Relative Aging

The positive semigroup formulation provides a way to measure the relative aging of one lifetime distribution to another. In this section, we fix an interval \([a, b)\) with \(-\infty \lt a \lt b \le \infty\). As usual, the collection \(\ms B\) of Borel measurable subsets of \([a, b)\) is the reference \(\sigma\)-algebra and Lebesgue measure \(\lambda\) is the reference measure.

Let \(\ms V\) denote the collection of random variables \(X\) in \([a, b)\) with the support assumption that \(\lambda(A) \gt 0\) if and only if \(\P(X \in A) \gt 0\) for measurable \(A \in \ms B\).

So then \(X \in \ms V\) has an absolutely continuous distribution with respect to \(\lambda\) with density function \(f\). Let \(F\) denote the reliability function of \(X\) for \(([a, b), \le)\). Then \(r = f / F\) is the rate function of \(X\) for \(([a, b), \le, \lambda)\) and the cumulative rate function \(R\) of \(X\) for \(([a, b), \le)\) is given by \[R(x) = \int_a^x r(t) \, dt = -\ln[F(x)], \quad x \in [a, b)\] Hence as in the standard setting of Section 1, \(F = e^{-R}\).

Let \(X \in \ms V\). Then \(R\) is a homeomorphism from \([a, b)\) onto \([0, \infty)\)

Define the operator \(\oplus\) by \(x \oplus y = R^{-1}[R(x) + R(y)]\) for \(x, \, y \in [a, b)\).
Define the measure \(\mu \) by \(d \mu(x) = r(x) \, dx\) for \(x \in [a, b)\)

Then \(([a, b), \oplus, \mu)\) is a positive semigroup isomorphic to \(([0, \infty), +, \lambda)\) and \(X\) has an exponential distribution on \(([a, b), \oplus)\).

Details:

These results follow from our assumptions and the results in Section 2.

Note that the graph associated with \(([a, b), \oplus)\) is still \(([a, b), \le)\) since \(R\) is strictly increasing.

Suppose now that \(X, \, Y \in \ms V\) have rate functions \(r\) and \(s\), respectively for \(([a, b), \le, \lambda)\) and cumulative rate functions \(R\) and \(S\), respectively, for \(([a, b), \le)\). Let \(\oplus_R\) and \(\oplus_S\) denote the semigroup operations corresponding to \(R\) and \(S\), respectively. Finally, let \(\mu\) and \(\nu\) denote the invariant measures for the semigroups \(([a, b), \oplus_R)\) and \(([a, b), \oplus_S)\), respectively. A natural way to study the aging of \(X\) relative to \(Y\) is to study the aging of \(X\) on the semigroup \(([a, b), \oplus_S, \nu)\) on which \(Y\) has an exponential distribution. The following meta-definition will help with our terminology in this section.

A function or property of \(X\) relative to \(Y\) is defined to be that function or property of \(X\) relative to the measured semigroup \(([a, b), \oplus_S, \nu)\).

Suppose that \(X, \, Y \in \ms V\). Then relative to \(Y\),

The reliability function of \(X\) is \(F = e^{-R}\).
The density function of \(X\) is \(e^{-R} r / s\).
The rate function of \(X\) is \(r / s\).
The cumulative rate function of \(X\) is \(R\).
The average rate function of \(X\) is \(R / S\).

Details:

Note that all of the functions of \(X\) relative to \(Y\) are expressed in terms of \(R\) and \(S\) and their derivatives \(r\) and \(s\).

