NBUE: New Better than Used in Expectation

• NBUE is a class of nonnegative distributions defined by a mean residual life that never exceeds the overall mean, generalizing exponential models in reliability theory.
• The framework leverages moment bounds and the TTT-transform to compare lifetime distributions, providing strong theoretical and practical testing tools.
• Statistical testing for NBUE employs various test statistics (e.g., Hollander–Proschan and generalized T0(j)) with bootstrap methods for improved empirical size control.

A nonnegative random variable $X$ is said to be New Better than Used in Expectation (NBUE) if, at every age $t\ge0$, the expected remaining lifetime of a unit of age $t$ does not exceed the expected lifetime of a brand‐new unit. The NBUE class of distributions plays a central role in nonparametric aging theory, generalizing exponential-type models and providing a framework for positive aging that is weaker than monotonic properties such as increasing failure rate (IFR) (Anis, 11 Jan 2026). NBUE is closely tied to survival analysis, reliability theory, and orderings for comparison of lifetime distributions.

1. Formal Definition and Equivalent Characterizations

Let $X$ be a nonnegative random variable with distribution function $F$, survival function $\overline{F}(x)=1-F(x)$, and finite mean $\mu=E[X]$. The mean residual life (MRL) function

$$e_F(t) = E[X-t\,|\,X>t] = \frac{1}{\overline{F}(t)}\int_t^\infty \overline{F}(u)\,du$$

measures the expected time-to-failure for a unit that has survived to age $t$.

The NBUE property is defined by

$$F \in \mathrm{NBUE} \quad \Leftrightarrow \quad \forall t\ge0:\ e_F(t) \le \mu$$

meaning “an item of age $t$ has no larger expected remaining life than a brand-new one” (Anis et al., 2012).

Alternative formulations include:

• For all $t\ge0$, $$\int_t^\infty \overline{F}(u)\,du \le \mu\,\overline{F}(t)$$.
• $e_F(t)$ is a nonincreasing function of $t$ (Anis et al., 2012).
• NBUE is equivalent to $X/\mu \ge_\mathrm{ttt} E$ in the Total-Time-on-Test (TTT) order, with $E\sim \mathrm{Exp}(1)$ (Lando et al., 2023).
• The scaled TTT curve $S_F(p) = T_F(p)/\mu$ never falls below the diagonal on $[0,1]$, where $T_F(p)=\int_0^{F^{-1}(p)} [1-F(x)]\,dx$.

2. Key Properties: Moment Bounds and Comparisons

NBUE distributions sit strictly between the exponential family (constant MRL) and stronger monotonic aging classes such as IFR or DMRL, but strict containments hold ($\mathrm{NBUE} \supset \mathrm{IFR}$, $\mathrm{NBUE} \supset \mathrm{DMRL}$) (Anis, 11 Jan 2026). Every IFR law is NBUE, but mixtures violating IFR may still be NBUE.

A fundamental exponential-comparison lemma (Marshall–Proschan):

$$\int_x^\infty \overline{F}(t)\,dt \le \mu\,e^{-x/\mu}$$

for all $x\ge0$, provides strong tail control.

Moment bounds are central:

• For all $r>0$,
  • $r\ge1$: $\mu_r := E[X^r] \le \Gamma(r+1)\mu^r$
  • $0$\mu_r \ge \Gamma(r+1)\mu^r$

Equality for all $r$ characterizes the exponential family. NBUE distributions are uniquely determined by their entire moment sequence, as Carleman's criterion is automatically satisfied due to exponentially bounded tails (Anis, 11 Jan 2026).

3. Statistical Testing of NBUE

Testing NBUE properties commonly arises with the null hypothesis of exponentiality ($H_0: F$ exponential) against the NBUE alternative ($H_1: F$ NBUE).

Multiple nonparametric and scale-invariant test statistics exist (Anis et al., 2012):

• Hollander–Proschan $T_1$
• Generalized Hollander–Proschan $T_0(j)$
• Koul $T_2$
• Coefficient of variation $T_3$
• Aly $T_4$
• Quantile dispersion $T_5$
• Dispersion of residual life $T_6$
• Right-spread $T_7$
• Equilibrium life $T_8$

The generalized Hollander–Proschan statistic $T_{n,j}$ (for real $j>0$) is preferred for broad NBUE alternatives; closed-form exact null distributions are derivable for small samples (Anis et al., 2012):

$$T_{n,j} = \frac{Y_j(F_n)}{\bar X_n},\quad Y_j(F_n) = \frac{1}{j(j+1)n^{j-1}} \sum_{k=1}^n \left[(n-k+1)^j-(n-k)^j\right] X_{(k)}$$

Exact critical regions based on these distributions yield much improved empirical size control in moderate and smaller samples.

