In this section with classify the tail weight of a distribution based on the hazard rate function and the mean excess loss function.
Classification Based on the Hazard Rate Function
Another way to classify the tail weight of a distribution is by using the hazard rate function:
h(x) = f (x)
S(x) = F0(x)
1 − F (x) = − d
dx[ln S(x)] = −S0(x) S(x).
By the existence of moments, the Pareto distribution is considered heavy-tailed. Its hazard rate function is
h(x) = f (x) S(x) = α
x + θ.
Note that h0(x) = −(x+θ)α 2 < 0 so that h(x) is nonincreasing. Thus, it makes sense to say that a distribution is considered to be heavy-tailed if the hazard rate function is nonincreasing. Likewise, the random variable X with pdf f (x) = xe−x for x > 0 and 0 otherwise has a light-tailed distribution according to the existence of moments (See Problem9.1). Its hazrad function is h(x) = x+1x which is a nondecreasing function. Hence, a nondecreasing hazard rate function is an indication of a light-tailed distribution.
Example 10.1
Let X be a random variable with survival function f (x) = x12 if x ≥ 1 and 0 otherwise. Based on the hazard rate function of the distribution, decide whether the distribution is heavy-tailed or light-tailed.
Solution.
The hazard rate function is
h(x) = −S0(x)
S(x) = −−2
x3 1 x2
= 2 x. Hence, for x ≥ 1,
h0(x) = −2 x2 < 0
which shows that h(x) is decreasing. We conclude that the distribution of X is heavy-tailed
Remark 10.1
Under this definition, a constant hazard function can be called both non-increasing and nondecreasing. We will refer to distributions with constant hazard function as medium-tailed distribution. Thus, the exponential random variable which was classified as light-tailed in Example9.1, will be referred to as a medium-tailed distribution.
The next result provides a criterion for testing tail weight based on the probability density function.
Theorem 10.1
If for a fixed y ≥ 0, the function f (x+y)f (x) is nonincreasing (resp. nondecreas-ing) in x then the hazard rate function is nondecreasing (resp. nonincreas-ing).
Thus, if f (x+y)f (x) is nondecreasing in x for a fixed y, then h(x) is a nonincreas-ing in x. Likewise, if f (x+y)f (x) is nonincreasing in x for a fixed y, then h(x) is a nondecreasing in x
Example 10.2
Using the above theorem, show that the Gamma distribution with parame-ters θ > 0 and 0 < α < 1 is heavy-tailed.
for 0 < α < 1. Thus, the hazard rate function is nonincreasing and the distribution is heavy-tailed
Next, the hazard rate function can be used to compare the tail weight of two
distributions. For example, if X and Y are two distributions with increasing (resp. decreasing) hazard rate functions, the distributions of X has a lighter (resp. heavier) tail than the distribution of Y if hX(x) is increasing (resp.
decreasing) at a faster rate than hY(x) for a large value of the argument.
Example 10.3
Let X be the Pareto distribution with α = 2 and θ = 150 and Y be the Pareto distribution with α = 3 and θ = 150. Compare the tail weight of these distributions using
(a) the relative tail weight measure;
(b) the hazard rate measure.
Compare your results in (a) and (b).
Solution.
(a) Note that both distributions are heavy-tailed using the hazard rate anal-ysis. However, h0Y(x) = −(x+150)2 2 < h0X(x) = −(x+150)1 2 so that hY(x) de-creases at a faster rate than hX(x). Thus, X has a lighter tail than X.
(b) Using the relative tail weight, we find
x→∞lim fX(x)
fY(x) = lim
x→∞
2(150)2
x + 150)2 ·(x + 150)4 3(150)4 = ∞.
Hence, X has a heavier tail than Y which is different from the result in (a)!
Remark 10.2
Note that the Gamma distribution is light-tailed for all α > 0 and θ > 0 by the existence of moments analysis. However, the Gamma distribution is heavy-tailed for 0 < α < 1 by the hazard rate analysis. Thus, the concept of light/heavy right tailed is somewhat vague in this case.
Classification Based on the Mean Excess Loss Function
A fourth measure of tail weight is the mean excess loss function as introduced in Section 5. For a loss random variable X, the expected amount by which loss exceeds x, given that it does exceed x is
e(x) = eX(x) = E[X − x|X > x] = E(X) − E(X ∧ x) 1 − F (x) .
In the context of life contingency models (See [3]), if X is the random variable representing the age at death and if T (x) is the continuous random variable
representing time until death of someone now alive at age x then e(x) is denoted by ˚e(x) = E[T (x)] = E[X − x|X > x]. In words, for a newborm alive at age x, ˚e(x) is the average additional number of years until death from age x, given that an individual has survived to age x. We call ˚e(x) the complete expectation of life or the residual mean lifetime.
Viewed as a function of x, an increasing mean excess loss function is an indication of a heavy-tailed distribution. On the other hand, a decreasing mean excess loss function indicates a light-tailed distribution.
Next, we establish a relationship between e(x) and the hazard rate function.
We have
But one of the characteristics of the hazard rate function is that it can generate the survival function:
SX(x) = e−
Rx 0 h(t)dt. Hence, we can write
e(x) = increasing function of x (and therefore e(x) is increasing) then the hazard rate function is decreasing and consequently the distribution is heavy-tailed.
Likewise, if the SXS(x+y)
X(x) is a decreasing function of x (and therefore e(x) is decreasing) then the hazard rate function is increasing and consequently the distribution is light-tailed.
Example 10.4
Let X be a random variable with pdf f (x) = 2xe−x2 for x > 0 and 0 otherwise. Show that the distribution is light-tailed by showing SXS(x+y)
X(x) is a decreasing function of x.
Solution.
We have SX(x) =R∞
x 2te−t2dt = e−x2. Thus, for a fixed y > 0, we have SX(x + y)
SX(x) = e−2xy− y2
whose derivative with respect to x is d
dx
SX(x + y) SX(x)
= −2ye−2xy−y2 < 0.
That is, SXS(x+y)
X(x) is a decreasing function of x
Practice Problems
Problem 10.1
Show that the Gamma distribution with parameters θ > 0 and α > 1 is light-tailed.
Problem 10.2
Show that the Gamma distribution with parameters θ > 0 and α = 1 is medium-tailed.
Problem 10.3
Let X be the Weibull distribution with probability density function f (x) =
τ xτ −1e−(xθ)τ
θτ . Using hazard rate analysis, show that the distribution is heavy-tailed for 0 < τ < 1 and light-heavy-tailed for τ > 1.
Problem 10.4
Let X be a random variable with pdf f (x) = 2xe−x2 for x > 0 and 0 otherwise. Determine the tail weight of this distributions using Theorem 10.1.
Problem 10.5
Using Theorem 10.1, show that the Pareto distribution is heavy-tailed.
Problem 10.6
Show that the hazard rate function of the Gamma distribution approaches
1
θ as x → ∞.
Problem 10.7
Show that limx→∞e(x) = limx→∞ 1 h(x). Problem 10.8
Find limx→∞e(x) where X is the Gamma distribution.
Problem 10.9
Let X be the Gamma distribution with 0 < α < 1 and θ > 0. Show that e(x) increases from αθ to θ.
Problem 10.10
Let X be the Gamma distribution with α > 1 and θ > 0. Show that e(x) decreases from αθ to θ.
Problem 10.11
Find limx→∞e(x) where X is the Pareto distribution with parameters α and θ and conclude that the distribution is heavy-tailed.
Problem 10.12
Let X be a random variable with pdf f (x) = (1+x)1 2 for x > 0 and 0 other-wise. Find an expression of limx→∞e(x).