Basic duration distributions
In this section we introduce some parametric families of duration distributions. Exponential family
The exponentially distributed durations feature no duration dependence. In consequence of the time-independent durations, the hazard function is constant, X(y) = X. The cdf is given by the expression F(y) = 1 – exp(-Xy), and the survivor function is S( y) = exp(-Xy).
The density is given by:
f(У) = X exp^Xy^ У > °. (21.5)
This family is parametrized by the parameter X taking strictly positive values.
An important characteristic of the exponential distributions is that the mean and standard deviation are equal, as implied by EY = |, VY = ^ . In empirical research the data violating this condition are called over – or under-dispersed depending on whether the standard deviation exceeds the mean or is less than the mean.
This family of distributions depends on two positively valued parameters a and v. The density is given by:
f( У) = [avyv-1exp(- ay)]/r(v), (21.6)
where r(v) = f 0 exp(-y)yv-1dy. When v = n is integer valued, this distribution may be obtained by summing n independent exponentially distributed durations with parameter X = a. In such a case Г(п) = (n – 1)!.
The form of the hazard function depends on the parameter v.
1. If v > 1, the hazard function is decreasing and approaching asymptotically a.
2. If v = 1, the hazard function is a constant, and the model reduces to the exponential model.
3. If v < 1, the hazard function is decreasing from +^> and approaches an asymptote at a.
The gamma model is quite often employed in practice. The mean and variance of gamma distributed durations are EY = a, VY = a.
This family of distributions also depends on two positive parameters a and b. The density is:
f( y) = abyb-1exp(- ayb). (21.7)
The formula of the survivor function is S( y) = exp(- ayb). The behavior of the hazard function X(y) = abyb-1 is determined by b. It is increasing for b > 1 at either a growing or diminishing rate, and it is decreasing for values of b < 1.
These distributions, contrary to those discussed above, have a nonmonotone hazard function which is first increasing, and next decreasing in y. Therefore they can be used for the analysis of bankruptcy rates. The lognormal duration distribution is such that log Y follows a normal distribution with mean m and variance о2. The density is a function of the normal density denoted by ф:
The survivor function is S(y) = 1 – Ф (logya m), where Ф denotes the cdf of a standard normal. The hazard function can be written as the ratio:
= 1 [1/дф(х)]
y 1 – ф(х)’
log y – m
where x = – a.