NWBUE: New Worse then Better than Used in Expectation

• The paper introduces NWBUE distributions, where the MRL rises initially (≥ μ) and then declines (≤ μ) after a change-point x₀.
• It outlines formal definitions and integral inequalities capturing the non-monotonic behavior, generalizing the classical NBUE property.
• It demonstrates that moment inequalities for NWBUE are generally looser than for NBUE and require tailored analytical treatment.

New Worse then Better than Used in Expectation (NWBUE) describes a class of life distributions that extends traditional monotonic ageing properties to encompass non-monotonic mean residual life (@@@@1@@@@) behavior. Specifically, an NWBUE distribution permits an initial phase in which the expected remaining lifetime at age $t$ (the MRL function) is at least as large as at the origin, followed by a "change-point" after which the MRL is at most the initial mean. This class generalizes the classical New Better than Used in Expectation (NBUE) distributions and is of particular interest in renewal theory, non-monotonic ageing models, and applications where non-monotonic residual life patterns are observed (Anis, 11 Jan 2026).

1. Formal Definition and Equivalent Formulations

Let $X$ be a non-negative random variable with cumulative distribution function $F$, survival function $\bar F(x) = 1 - F(x)$, and finite mean $\mu = \mathbb{E}[X] < \infty$. The mean residual life at age $t$ is defined as

$$e_F(t) = \mathbb{E}[X-t \mid X>t] = \frac{1}{\bar F(t)} \int_t^\infty \bar F(x) \, dx, \quad t \ge 0.$$

$X$ is said to be New Worse then Better than Used in Expectation with change-point $x_0$ (NWBUE$(x_0)$) if there exists $x_0 \ge 0$ such that

$\begin{cases} e_F(t) \ge \mu, & 0 \le t < x_0, \[6pt] e_F(t) \le \mu, & t \ge x_0. \end{cases}$

An equivalent "integral-inequality" form is:

$$\forall\, t < x_0:\quad \int_t^\infty \bar F(x)\, dx \ge \mu\, \bar F(t), \qquad \forall\, t \ge x_0:\quad \int_t^\infty \bar F(x)\, dx \le \mu\, \bar F(t).$$

For $x_0 = 0$, this reduces to the NBUE property. In renewal-theoretic terms, NWBUE$(x_0)$ signifies that the expected residual life of the renewal process increases above the initial mean up to $x_0$, then decreases below thereafter; all NBUE shock-model interpretations extend accordingly (Anis, 11 Jan 2026).

2. Comparison with NBUE and Related Properties

NBUE is a classical monotonic ageing property, signifying that $e_F(t) \le \mu$ for all $t \ge 0$. NWBUE$(x_0)$ distributions admit a single "worse-than-used" phase (with residual life above the mean) before the change-point, then transition to "better-than-used" (with residual life no greater than the mean).

The following relationships hold:

• $F$ is NBUE if and only if $F$ is NWBUE$(x_0)$ with $x_0=0$.
• Structural properties such as closure under scale transformations, weak convergence, and shock-model parallels transfer from NBUE to NWBUE provided the change-point is tracked.

A fundamental distinction arises in moment behavior: while many NBUE properties extend to NWBUE, the standard moment bounds do not, as demonstrated by explicit counterexamples (see Section 4 below) (Anis, 11 Jan 2026).

3. Moment Inequalities and Their Limitations

Let $\mu_r = \mathbb{E}[X^r]$ with $r \ge 1$. The following results characterize the mitra–Basu moment bounds for NWBUE$(x_0)$ [2]:

1.

$$\mu_r \le r\, e^{x_0/\mu} \int_{x_0}^\infty x^{r-1} e^{-x/\mu} dx;$$

2.

$$\mu_r \le x_0^r + \mu^r\, \Gamma(r+1) \sum_{j=0}^{r-1} \frac{(x_0/\mu)^j}{j!};$$

1. In particular,

$\mu_r \le \mu^r\, \Gamma(r+1)\, e^{x_0/\mu}.$

For the NBUE case ($x_0=0$), the established bounds are sharper:

$$\mu_r = \mathbb{E}[X^r] \begin{cases} \le \Gamma(r+1)\, \mu^r, & r \ge 1, \ \ge \Gamma(r+1)\, \mu^r, & 0 < r < 1. \end{cases}$$

The moment bounds for NWBUE are generally looser than for NBUE, and interchanging them leads to errors. The NWBUE inequalities are valid, but not necessary, for NBUE, and vice versa (Anis, 11 Jan 2026).

4. Illustrative Examples and Counterexamples

The discrepancy between NBUE and NWBUE moment bounds is demonstrated by explicit constructions:

Example	NWBUE Class	Bound Violated	Details
MRL: $e_F(x) = 5 + x$ ($0\le x\le 1$), etc.	NWBUE$(10)$	NBUE bound ($r=2$)	$\mu=5$, $\mathbb{E}[X^2]=54.1210 > 50$ NBUE bound
$X \sim$ Weibull(scale=1, shape=2)	NBUE	NWBUE bound (some $x_0$)	$\mu = \frac{\sqrt{\pi}}{2}$, $\mathbb{E}[X^2]=1$, NWBUE bounds stricter for $x_0>0$

These concrete examples show that the moment inequalities appropriate for one class fail for the other, establishing the necessity of class-specific bounds (Anis, 11 Jan 2026).

5. Connections with Other Ageing Classes and Corrections

Relationships with non-monotonic ageing classes, such as the Increasing-then-Decreasing Mean Residual Life (IDMRL) class, require careful qualification. Mitra–Basu [2] asserted "every IDMRL$(t_0)$ law is NWBUE$(t_0^\star)$"—this is generally false. The corrected result is:

Theorem: If $F$ is IDMRL with turning point $\tau_0$ (i.e., $e_F$ increases on $[0, \tau_0]$ and then decreases), and if there exists $x^\star > \tau_0$ such that $e_F(x^\star) = \mu$, then $F$ is NWBUE with change-point $x_0 = x^\star$. If no such $x^\star$ exists, the MRL remains strictly above the mean, so $F$ is NBUE (i.e., NWBUE$(0)$).

An explicit NWBUE$(3)$ example is:

$\bar F(x) = \begin{cases} \dfrac{4}{(2+x)^2}, & 0 \le x < 1, \[6pt] \dfrac{2}{27}(7-x), & 1 \le x < 3, \[6pt] \dfrac{2x(3+x)^2}{729} e^{3-x}, & x \ge 3, \end{cases}$

with $e_F(0) = 2$, $e_F(3) = 2$, and the MRL increasing then decreasing, confirming the NWBUE$(3)$ property (Anis, 11 Jan 2026).

6. Structural Properties and Analytical Implications

The NWBUE class encompasses distributions with non-monotonic MRL, characterized by a single change-point $x_0$. This generalization accommodates systems where initial "worsening" (i.e., higher than mean residual life) is observed, followed by an eventual transition to "better-than-used" behavior. Significant structural properties—closure under scaling, convergence, and analogs of NBUE's shock-model formulation—remain intact for NWBUE, conditioned on tracking the change-point.

Nevertheless, the transition from monotonic to non-monotonic ageing introduces substantial complexity in the analysis of statistical functionals, particularly higher moments and residual life inequalities. Established NBUE bounds are not generally valid for NWBUE distributions and vice versa, necessitating careful selection of the correct class for analytical and applied work. Corrections to prior literature clarifying these subtleties have been presented (Anis, 11 Jan 2026).