The order associated with \((S, \oplus_S)\) is the ordinary order \(\le\) so the reliability function of \(X\) relative to \(Y\) is still \(F = e^{-R}\).
Using the notation of differentials, note that \[f(x) \, dx = \frac{f(x)}{s(x)} \, d\nu(x) = F(x) \frac{r(x)}{s(x)} \, d\nu(x)\] so \(F r / s = e^{-R} r / s\) is the density of \(X\) relative to \(Y\).
From (a) and (b), the rate function of \(X\) relative to \(Y\) is \[\frac{F r / s}{F} = \frac{r}{s}\]
From (c) and a change of variables, the cumulative rate function of \(X\) relative to \(Y\) is given by \[\int_a^x \frac{r(t)}{s(t)} \, d\nu(t) = \int_a^x r(t) \, dt = R(x), \quad x \in [a, b)\]
From (d), the average rate function of \(X\) relative to \(Y\) is given by \[\frac{R(x)}{\nu[a, x)} = \frac{R(x)}{S(x)}, \quad x \in [a, b)\]

Note that the reliability function and cumulative rate function of \(X\) relative to \(Y\) are still \(F = e^{-R}\) and \(R\), respectively, because the associated graph is just \(([a, b), \le)\). Thus, these functions are graph invariants. For the results that follow, we use the notation established above for variables \(X\) and \(Y\) in \(\ms V\). First we give another point of view.

Suppose again that \(X, \, Y \in \ms V\). Reliability properties of \(X\) relative to \(Y\) are equivalent to those properties of \(S(X)\) on the standard semigroup \(([0, \infty), +)\), and this variable has cumulative rate \(R \circ S^{-1}\).

Details:

This follows from since the random variable \(S(X)\) has cumulative rate function \(R \circ S^{-1}\) for \(([0, \infty), \le)\): \[\P[S(X) \ge t] = \P[X \ge S^{-1}(t)] = \exp\left[-R[S^{-1}(t)]\right], \quad t \in [0, \infty)\]

\(X\) is exponential relative to \(Y\) if and only if the semigroup operators \(\oplus_R\) and \(\oplus_S\) are the same. The exponential relation defines an equivalence relation on \(\ms V\). That is, for \(X, \, Y, \, Z \in \ms V\),

\(X\) is exponential relative to \(X\).
If \(X\) is exponential relative to \(Y\) then \(Y\) is exponential relative to \(X\).
If \(X\) is exponential relative to \(Y\) and \(Y\) is exponential relative to \(Z\) then \(X\) is exponential relative to \(Z\).

Details:

Note that \(X\) is exponential relative to \(Y\) if and only if \(S = c R\) for some \(c \in (0, \infty)\). So then \(S^{-1} = (1 / c) R^{-1}\) and hence the operators \(\oplus_R\) and \(\oplus_S\) are the same.

Next we consider the incresaing failure rate (IFR) and decreasing failure rate (DFR) properties.

Suppose that \(X, \, Y \in \ms V\). The following are equivalent:

\(X\) is IFR relative to \(Y\).
\(Y\) is DFR relative to \(X\).
\(R\) is convex on \(([a, b), \oplus_S)\)
\((x \oplus_S h) \oplus_R y \le (y \oplus_S h) \oplus_R x \text{ for all } x, \, y, \, h \in [a, b) \text{ with } x \le y\)

Details:

The rate function of \(X\) relative to \(Y\) is \(r / s\) and the rate function of \(Y\) relative to \(X\) is \(s / r\), and of course \(r / s\) is increasing if and only if \(s / r\) is decreasing. Also, \(X\) is IFR relative to \(Y\) if and only if the distribution with cumulative rate function \(R \circ S^{-1}\) is IFR for \(([0, \infty), \le)\). This in turn is equivalent to the convexity of \(R \circ S^{-1}\) on \([0, \infty)\). This means that \[ R[S^{-1}(u + t)] - R[S^{-1}(u)] \le R[S^{-1}(v + t)] - R[S^{-1}(v)], \quad u, \, v, \, t \in [0, \infty), \, u \le v \] Let \(x = S^{-1}(u)\), \(y = S^{-1}(v)\), and \(h = S^{-1}(t)\). Then equation above becomes \[R(x \oplus h) - R(x) \le R(y \oplus h) - R(y), \quad x, \, y, \, h \in [a, b), \, x \le y\] This means that \(R\) is convex on \(([a, b), \oplus_S)\). Equivalently we have \[R(x \oplus_S h) + R(y) \le R(y \oplus_S h) + R(x), \quad x, \, y, \, h \in [a, b), \, x \le y\] Applying \(R^{-1}\) to both sides of this inequality gives the inequality in (d). Conversely, we can work backwords from (d) to the convexity condition in the first displayed equation.

