Hazard Rate and Future Lifetime for the Generalized Normal Distribution

Abstract

The target of this paper is to discuss a generalized form of the well-known Law of Frequency Error. This particular Law of Frequency of Errors is what is known as “Gaussian” or “Normal” distribution and appeared to have an aesthetic appeal to all the branches of Science. The Generalized Normal Distribution is presented as a basis to our study. We derive also the corresponding hazard function as well as the future lifetime of the Generalized Normal Distribution (GND), while new results are also presented. Moreover, due to some of the important distribution the GND family includes, specific results can also be extracted for some other distributions.

Appendix

Proof (of Proposition 2)

Considering the definition of the cumulative hazard function $H_X := \int h_X$ for an r.v. X, and the fact that

$$\displaystyle \begin{aligned}\tfrac{\,{\mathrm{d}}\,}{\,{\mathrm{d}}\, x}\big(\log S_X\big) = \frac{S^{\prime}_X}{S_X} = -\frac{f_X}{S_X} = -h_X, \notag\end{aligned} $$

where S _X := 1 − F _X denotes the survival function for a r.v. X, we obtain that

and applying (8) and (9) we finally derive (16) and (17), respectively.

Proof (of Proposition 3)

Recall that the future lifetime at time $t_0\in \mathbb {R}_+$ is defined to be the time remaining until failure (or death), given survival until time t ₀. Let $X_{t_0}$, or X ₀, be an r.v. describing the future lifetime of a system described by an r.v. X at time t ₀, i.e. X ₀ := X − t ₀ provides the time to failure (of a system with r.v. X) at, or before, time t + t ₀ given survival until time t ₀. The c.d.f. of X ₀, which is the probability of failure at, or before, time t + t ₀ given survival until time t ₀, is then written in the form

$$\displaystyle \begin{aligned}F_{X_0}(t) := \Pr\big(X\le t+t_0\;\big|\;X >t_0\big) = \frac{\Pr\big(t < X\le t+t_0\big)}{\Pr\big(X > t_0\big)} = \frac{F_X(t_0+t)-F_X(t_0)}{S_X(t_0)},\end{aligned} $$

(35)

for $t\in \mathbb {R}_+$, while the future lifetime probability density of X ₀ is then

$$\displaystyle \begin{aligned} f_{X_0}(t) := \tfrac{\,{\mathrm{d}}\,}{\,{\mathrm{d}}\, t}F_{X_0}(t) = \frac{f_X(t+t_0)}{S_X(t_0)},\quad t\in\mathbb{R}_+.\end{aligned} $$

(36)

Assuming now that $X\sim \mathscr N_\gamma \big (\mu ,\sigma ^2\big )$ and $t_0 := x_0\in \mathbb {R}$, the expressions (29) and (30) are derived from (36) and (35), through (1) and (8), respectively.

The corresponding expected future lifetime of X ₀ at $x_0\in \mathbb {R}$ is then given, according to (36), by

(37)

Using the linear transformation u = u(x) := x + x ₀, $x\in \mathbb {R}$, we obtain

and thus (31) is derived by substituting (8) to the above.

Proof (of Proposition 4)

The p.d.f. f _T of r.v. T := |X| can be easily expressed as f _T = 2f _X, where f _X denotes the p.d.f. of the r.v. $X\sim \mathscr N_\gamma \big (0,\sigma ^2\big )$. The corresponding c.d.f. F _T can also be expressed through the c.d.f. F _X, as

(38)

and through (8) we obtain

$$\displaystyle \begin{aligned}F_T(t) = 1-\frac{\operatorname{\Gamma}\big(g,g(t/\sigma)^{1/g}\big)}{\operatorname{\Gamma}(g)},\quad t\in\mathbb{R}_+,\end{aligned} $$

(39)

while the survival function S _T of T is given by

$$\displaystyle \begin{aligned}S_T(t) = \frac{\operatorname{\Gamma}\big(g,g(t/\sigma)^{1/g}\big)}{\operatorname{\Gamma}(g)},\quad t\in\mathbb{R}_+.\end{aligned} $$

(40)

According now to (36) and (35) we easily derive respectively, through (1), (39), and (40), the requested expressions (32) and (33).

The corresponding expected future lifetime of T ₀ at $t_0\in \mathbb {R}$ is given, similarly to (37), by

where u = u(t) := t + t ₀. $t\in \mathbb {R}_+$. Integrating by parts, the above is written as

(41)

where

and u = u(t) := g(t∕σ)^1∕g, $t\in \mathbb {R}_+$. Integration by parts yields

(42)

Recall the known limit

$$\displaystyle \begin{aligned}\underset{u\to+\infty}{\lim}\;\frac{\operatorname{\Gamma}(g,u)}{u^{g-1}\,{\mathrm{e}}^{-u}} = 1,\end{aligned} $$

(43)

which implies that

$$\displaystyle \begin{aligned}\underset{u\to+\infty}{\lim}u^g\operatorname{\Gamma}(g,u) = \underset{u\to+\infty}{\lim}\frac{u^{2g-1}}{e^u} = 0, \notag\end{aligned} $$

and through (22), the definite integral I(t ₀) in (42) can be written successively as

(44)

Splitting now the definite integral of (44) into $\int _{0}^{+\infty }-\int _{0}^{u_0}$ and apply the definitions of the gamma and the lower incomplete gamma functions respectively, we obtain

$$\displaystyle \begin{aligned}I(t_0) = -(u_0/g)^g\sigma\operatorname{\Gamma}(g,u_0)+g^{-g}\sigma\big[\operatorname{\Gamma}(2g)-\gamma(2g,u_0)\big], \notag\end{aligned} $$

and using (22) again,

$$\displaystyle \begin{aligned}I(t_0) = -(u_0/g)^g\sigma\operatorname{\Gamma}(g,u_0)+g^{-g}\sigma\operatorname{\Gamma}(2g,u_0).\end{aligned} $$

(45)

Finally, substituting u ₀ := u(t ₀) = g(t ₀∕σ)^1∕g into (45), and then applying (45) into (41), the requested expected future lifetime of T ₀ at time $t_0\in \mathbb {R}_+$ is then given by (34).