For goodness-of-fit or composite tests, the TTT-transform based approach with bootstrap critical values is widely used:

$T_{n,r} = \sqrt{n}\|\widehat\delta_+\|_r$

where $\widehat\delta_+(p) = \max\{0,\,p-S_{F_n}(p)\}$ measures departures from NBUE, and critical values $c_{n,\alpha}^*$ are obtained by nonparametric bootstrap (Lando et al., 2023). Power and level control are validated by extensive simulations.

4. Weak Convergence, Closure, and Uniqueness

NBUE is preserved under weak limits provided means remain bounded (Anis, 11 Jan 2026). For a sequence of NBUE laws $F_n$ with $F_n \Rightarrow F$ (in distribution), bounded $\mu_n\to\mu$, then $F$ is NBUE and $r$-th moments converge:

$$\lim_{n\to\infty}\int x^r\,dF_n(x) = \int x^r\,dF(x)$$

Uniform integrability is guaranteed by the moment bounds. Moment-determinacy ensures that convergence of all moments implies weak convergence and NBUE property for the limit. This result establishes the stability of NBUE class under empirical and theoretical limits.

5. Practical Recommendations and Implementation

Test selection for NBUE alternatives depends on sampling regime and suspected underlying models (Anis et al., 2012):

• For Weibull/Gamma alternatives:
  • Small $n\le10$: $T_0(j=0.25)$
  • Moderate $n=15$–$25$: quantile dispersion $T_5$
  • Large $n>30$: $T_0(j=1)$
• For linear-failure-rate alternatives:
  • $n\le25$: $T_0(j=0.25)$
  • $n>30$: Aly's test $T_4$.

Tests $T_8$ (equilibrium life) are not recommended, as power is poor. For all tests based on $T_{n,j}$, exact critical value tables are preferable in small samples due to slow convergence to asymptotic normality (Anis et al., 2012). For bootstrap-based TTT approaches, broad alternatives are better detected with $r=1$ ($L^1$), while $r=\infty$ (supremum-type) is sensitive to localized departures (Lando et al., 2023).

Implementation involves order statistics, TTT transforms, and extensive resampling or reference to published cutoff tables. Computational complexity remains linear in $n$ per replicate for bootstrap tests; overall cost scales with number of resamples $K$.

6. Illustrative Examples and Boundaries

Exponential distributions are precisely the NBUE laws with constant MRL and equality in all moment bounds.

• Weibull($2,1$): IFR and NBUE, with mean $\mu=\sqrt{\pi}/2$, and $E[X^2]=1\leq 2\mu^2$ per NBUE moment bound (Anis, 11 Jan 2026).
• Mixtures of exponentials may be NBUE but fail IFR.
• NBUE strictly contains DMRL and IFR: there exist NBUE distributions with nonmonotonic hazard rates.

Moment bounds $\mu_r=\Gamma(r+1)\mu^r$ for all $r$ uniquely characterize the exponential family within NBUE.

7. Extensions and Related Classes

The non-monotonic extension NWBUE (New Worse then Better than Used in Expectation) admits a single turning-point in mean residual life, violating NBUE moment bounds (Anis, 11 Jan 2026). Classical NBUE properties (moment bounds, exponential-comparison lemma, closure, and moment-determination) do not persist in NWBUE, as shown by counterexamples.

NBUE is central to nonparametric reliability analysis and stochastic comparison; it is used in the context of total time on test ordering and excess wealth orderings, where bootstrap-based nonparametric inference yields consistent and powerful tests for NBUE hypotheses and related dominance properties (Lando et al., 2023).

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Statistic	Description	Notable Properties
$T_0(j)$	Generalized Hollander–Proschan	Exact null CDF, moment bounds, recommended
$T_1$	Hollander–Proschan	Asymptotic normality, critical value tables
TTT-transform	Total-Time-on-Test curve	Bootstrap critical values, efficient

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NBUE theory provides rigorous foundations for modeling and testing positive aging in nonnegative lifetimes, with sharp inequalities, characterizations, and statistical methods that are broadly applicable across reliability, survival analysis, and nonparametric inference.