The IFR relation defines a partial order on \(\ms V\), modulo the exponential equivalence in . That is, for \(X, \, Y, \, Z \in \ms V\),

\(X\) is IFR relative to \(X\).
If \(X\) is IFR relative to \(Y\) and \(Y\) is IFR relative to \(Z\) then \(X\) is IFR relative to \(Z\).
If \(X\) is IFR relative to \(Y\) and \(Y\) is IFR relative to \(X\) then \(X\) is exponential relative to \(Y\).

Details:

Let \(R, \, S, \, T\) denote the cumulative rate functions of \(X, \, Y, \, Z\), respectively. The characterization of the IFR property in terms of convexity is the best one to use.

\(R \circ R^{-1}\) is the identity function and is trivially convex. Hence \(X\) is IFR relative to \(X\).
If \(X\) is IFR relative to \(Y\) and \(Y\) is IFR relative to \(Z\) then \(R \circ S^{-1}\) and \(S \circ T^{-1}\) are convex (and of course increasing). Hence \((R \circ S^{-1}) \circ (S \circ T^{-1}) = R \circ T^{-1}\) is convex and hence \(X\) is IFR relative to \(Z\).
If \(X\) is IFR relative to \(Y\) and \(Y\) is IFR relative to \(X\) then \(R \circ S^{-1}\) and \(S \circ R^{-1}\) are convex. But \(S \circ R^{-1} = (R \circ S^{-1})^{-1}\) and hence \(R \circ S^{-1}\) is both convex and concave. Hence this function is linear and so \(X\) is exponential relative to \(Y\).

Next we consider the increasing failure rate average (IFRA) and decreasing failure rate average (DFRA) properties.

Suppose again that \(X, \, Y \in \ms V\). The following are equivalent:

\(X\) is IFRA relative to \(Y\).
\(Y\) is DFRA relative to \(X\).

Details:

The average rate function of \(X\) relative to \(Y\) is \(R / S\) and the average rate function of \(Y\) relative to \(X\) is \(S / R\). Of course, \(R / S\) is increasing if and only if \(S / R\) is decreasing.

The IFRA relation defines a partial order on \(\ms V\), modulo the exponential equivalence in . That is, for \(X, \, Y, \, Z \in \ms V\),

\(X\) is IFRA relative to \(X\).
If \(X\) is IFRA relative to \(Y\) and \(Y\) is IFRA relative to \(Z\) then \(X\) is IFRA relative to \(Z\).
If \(X\) is IFRA relative to \(Y\) and \(Y\) is IFRA relative to \(X\) then \(X\) is exponential relative to \(Y\).

Details:

Once again, let \(R, \, S, \, T\) denote the cumulative rate functions of \(X, \, Y, \, Z\) respectively.

\(R / R = 1\) is trivially increasing so \(X\) is IFRA relative to \(X\).
If \(X\) is IFRA relative to \(Y\) and \(Y\) is IFRA relative to \(Z\) then \(R / S\) and \(S / T\) are increasing.Hence \((R / S) (S / T) = R / T\) is increasing so \(X\) is IFRA relative to \(Z\).
If \(X\) is IFRA relative to \(Y\) and \(Y\) is IFRA relative to \(X\) then \(R / S\) and \(S / R\) are both increasing. Hence \(R / S\) is constant, so \(X\) is exponential relative to \(Y\).

Finally, we consider the new better than used (NBU) againg property and the complementary new worst than used (NWU) property.

Suppose again that \(X, \, Y \in \ms V\). The following are equivalent:

\(X\) is NBU relative to \(Y\).
\(Y\) is NWU relative to \(X\).
\(x \oplus_R y \le x \oplus_S y\) for \(x, \, y \in [a, b)\).

