# The Decreasing Hazard Rate Phenomenon: A Review of Different Models, with a Discussion of the Rationale behind Their Choice

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- always decreasing in time (the class of such models is denoted as “decreasing hazard rate”, DHR);
- -
- first increasing, then decreasing in time (the class of such models is denoted as “first increasing, then decreasing hazard rate”, IDHR).

- the Reliability function (RF);
- the above introduced hazard rate function (HRF);
- the Mean Time to Failure (MTTF);
- the “residual reliability function” (RRF).

^{+}, the quantity h(x)Δx is equal to the conditional probability that the failure takes place within the interval (x, x + Δx), provided that the component has reached an age x (i.e., it did not fail—or survived—until x); namely h(x)Dx can be viewed as the instantaneous failure (conditional) probability of a component having age x.

^{+}) = 1, an integral relationship can be derived that lets the RF, R(t), be explained as a function of the instantaneous HRF:

^{β})

^{β−1}

^{β−1}exp(−αt

^{β})

## 2. A Premise to the Review of the Basic DHR or IDHR Lifetime Models on the Basis of Stress-Strength Models

## 3. Birnbaum-Saunders (BS) Model

^{2}are equal, respectively, to:

^{2}/2)

^{2}= β

^{2}α

^{2}[1 + (5/4)α

^{2}]

_{k}; k = 1,2,…}of stresses being RV from a common distribution– and denotes as X

_{n}the total stress after n cycles, then it holds:

**N**(= set of natural numbers):

_{n}> y)

_{n}< y) (i.e., “the total stress after n cycles is smaller than y”); then, considering the complementary events, the above relation is obtained.

_{K}, are independent Gaussian RV, with N(θ,σ) distribution (of course θ > 0), or that the stress number is so high that the central limit theorem [45] is valid. As a consequence, W

_{n}is (or converges towards, as n diverges) a N(θn, σ n) RV, so that:

_{n}≤ y) = 1 − Φ [(y − θn)/σ √n] = Φ [(Φ θn − y)/σ √n]

## 4. Exponential Model

- (1)
- X(t) is a stochastic process which can be described as a “shock type” stress, the shocks (whose amplitudes are denoted as Z
_{k}) occurring at the time instants T_{k}; - (2)
- The device fails only because of the occurrence of a stress, i.e., at the time t = T
_{k}when stress amplitude Z_{k}is greater than Strength Y(t) = Y(T_{j}); of course, such failure time is a RV.

_{j}< Y

_{j}) must be verified for every index j = 1,…,N(t), being Z

_{k}= X(T

_{k}), and Y

_{k}= Y(T

_{k}), T

_{k}being the RV “time of k-th stress occurrence”.

_{n}= [N(t) = n]:

_{n}) = P[(X

_{1}< Y

_{1})∩(X

_{2}< Y

_{2})∩…∩(X

_{n}< Y

_{n})| A

_{n}]

_{n}) is denoted as R

_{n}hereafter.

_{n}, the RF R(t) can be attained—in terms of the R

_{n}’s and of the distribution of the point process N(t)—resorting to the total probability theorem, as follows:

_{j}(j = 1,..n,..) and Y

_{j}(j = 1,..n,..) are statistically independent of each other and of N(t), then:

_{n}as:

- the shock amplitudes, i.s., the RV Z
_{j}, are independent with common CDF $G\left(z\right)=P\left({Z}_{j}<z\right),\forall \mathrm{j}=1,2,\dots ,n,\dots $, (independent of time), and PDF g(z); - the Y
_{j}are independent with common CDF $F\left(y\right)={F}_{y}\left(y\right)=P\left({Y}_{j}<y\right),\forall \mathrm{j}=1,2,\dots ,n,\dots $, (independent of time), and PDF f(y);

_{n}= w

^{n}

_{j}RV (and the same for Y and Y

_{j})—evaluated as using the same approach when the SS models were introduced (Equation (14)):

_{j}is smaller than the generic stress Z

_{j}. In case that the strength is a constant, y, then θ = P(Z > y) = 1 − G(z). In any case, the above RF of (41) is clearly an Exponential one, i.e., it may be expressed as:

## 5. Inverse Gaussian (IG) Model

^{2}= μ/λ

## 6. Inverse Weibull (IW) Model

^{2}Γ(1 − 2/β) − μ

^{2}

#### 6.1. Derivation of the IW Model from a SS Model: Case of Stress Being a Weibull RV

^{β}]

^{b}) by denoting: α = 1/θ

^{β}.

_{X}(x) = P(X ≤ x) = 1 − exp[−(x/υ)

^{ϖ}] (υ,ϖ) > 0)

^{η}, (η,κ > 0)

_{X}[y(t)] = 1 − exp{− [κ /(υ t

^{η})]

^{ϖ}} = 1 − exp{−[1/αt]

^{β}}

^{1/η}:

^{β}}

#### 6.2. Derivation of the IW Model from a SS Model: Case of Stress Being a Weibull Stochastic Process

_{X}(x,t) for evidencing time-dependency—characterized by time-dependent scale parameter θ = θ (t) (and a constant shape parameter β > 0):

_{X}(x,t) = P[X(t) ≤ x] = 1 − exp{−[x/θ(t)]

^{β}} (β > 0)

^{m}(k,m > 0)

^{m}Γ(1 + 1/β)

^{β}} = 1 − exp{−[y/kt

^{m}]

^{β}} = 1 − exp{−[1/ηt]

^{γ}}

#### 6.3. Derivation of the IW Model from a Linear Stress Process

## 7. Log-Logistic (LL) Model

^{β}]; F(y) = 1 − exp[−(y/α)

^{β}]

^{m}

^{b}]

^{1/m}.

