# Flexible Birnbaum–Saunders Distribution

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## Abstract

**:**

## 1. Introduction

#### 1.1. Asymmetry

**Lemma**

**1.**

#### 1.2. Bimodality

**Lemma**

**2.**

#### 1.3. BS Model

## 2. Results in Flexible Birnbaum-Saunders

#### 2.1. Interpretation of Parameters.

- (i)
- $\lambda $ positive versus $\lambda $ negative.
- (ii)

#### 2.2. Properties

**Proposition**

**1.**

**Proof.**

#### 2.2.1. Effect of $\lambda $.

**Corollary**

**1.**

**Proof.**

#### 2.2.2. Effect of $\delta $.

**Corollary**

**2.**

#### 2.2.3. Shape of ${f}_{T}(\xb7)$.

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

- 1.
- $0<{t}_{1}^{*}<\beta $ solution of$${a}_{{t}_{1}}=\delta \phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.277778em}{0ex}}\lambda \frac{\varphi \left(\lambda {a}_{{t}_{1}}\right)}{\Phi \left(\lambda {a}_{{t}_{1}}\right)}+\frac{{a}_{{t}_{1}}^{\u2033}}{{\left\{{a}_{{t}_{1}}^{\prime}\right\}}^{2}}\phantom{\rule{0.277778em}{0ex}}.$$
- 2.
- ${t}_{2}^{*}>\beta $ solution of$${a}_{{t}_{2}}=-\delta \phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.277778em}{0ex}}\lambda \frac{\varphi \left(\lambda {a}_{{t}_{2}}\right)}{\Phi \left(\lambda {a}_{{t}_{2}}\right)}+\frac{{a}_{{t}_{2}}^{\u2033}}{{\left\{{a}_{{t}_{2}}^{\prime}\right\}}^{2}}\phantom{\rule{0.277778em}{0ex}},$$

**Proof.**

**Remark**

**1.**

- 1.
- Let $Z\sim SN\left(\lambda \right)$, $\lambda \in \mathbb{R}$. Then Z is unimodal and the mode, ${z}^{*}$, is given by the solution of the non-linear equation$$z=\lambda \frac{\varphi \left(\lambda z\right)}{\Phi \left(\lambda z\right)}\phantom{\rule{0.277778em}{0ex}}.$$
- 2.
- Let $T\sim BS(\alpha ,\beta )$, $\alpha ,\beta >0.$ Then T is unimodal and the mode, ${t}^{*}$, is given by the solution of the non-linear equation$$-{a}_{t}{\left\{{a}_{t}^{\prime}\right\}}^{2}+{a}_{t}^{{}^{\u2033}}=0.$$

**Theorem**

**1.**

- (i)
- Let ${t}_{p}$ be the pth quantile of T, $0<p<1$.$${t}_{p}=\beta {\left(\frac{\alpha}{2}{z}_{p}+\sqrt{{\left(\frac{\alpha}{2}{z}_{p}\right)}^{2}+1}\right)}^{2}$$
- (ii)
- $kT\sim FBS(\alpha ,k\beta ,\delta ,\lambda )$ for $k>0$.
- (iii)
- ${T}^{-1}\sim FBS(\alpha ,{\beta}^{-1},\delta ,-\lambda )$.

**Proof.**

#### 2.2.4. Lifetime Analysis

**Proposition**

**4.**

- (i)
- The survival function is $S\left(t\right)=P[T>t]=1-{F}_{T}\left(t\right)$ with ${F}_{T}(\xb7)$ given in (9).
- (ii)
- The hazard function, $r\left(t\right)=f\left(t\right)/S\left(t\right)$, is$$r\left(t\right)=\left\{\begin{array}{cc}\frac{{c}_{\delta}{a}_{t}^{\prime}\varphi \left(\right|{a}_{t}|+\delta )\Phi \left(\lambda {a}_{t}\right)}{1-{c}_{\delta}{\Phi}_{B{N}_{\lambda}}\left(\frac{\lambda \delta}{\sqrt{1+{\lambda}^{2}}},{a}_{t}-\delta \right)},\hfill & \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0<t<\beta \hfill \\ \frac{{c}_{\delta}{a}_{t}^{\prime}\varphi \left(\right|{a}_{t}|+\delta )\Phi \left(\lambda {a}_{t}\right)}{1-{c}_{\delta}\left[{\Phi}_{B{N}_{\lambda}}\left(\frac{\lambda \delta}{\sqrt{1+{\lambda}^{2}}},-\delta \right)+{\Phi}_{B{N}_{\lambda}}\left(-\frac{\lambda \delta}{\sqrt{1+{\lambda}^{2}}},{a}_{t}+\delta \right)-{\Phi}_{B{N}_{\lambda}}\left(-\frac{\lambda \delta}{\sqrt{1+{\lambda}^{2}}},\delta \right)\right]},\hfill & \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}if\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge \beta \phantom{\rule{4.pt}{0ex}}\hfill \end{array}\right.$$