Details:

Starting with the definition of \(X\) being NBU relative to \(Y\) and ending with the definition of \(Y\) being NWU relative to \(X\), we have the following equivalent conditons: \begin{align*} F(x \oplus_S y) & \le F(x) F(y), \quad x, \, y \in [a, b) \\ R(x \oplus_S y) & \ge R(x) + R(y), \quad x, \, y \in [a, b) \\ x \oplus_S y & \ge x \oplus_R y, \quad x, \, y \in [a, b) \\ S(x \oplus_R y) & \le S(x) + S(y), \quad x, \, y \in [a, b) \\ G(x \oplus_R y) & \ge G(x) G(y), \quad x, \, y \in [a, b) \end{align*}

The NBU relation defines a partial order on \(\ms V\), modulo the exponential equivalence in . That is, for \(X, \, Y, \, Z \in \ms V\),

\(X\) is NBU relative to \(X\).
If \(X\) is NBU relative to \(Y\) and \(Y\) is NBU relative to \(Z\) then \(X\) is NBU relative to \(Z\).
If \(X\) is NBU relative to \(Y\) and \(Y\) is NBU relative to \(X\) then \(X\) is exponential relative to \(Y\).

Details:

The proof is trivial using the characterization of the NBU property in terms of the semigroup operator in part (c) of .

The relative aging properties for random variables in \(\ms V\) are related as follows: \[\text{IFR} \implies \text{IFRA} \implies \text{NBU}\] Equivalently, the NBU partial order extends the IFRA partial order, which in turn extends the IFR partial order (modulo the exponential equivalence in ).

Details:

The three again properties of \(X\) relative to \(Y\) are equivalent to the aging properties of \(S(X)\) for the standard graph \(([0, \infty], +, \lambda)\). It's well known that the aging properties are related as above in this standard setting.

In the following subsections, we consider several two-parameter families of distributions. In each case, \(\alpha\) is the exponential rate parameter while \(\beta\) is a shape parameter, which can be thought of in this context as an aging parameter, a parameter that determines the relative aging.

Weibull Distributions

The base interval is \([0, \infty)\). Define \(\varphi: [0, \infty) \to [0, \infty)\) by \(\varphi(x) = x^\beta\) for \(x \in [0, \infty)\) where \(\beta \in (0, \infty)\) is a parameter.

The corresponding semigroup operator \(\oplus\) is given by \(x \oplus y = (x^\beta + y^\beta)^{1/\beta}\) for \(x, \, y \in [0, \infty)\).
The exponential distribution on \(([0, \infty), \oplus)\) with rate \(\alpha \in (0, \infty)\) has reliability function \(F\) given by \(F(x) = \exp(-\alpha x^\beta)\) for \(x \in [0, \infty)\). This is a Weibull distribution with shape parameter \(\beta\).

Interestingly, if \(\beta \ge 1\) then \(x \oplus y\) is the \(\beta\)-norm of \((x, y)\) thought of as a vector in \(\R^2\). In the usual formulation, \(\alpha\) is written as \(\alpha = R(c) = c^\beta\) where \(c \in (0, \infty)\) is the scale parameter. But in our setting, \(\alpha\) is the rate parameter and \(\beta\) the aging parameter.

Suppose that \(X\) has the Weibull distribution above with rate parameter \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Relative to the standard graph \(([0, \infty), \le, \lambda)\),

The density function \(f\) of \(X\) is given by \(f(x) = \alpha \beta x^{\beta - 1} \exp(-\alpha x ^\beta)\) for \(x \in (0, \infty)\).
The rate function \(r\) of \(X\) is given by \(r(x) = \alpha \beta x^{\beta - 1}\) for \(x \in (0, \infty)\).
The cumulative rate function \(R\) of \(X\) is given by \(R(x) = \alpha x^\beta\) for \(x \in [0, \infty)\).
The average rate function \(\bar r\) of \(X\) is given by \(\bar r(x) = \alpha x^{\beta - 1}\) for \(x \in (0, \infty)\).
\(X\) is DFR if \(0 \lt \beta \lt 1\), has constant rate if \(\beta = 1\), and is IFR if \(\beta \gt 1\).