^{b}t

^{b}

^{−1}/[1 + (λt)

^{b}]

## 8. Lognormal Model

- (i)
- if the shape parameter β of the LN(α,β) model is low enough (more precisely, in practice if β < 0.3) the LN PDF tends to symmetry and may approximate well also a Normal model with the same mean (as can be shown in an analytical way, resorting to the series expansion of y = exp(x) for x→0);
- (ii)
- the CV, ν = σ/μ, varies over a broad interval: in more detail, ν = 1 when β = 0.8325, as for the Exponential model, to which the LN model is so close in this case that the two models cannot almost be distinguished one from another; this motivates the applications of the LN model for microelectronic components [118]. Also Section 10 will show an application of the LN model for electronics components.

#### 8.1. Stress Process with Linear Function Yielding the Lognormal Model

#### 8.2. Stress Process with Log-Linear Function Yielding the Lognormal Model

#### 8.3. Stress Process with Power Function Yielding to the Lognormal Model

^{c}

^{c}is also an LN RV, regardless of the value of the real exponent c.

## 9. A Brief Account of Mixture Models Leading to DHR or IDHR Models

_{b}is the “base” HRF (it is a positive constant for an Exponential model, in all other cases is a time function as in [79]); p (p > 1) is an integer value depending on the component;

_{1}, h

_{2},h

_{p}are the so-called “environment factors”. In practice, the environment factors are positive constants which account for all the factors (environment, production, quality, applied voltage and others) affecting the final value of the HRF, so they are to be considered, in the most general case, as RV in order to account for the environment randomness for electronics devices already discussed in Section 1. If the product h

_{1}h

_{2..}, h

_{p}is denoted by Ω, the above model (79) is obtained.

^{b−1}, (b > 0)

## 10. On the Consequences of Mistaken Model Identification in Terms of the Hazard Rate Function. A Numerical Example from a Real Dataset

^{4}, the parameters of the above expression are so estimated from the real data, as reported in [70]: ξ = 2.3054; δ = 0.149415.

^{β}]

^{4}): θ = 10.5464, β = 7.890.

^{4}h). In order to better appreciate the LN HRF behavior, since its values are much smaller than those of the Weibull one, the curve of the LN HRF alone is also reported in Figure 4. As apparent, for large times the LN HRF tends to become nearly constant (and very slowly decreasing for very large times), while the Weibull one sharply increases.

## 11. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Glossary

BS | Birnbaum–Saunders (distribution) |

CDF | Cumulative distribution function |

CLT | Central limit theorem |

CV | Coefficient of variation |

DHR | Decreasing hazard rate |

D[Y] | Standard deviation of the RV Y |

E[Y] | Expectation of the RV Y |

F( ) | Generic CDF |

f( ) | Generic PDF |

f(x), F(x) | PDF and CDF of Stress |

g(y), G(y) | PDF and CDF of Strength |

HRF | Hazard Rate Function |

IDHR | First increasing, then decreasing hazard rate |

IG | Inverse Gaussian (distribution) |

IHR | Increasing hazard rate |

IW | Inverse Weibull (distribution) |

LL | Log-logistic (distribution) |

LN | Lognormal (distribution) |

MTTF | Mean Time to Failure |

N(α,β) | Normal (Gaussian) random variable with mean α and standard deviation β |

Probability density function | |

RF | Reliability function |

RM | Reliability model |

R(t) | Reliability function at mission time t |

RRF | Residual reliability function |

R(t|s) | Residual reliability function at mission time t, for a device aged s time units |

RV | Random variable |

SD, σ | Standard deviation |

SS | Stress-Strength |

Var, σ^{2} | Variance |

W(t) | Wear process at time t acting on a device |

W(α,β) | Weibull model with CDF: F(x) = 1 − exp(−αx^{β}) |

W’(θ,β) | Weibull alternative form, with CDF: F(x) = 1 − exp(−(x/θ)^{β}) |

Γ( ) | Euler’s Gamma function |

μ | Mean value (Expectation) |

Φ(z) | Standard normal CDF |

φ(z) | Standard normal PDF |

Remark: the symbol “log” always denotes natural logarithm. |

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**Figure 1.**The RF curves of the LN and Weibull models for lumen data after [70].

**Figure 2.**The PDF curves of the LN and Weibull models for lumen data after [70].

**Table 1.**Values of some p-quantiles of the LN and the Weibull reliability models for lumen data after [70] (for p = 0.05, 0.50, 0.95).

p | LN | Weibull |
---|---|---|

0.05 | 7.8428 | 7.2379 |

0.50 | 10.0278 | 10.0677 |

0.95 | 12.8215 | 12.1199 |

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**MDPI and ACS Style**

Chiodo, E.; Mazzanti, G.
The Decreasing Hazard Rate Phenomenon: A Review of Different Models, with a Discussion of the Rationale behind Their Choice. *Electronics* **2021**, *10*, 2553.
https://doi.org/10.3390/electronics10202553

**AMA Style**

Chiodo E, Mazzanti G.
The Decreasing Hazard Rate Phenomenon: A Review of Different Models, with a Discussion of the Rationale behind Their Choice. *Electronics*. 2021; 10(20):2553.
https://doi.org/10.3390/electronics10202553

**Chicago/Turabian Style**

Chiodo, Elio, and Giovanni Mazzanti.
2021. "The Decreasing Hazard Rate Phenomenon: A Review of Different Models, with a Discussion of the Rationale behind Their Choice" *Electronics* 10, no. 20: 2553.
https://doi.org/10.3390/electronics10202553