . SEIFRA. 2) IFR ») PERAL Les S&IFRA- I - I log S(E) & t- .) – log S(x) { lety log S(x+y) 320, 420. - – log S (x) < x las 5 (243) :) log $(re) < lng (2) -

4) w Again, let SE IARA LEOMRE -- log S(E) at - - log S(x) IFR » - aty log s (nty), aso I Let SED = -log S (Y) by log S(x+y). (*) 2014 =) Log. S(y) I log S(x+y) Adding (*) & () wegen deg S(x+y) < $(oc) (y) & SENBU IFRA 3 NBU. Let S EFFR. S (t +x) S(+2+2), tiste, aso SCH S(tz)

I m(+1) >m(+2) 6. & FEDMR? PER DMRL. Let S EDMRL un m(o), m(t).