Part (c) is the reason for the fame of the Weibull distribution in classical reliability theory. But we can make a stronger statement using .

Suppose that \(X_1, \, X_2\) have Weibull distributions with common rate parameter \(\alpha \in(0, \infty)\) and with aging parameters \(\beta_1, \beta_2 \in (0, \infty)\), respectively.

The rate function \(r_1 / r_2\) of \(X_1\) relative to \(X_2\) is given by \[\frac{r_1(x)}{r_2(x)} = \frac{\beta_1}{\beta_2} x^{\beta_1 - \beta_2}, \quad x \in (0, \infty)\]
\(X_1\) is IFR relative to \(X_2\) if and only if \(\beta_1 \ge \beta_2\).

Open the simulation of the Weibull distribution. Vary the rate and aging parameters and note the shape of the probability density function. Run the simulation and compare the empirical density function to the probability density function.

Gompertz Distributions

The base interval is \([0, \infty)\). Define \(\varphi: [0, \infty) \to [0, \infty)\) by \(\varphi(x) = e^{\beta x} - 1\) for \(x \in [0, \infty)\) where \(\beta \in (0, \infty)\) is a parameter.

The corresponding semigroup operator \(\oplus\) is given by \[x \oplus y = \frac{1}{\beta} \ln(e^{\beta x} + e^{\beta y} - 1), \quad x, \, y \in [0, \infty)\]
The exponential distribution on \(([0, \infty), \oplus)\) with rate \(\alpha \in (0, \infty)\) has reliability function \(F\) given by \[F(x) = \exp[-\alpha (e^{\beta x} - 1)], \quad x \in [0, \infty)\] This is the Gompertz distribution with parameters \(\alpha\) and \(\beta\).

In the usual setting, \(\alpha\) is a shape parameter and \(1 / \beta\) a scale parameter, but once again in our setting \(\alpha\) is the rate parameter and \(\beta\) the aging parameter.

Suppose that \(X\) has the Gompertz distribution above with rate parameter \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Relative to the standard graph \(([0, \infty), \le, \lambda)\),

The density \(f\) function of \(X\) is given by \(f(x) = \alpha \beta e^{\beta x} \exp\left[-\alpha (e^{\beta x} - 1)\right]\) for \(x \in (0, \infty)\).
The rate function \(r\) of \(X\) is given by \(r(x) = \alpha \beta e^{\beta x}\) for \(x \in (0, \infty)\).
The cumulative rate function \(R\) of \(X\) is given by \(R(x) = \alpha(e^{\beta x} - 1)\) for \(x \in [0, \infty)\).
The average rate function \(\bar r\) of \(X\) is given by \(\bar r(x) = \alpha (e^{\beta x} - 1) / x\) for \(x \in (0, \infty)\).
\(X\) is IFR for all \(\beta \in (0, \infty)\).

Not only does the Gompertz distribution have increasing failure rate (for the standard graph), the failure rate function increases exponentially. For this reason, the Gompertz distribution is widely used in actuarial science. But once again, we can make a stronger statment in terms of relative aging via proposition .

Suppose that \(X_1, \, X_2\) have Gompertz distributions with common rate parameter \(\alpha \in (0, \infty)\) and with aging parameters \(\beta_1, \, \beta_2 \in (0, \infty)\), respectively.

The rate function \(r_1 / r_2\) of \(X_1\) relative to \(X_2\) is given by \[\frac{r_1(x)}{r_2(x)} = \frac{\beta_1}{\beta_2} e^{(\beta_1 - \beta_2) x}, \quad x \in [0, \infty)\]
\(X_1\) is IFR relative to \(X_2\) if and only if \(\beta_1 \ge \beta_2\).

Open the simulation of the Gompertz distribution. Vary the rate and aging parameters and note the shape of the probability density function. Run the simulation and compare the empirical density function to the probability density function.