**Remark**

**2.**

- 1.
- $r\left(t\right)$ corresponding to Figure 1a,b. These are, first, quickly increasing, later decreasing more slowly or even in a flat way. It can be applied in practical situations in which the risk of failure increases quickly until certain point in which its behaviour becomes flatter. As [23] points out, the flat area is very interesting in survival analysis and reliability contexts.
- 2.
- $r\left(t\right)$ corresponding to Figure 2a,b are increasing-decreasing-increasing. This kind of hazard functions has been recently introduced and discussed in literature, due to its interest in reliability of systems, see for instance [23] or [24] (and references therein). In plot for Figure 2b, $r\left(t\right)$ is (quickly) increasing—or (quickly) decreasing. On the other hand, for Figure 2a the initial effect increasing—decreasing is less accentuated.

## 3. Moments and Maximum Likelihood Estimation

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.1. Maximum Likelihood Estimators

#### 3.2. Expected and Observed Information Matrices

## 4. Numerical Illustrations

#### 4.1. Nickel Concentration

#### 4.1.1. FBS versus the BS and SBS distributions

#### 4.1.2. FBS versus a Mixture of Normal Distributions

#### 4.1.3. FBS versus a Mixture of Log-Normal Distributions

**Remark**

**3.**

#### 4.2. Air Pollution

#### 4.2.1. FBS versus the BS and SBS Distributions

#### 4.2.2. FBS versus the Extended BS (EBS) Model

## 5. Conclusions

- (i)
- the FBS model provides consistently better fits than the BS and SBS models (they can be considered relevant precedents of our proposal)
- (ii)
- the FBS distribution can improve the fit provided by other competing models designed to deal with bimodality (such as a mixture of normal distributions). It can also perform better for unimodal situations in which a generalized BS model with skewness parameters must be applied, such as the EBS model proposed in [16]. We highlight that in both situations FBS provides a better fit with a more parsimonious model (less number of parameters), and the problem of identifiability of mixtures can be circumvented.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of Proposition 1 (cdf in the FBS model).**

**Proof**

**of Proposition 3 (Modes in the FBS model).**

**Remark**

**A1**

**.**In order to illustrate the use of Equations (12) and (13) next cases are considered.

- 1.
- Consider the pdf given in Figure 1a, case $\alpha =0.75$, $\beta =1$, $\lambda =1$, $\delta =0.75$. In this setting there do not exist ${t}_{1}^{*}\in (0,\beta )$ and ${t}_{2}^{*}>\beta $ satisfying (12) and (13), respectively. It can be checked than the distribution is unimodal and the mode is at β.
- 2.
- 3.