Lomax Distributions

The base interal is \([0, \infty)\). Define \(\varphi: [0, \infty) \to [0, \infty)\) by \(\varphi(x) = \ln(x + \beta) - \ln(\beta)\) for \(x \in [0, \infty)\) where \(\beta \in (0, \infty)\) is a parameter.

The corresponding semigroup operator \(\oplus\) is given by \[x \oplus y = x + y + \frac{x y}{\beta}, \quad x, \, y \in [0, \infty) \]
The exponential distribution on \(([0, \infty), \oplus)\) with rate \(\alpha \in (0, \infty)\) has reliability function \(F\) given by \[F(x) = \left(\frac{\beta}{x + \beta}\right)^\alpha, \quad x \in [0, \infty)\] This is the Lomax distribution with shape parameter \(\alpha\) and scale parameter \(\beta\).

The Lomax distribution is also known as the Pareto type 2 distribution. Again in our setting, \(\alpha\) is the rate parameter and \(\beta\) the aging parameter.

Suppose that \(X\) has the Lomax distribution with rate parameter \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Relative to the standard graph \(([0, \infty), \le)\),

The density function \(f\) of \(X\) is given by \(f(x) = \alpha \beta^\alpha / (x + \beta)^{\alpha + 1}\) for \(x \in (0, \infty)\).
The rate function \(r\) of \(X\) is given by \(r(x) = \alpha /(x + \beta)\) for \(x \in (0, \infty)\).
The cumulative rate function \(R\) of \(X\) is given by \(R(x) = \alpha [\ln(x + \beta) - \ln(\beta)]\) for \(x \in [0, \infty)\).
The average rate function \(\bar r\) of \(X\) is given by \(\bar r(x) = \alpha [\ln(x + \beta) - \ln(\beta)] / x\) for \(x \in (0, \infty)\).
\(X\) is DFR for all \(\beta \in (0, \infty)\).

Although the Lomax distributions are DFR, we can once again give a stronger statement in terms of relative aging.

Suppose that \(X_1, \, X_2\) have Lomax distributions with common rate parameter \(\alpha \in (0, \infty)\) and with aging parameters \(\beta_1, \, \beta_2 \in (0, \infty)\), respectively.

The rate function \(r_1 / r_2\) of \(X_1\) relative to \(X_2\) is given by \[\frac{r_1(x)}{r_2(x)} = \frac{x + \beta_2}{x + \beta_1}, \quad x \in [0, \infty)\]
\(X_1\) is IFR relative to \(X_2\) if and only if \(\beta_1 \ge \beta_2\).

Open the simulation of the Lomax distribution. Vary the rate and aging parameters and note the shape of the probability density function. Run the simulation and compare the empirical density function to the probability density function.

Modified Beta Distributions

The base interval is \([0, 1)\). Define \(\varphi: [0, 1) \to [0, \infty)\) by \(\varphi(x) = -\ln(1 - x^\beta)\) for \(x \in [0, 1)\) where \(\beta \in (0, \infty)\) is a parameter.

The corresponding semigroup operator \(\oplus\) is given by \(x \oplus y = (x^\beta + y^\beta - x^\beta y^\beta)^{1/\beta}\) for \(x, \, y \in(0, 1)\).
The exponential distribution on \(([0, 1), \oplus)\) with rate \(\alpha \in (0, \infty)\) has reliability function \(F\) given by \(F(x) = \left(1 - x^\beta\right)^\alpha\) for \(x \in [0, 1)\).

Note that if \(\alpha = 1\) or \(\beta = 1\), the distribution is beta, and in particular, if \(\alpha = \beta = 1\), the distribution is uniform. We will refer to this class of distributions as modified beta distributions, and once again, \(\alpha\) is the rate parameter and \(\beta\) the aging parameter.

Suppose that \(X\) has the modified beta distribution with rate parameter \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Relative to the standard graph \(([0, 1), \le)\),

The density function \(f\) of \(X\) is given by \(f(x) = \alpha \beta x^{\beta - 1} (1 - x^\beta)^{\alpha - 1}\) for \(x \in (0, 1)\).
The rate function \(r\) of \(X\) is given by \(r(x) = \alpha \beta x^{\beta -1} / (1 - x^\beta) \) for \(x \in (0, 1)\).
The cumulative rate function \(R\) of \(X\) is given by \(R(x) = -\alpha \ln(1 - x^\beta)\) for \(x \in [0, 1)\).
The average rate function \(\bar r\) of \(X\) is given by \(\bar r(x) = -\alpha \ln(1 - x^\beta) / x\) for \(x \in (0, 1)\).
\(X\) is IFR if \(\beta \gt 1\) but neither IFR or DFR if \(0 \lt \beta \lt 1\).

Details:

For part (e), note that \(r\) is increasing if \(\beta \gt 1\). If \(0 \lt \beta \lt 1\), \(r\) decreases and then increases with minimum at \(x = (1 - \beta)^{1/\beta}\).

Suppose that \(X_1, X_2\) have modified beta distributions with common rate parameter \(\alpha \in (0, \infty)\) and with aging parameters \(\beta_1, \, \beta_2 \in (0, \infty)\) respectively.

The rate function \(r_1 / r_2\) of \(X_1\) relative to \(X_2\) is given by \[\frac{r_1(x)}{r_2(x)} = \frac{\beta_1 x^{\beta_1 - 1} (1 - x^{\beta_2})}{\beta_2 x^{\beta_2 - 1} (1 - x^{\beta_1})}, \quad x \in (0, 1)\]
\(X_1\) is IFR relative to \(X_2\) if and only if \(\beta_1 \ge \beta_2\).

Open the simulation of the modified Beta distribution. Vary the rate and aging parameters and note the shape of the probability density function. Run the simulation and compare the empirical density function to the probability density function.

The modified beta distribution can be obtained by a simple transformation of a beta random variable

Suppose that \(U\) has the beta distribution with left parameter 1 and right parameter \(\alpha \in (0, \infty)\). If \(\beta \in (0, \infty)\) then \(X = U^{1 / \beta}\) has the modified beta distribution with rate \(\alpha\) and aging parameter \(\beta\).

Details:

\(U\) has probability density function \(g\) given by \(g(u) = \alpha (1 - u)^{\alpha - 1}\) for \(u \in (0, 1)\). The transformation is \(x = u^{1 / \beta}\) with inverse \(u = x^\beta\). Hence \(X\) has density function \(f\) given by \[f(x) = g(u) \frac{du}{dx} = \alpha \beta x^{\beta - 1} (1 - x^\beta)^{\alpha - 1}, \quad x \in (0, 1)\]

From , the moments of the modified beta distrbution can be expressed simply in terms of the beta function \(B\).

Suppose that \(X\) has the modified beta distribution with rate \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Then \[\E(X^n) = \alpha B(1 + n / \beta, \alpha)\]

Details:

As in , we can take \(X = U^{1 / \beta}\) where \(U\) has the beta distribution parameters 1 and \(\alpha\). Then by the standard formula for the moments of the beta distribution, \[\E\left(X^n\right) = \E\left(U^{n / \beta}\right) = \frac{B(1 + n / \beta, \alpha)}{B(1, \alpha)} = \alpha B(1 + n / \beta, \alpha) \]

Suppose again that \(X\) has the modified beta disstribution with rate \(\alpha \in (0, \infty)\) and aging parameter \(\beta \in (0, \infty)\). Then \(X\) has quantile function \(Q\) given by \(Q(p) = \left[1 - (1 - p)^{1 / \alpha}\right]^{1 / \beta}\) for \(p \in [0, 1]\).

Details:

This follows by solving \(1 - \left(1 - x^\beta\right)^\alpha = p\) for \(x\) in terms of \(p\). The expression on the left is the ordinary cumulative distribution function.

Strong Aging Properties

Consider an aging property, and the corresponding improvement property, for continuous distributions in the standard space \(([0, \infty), +, \lambda)\). We are interested in characterizing those aging properties that can be extended to relative aging properties for continuous distributions on an arbitrary interval \([a, b)\) with \(-\infty \lt a \lt b \le \infty\). Moreover, we want the relative aging property to define a partial order on the distributions, modulo the exponential equivalence in , just as the IFR, IFRA, and NBU properties do. Such a characterization would seem to describe strong aging properties.