**Proof**

**of Theorem 1 (pth quantile, change of scale and reciprocity).**

- (i)
- (ii)
- Note that the pdf of T can be rewritten as$${f}_{T}\left(t\right)={c}_{\delta}{a}_{t}^{\prime}(\alpha ,\beta )\varphi \left(\right|{a}_{t}(\alpha ,\beta )|+\delta )\Phi \left(\lambda {a}_{t}(\alpha ,\beta )\right)$$$${a}_{t}^{\prime}={a}_{t}^{\prime}(\alpha ,\beta )=\frac{\partial}{\partial t}{a}_{t}(\alpha ,\beta )=\frac{{t}^{-3/2}{\beta}^{-1/2}}{2\alpha}(t+\beta )\phantom{\rule{0.277778em}{0ex}}.$$Let $Y=kT$ with $k>0$. By applying the Jacobian technique ${f}_{Y}\left(y\right)=\left|J\right|{f}_{T}(\frac{y}{k};\alpha ,\beta ,\delta ,\lambda )$ with $\left|J\right|=\frac{1}{k}$. From (6), ${a}_{y/k}(\alpha ,\beta )={a}_{y}(\alpha ,k\beta )$, and from (A12)$$\left|J\right|{a}_{y/k}^{\prime}(\alpha ,\beta )=\frac{{y}^{-3/2}{\left(k\beta \right)}^{-1/2}}{2\alpha}(y+k\beta )={a}_{y}^{\prime}(\alpha ,k\beta )\phantom{\rule{0.277778em}{0ex}}.$$Therefore$${f}_{Y}\left(y\right)={c}_{\delta}{a}_{y}^{\prime}(\alpha ,k\beta )\varphi \left(\right|{a}_{y}(\alpha ,k\beta )|+\delta )\Phi \left(\lambda {a}_{y}(\alpha ,k\beta )\right),$$
- (iii)
- Let be $Y={T}^{-1}$. In this case $\left|J\right|={Y}^{-2}$, ${a}_{{y}^{-1}}(\alpha ,\beta )=-{a}_{y}(\alpha ,{\beta}^{-1})$, and $\left|J\right|{a}_{y-1}^{\prime}(\alpha ,\beta )={a}_{y}^{\prime}(\alpha ,{\beta}^{-1})$. Therefore$${f}_{Y}\left(y\right)=\left|J\right|{f}_{T}({y}^{-1};\alpha ,\beta ,\delta ,\lambda )={c}_{\delta}{a}_{y}^{\prime}(\alpha ,{\beta}^{-1})\varphi \left(|{a}_{y}(\alpha ,{\beta}^{-1})|+\delta \right)\Phi \left(-\lambda {a}_{y}(\alpha ,{\beta}^{-1})\right),$$

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**Figure 1.**FBS distributions for $\alpha =0.75,$ $\beta =1$ (both fixed). In (

**a**) $\lambda =1$ versus (

**b**) $\lambda =-1$. Increasing values of $\delta >0$: $\delta =0.75$ (red solid line), $1.5$ (green dashed line), $2.25$ (black dotted line) and $3.0$ (blue dashed and dotted line).

**Figure 2.**Flexible Birnbaum–Saunders (FBS) distributions for $\alpha =0.30,$ $\beta =0.75$ (both fixed). In (

**a**) $\lambda =0.5$ versus (

**b**) $\lambda =-0.5$. Decreasing values of $\delta <0$: $\delta =-0.75$ (red solid line), $-1.5$ (green dashed line), $-2.25$ (black dotted line) and $-3.0$ (blue dashed and dotted line).

**Figure 3.**Hazard function of the FBS distribution for plots corresponding to Figure 1 (a), (b): $\alpha =0.75$, $\beta =1$ (both fixed), $\delta =0.75$ (red solid line), 1.5 (green dashed line), 2.25 (black dotted line) and 3.0 (blue dashed dotted line), in (a) $\lambda =1$ versus (b) $\lambda =-1$. For plots corresponding to Figure 2 (a), (b): $\alpha =0.30$, $\beta =0.75$ (both fixed), $\delta =-0.75$ (red solid line), –1.5 (green dashed line), –2.25 (black dotted line) and –3.0 (blue dashed dotted line), in (a) $\lambda =0.5$ versus (b) $\lambda =-0.5$.

**Figure 4.**(

**a**) Plots for FBS, (solid line), MLN (dashed line), BS (dotted line) and SBS (dotted and dashed line) models. (

**b**) Empirical cdf with estimated FBS cdf (dashed line) and estimated BS cdf (dotted line).

**Figure 5.**(

**a**) Plots for FBS, (solid line), BS (dotted line), SBS (dashed line) and EBS (dotted and dashed line) models. (

**b**) Empirical cdf with estimated FBS cdf (dashed line).

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## Share and Cite

**MDPI and ACS Style**

Martínez-Flórez, G.; Barranco-Chamorro, I.; Bolfarine, H.; Gómez, H.W.
Flexible Birnbaum–Saunders Distribution. *Symmetry* **2019**, *11*, 1305.
https://doi.org/10.3390/sym11101305

**AMA Style**

Martínez-Flórez G, Barranco-Chamorro I, Bolfarine H, Gómez HW.
Flexible Birnbaum–Saunders Distribution. *Symmetry*. 2019; 11(10):1305.
https://doi.org/10.3390/sym11101305

**Chicago/Turabian Style**

Martínez-Flórez, Guillermo, Inmaculada Barranco-Chamorro, Heleno Bolfarine, and Héctor W. Gómez.
2019. "Flexible Birnbaum–Saunders Distribution" *Symmetry* 11, no. 10: 1305.
https://doi.org/10.3390/sym11101305