A strong aging property on \(([0, \infty), +, \lambda)\) satisfies the following conditions:

A distribution both ages and improves if and only if the distribution is exponential.
The distribution with cumulative rate function \(R\) ages if and only if the distribution with cumulative rate function \(R^{-1}\) improves.
If the distributions with cumulative rate functions \(R\) and \(S\) age, then the distribution with cumulative rate function \(R \circ S\) ages.
If a distribution is IFR then the distribution ages.

The last condition is to ensure that the property does capture some idea of aging, and to incorporate the fact that the IFR condition is presumably the strongest aging property. For the next definition, suppose again that \(-\infty \lt a \lt b \le \infty\).

Consider a strong aging property on \(([0, \infty), +, \lambda)\), and suppose that \(X\) and \(Y\) are random variables with continuous distributions on \([a, b)\) having cumulative rate functions \(R\) and \(S\), respectively. We say that \(X\) ages (improves) relative to \(Y\) if the distribution on \([0, \infty)\) with cumulative rate function \(R \circ S^{-1}\) ages (improves), respectively.

Consider a strong aging property. The corresponding aging relation defines a partial order on the equivalence class of continuous distributions on \([a, b)\), modulo the exponential equivalence in

Details:

Suppose that \(X, \, Y, \, Z\) are random variables with continuous distributions on \([a, b)\) and with cumulative rate functions \(R, \, S, \, T\), respectively. Of coure \(R \circ R^{-1}\) is the identity function on \([0, \infty)\) and is the cumulative rate function of the standard exponential distribution. Hence \(X\) ages relative to \(X\). Suppose that \(X\) ages relative to \(Y\) and that \(Y\) ages relative to \(Z\). Then the distributions on \([0, \infty)\) with cumulative rate functions \(R \circ S^{-1}\) and \(S \circ T^{-1}\) age. Hence from part (d) of definition , the distribution on \([0, \infty)\) with cumulative rate function \((R \circ S^{-1}) \circ (S \circ T^{-1})\) ages. But of course this function reduces to \(R \circ T^{-1}\) and hence \(X\) ages relative to \(Z\). Finally, suppose that \(X\) ages relative to \(Y\) and that \(Y\) ages relative to \(X\). Then the distributions on \([0, \infty)\) with cumulative rate functions \(R \circ S^{-1}\) and \(S \circ R^{-1}\) age. But \((S \circ R^{-1})^{-1} = R \circ S^{-1}\) and so the distribution with this cumulative rate function improves by (b) of . Hence by (a), this distribution is exponential.

The IFR, IFRA, and NBU properties are strong aging properties.

Details:

This follows from propositions , , and .

From our point of view, the conditions in definition are natural for a strong aging property. Conditions (a) and (d) are often used in the literature to justify aging properties, but Conditions (b) and (c) seem to have been overlooked, even though they are essential for the partial order result in . Not all of the common aging properties are strong.

Suppose that \(X\) has a continuous distribution on \([0, \infty)\)

\(X\) is new better than used in expectation (NBUE) iif \(E(X - t \mid X \ge t) \le E(X)\) for \(t \in [0, \infty)\).
\(X\) is new worse than used in expectation (NBUE) iif \(E(X - t \mid X \ge t) \ge E(X)\) for \(t \in [0, \infty)\).

Define \(R: [0, \infty) \to [0, \infty)\) by \[R(t) = \begin{cases} a t, \quad &0 \le t \lt 1 \\ a + b (t - 1), \quad &1 \le t \lt 2 \\ a + b (t - 1) + c (t - 2), \quad &t \ge 2 \end{cases}\] Positive constants \(a\), \(b\), and \(c\) can be chosen such that the distribution with cumulative rate function \(R\) is NBUE, but the distribution with cumulative rate function \(R^{-1}\) is not NWUE.