# Negative Binomial Kumaraswamy-G Cure Rate Regression Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Kumaraswamy Family of Distributions

**Definition.**

#### 2.2. The Unified Cure Rate Model

- If ${p}_{0}=1$, then ${S}_{pop}\left(t\right)=S\left(t\right)$;
- ${S}_{pop}\left(0\right)=1$;
- ${S}_{pop}\left(t\right)$ it is not increasing;
- ${lim}_{t\to \infty}{S}_{pop}\left(t\right)={p}_{0}$.

#### 2.3. Negative Binomial Distribution

#### 2.4. Negative Binomial Kumaraswamy-G Cure Rate Model

#### Negative Binomial Kumaraswamy Exponential Cure Rate Model

#### 2.5. Negative Binomial Kumaraswamy-G Regression Cure Rate Model

#### 2.6. Inference

`optim`function in R (R Core Team 2013).

#### 2.7. Simulation Studies

- Determine the desired parameter values, as well as the value of the cured fraction p;
- For each $i=1,\dots ,n$, generate a random variable ${M}_{i}\sim $ Bernoulli$(1-p)$;
- If ${M}_{i}=0$ set ${t}_{i}^{\prime}=\infty $. If ${M}_{i}=1$, take ${t}_{i}^{\prime}$ as the root of $F\left(t\right)=u$, where $u\sim $ uniform$(0,1-p)$;
- Generate ${u}_{i}^{\prime}\sim $ uniform$\left(0,max\left({t}_{i}^{\prime}\right)\right)$, for $i=1,\dots ,n$, considering only the finite ${t}_{i}^{\prime}$;
- Calculate ${t}_{i}=min\left({t}_{i}^{\prime},{u}_{i}^{\prime}\right)$. If ${t}_{i}<{u}_{i}^{\prime}$ set ${\delta}_{i}=1$, otherwise set ${\delta}_{i}=0$.

## 3. Real Data Application

`optim`was used to maximize the log-likelihood function. The algorithm “BFGS” was chosen for maximization. For computational stability, the observed times in each data set were divided by their maximum value. As the simulations results shows large values for deviation in small sample sizes, we are going to use 1000 bootstrap estimates for the deviations of the parameters.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**From the left to the right, top to bottom, the BerKumExp, PoiKumExp, GeoKumExp and NegBinKumExp distributions. The colors black, red, green and blue represent the nodule categories 1, 2, 3 and 4, respectively.

**Table 1.**Survival function ${S}_{pop}\left(t\right)$, density function ${f}_{pop}\left(t\right)$, and cured fraction of different distributions of latent causes.

Distribution | ${\mathit{S}}_{\mathit{pop}}\left(\mathit{t}\right)$ | ${\mathit{f}}_{\mathit{pop}}\left(\mathit{t}\right)$ | ${\mathit{p}}_{0}$ | $\mathit{A}\left(\mathit{s}\right)$ |
---|---|---|---|---|

Bernoulli$\left(\theta \right)$ | $1-\theta +\theta S\left(t\right)$ | $\theta f\left(t\right)$ | $1-\theta $ | $1-\theta +\theta s$ |

Binomial$(K,{\theta}^{*})$ | ${\left[1-{\theta}^{*}+{\theta}^{*}S\left(t\right)\right]}^{K}$ | $K{\theta}^{*}f\left(t\right){\left[1-{\theta}^{*}+{\theta}^{*}S\left(t\right)\right]}^{K-1}$ | ($1-{\theta}^{*}{)}^{K}$ | ${(1-{\theta}^{*}+{\theta}^{*}s)}^{K}$ |

Poisson$\left(\theta \right)$ | $exp\left[-\theta F\left(t\right)\right]$ | $\theta f\left(t\right)exp\left[-\theta F\left(t\right)\right]$ | ${e}^{-\theta}$ | $exp\left[\theta (1-s)\right]$ |

Geometric$\left(\theta \right)$ | ${\left[1+\theta F\left(t\right)\right]}^{-1}$ | $\theta f\left(t\right){\left[1+\theta F\left(t\right)\right]}^{-2}$ | $1/(1+\theta )$ | ${\left[1+\theta (1-s)\right]}^{-1}$ |

Negative Binomial$(\eta ,\theta )$ | ${\left[1+\eta \theta F\left(t\right)\right]}^{-1/\eta}$ | $\theta f\left(t\right){\left[1+\eta \theta F\left(t\right)\right]}^{-1-1/\eta}$ | ${(1+\eta \theta )}^{-1/\eta}$ | ${\left[1+\eta \theta (1-s)\right]}^{-1/\eta}$ |

**Table 2.**${S}_{pop}\left(t\right)$, ${f}_{pop}\left(t\right)$ and the cured fraction for different distributions of N.

Parametrization | Model | ${\mathit{S}}_{\mathit{pop}}\left(\mathit{t}\right)$ |
---|---|---|

$\eta \to 0$ | Poisson | $exp\left\{-\theta \left\{1-{\left[1-G{\left(t\right)}^{\lambda}\right]}^{\phi}\right\}\right\}$ |

$\eta =-1$ | Bernoulli | $1-\theta +\theta {\left[1-G{\left(t\right)}^{\lambda}\right]}^{\phi}$ |

$\eta =-1/m$ | Binomial | ${\left\{1-\frac{\theta}{m}+\frac{\theta}{m}{\left[1-G{\left(t\right)}^{\lambda}\right]}^{\phi}\right\}}^{m}$ |

$\eta =1$ | geometric | ${\left\{1+\theta \left\{1-{\left[1-G{\left(t\right)}^{\lambda}\right]}^{\phi}\right\}\right\}}^{-1}$ |

Parameters | Estimates | Std. Dev. | Inf 95% CI | Sup 95% CI |
---|---|---|---|---|

$\alpha $ | 1.8052 | 0.7308 | 0.6052 | 3.8146 |

$\lambda $ | 3.5177 | 1.2003 | 2.2506 | 6.6982 |

$\varphi $ | 0.4774 | 0.3695 | 0.1361 | 1.5992 |

${\beta}_{0}$ | −1.4788 | 0.2245 | −1.9330 | −1.0434 |

${\beta}_{1}$ | 0.2288 | 0.0505 | 0.1281 | 0.3251 |

${\beta}_{2}$ | 0.0045 | 0.0039 | −0.0025 | 0.0121 |

${p}_{1}$ | 0.6412 | 0.0420 | 0.5508 | 0.7171 |

${p}_{2}$ | 0.5506 | 0.0360 | 0.4769 | 0.6185 |

${p}_{3}$ | 0.4357 | 0.0364 | 0.3607 | 0.5040 |

${p}_{4}$ | 0.2896 | 0.0590 | 0.1780 | 0.3991 |

Parameters | Estimates | Std. Dev. | Inf 95% CI | Sup 95% CI |
---|---|---|---|---|

$\alpha $ | 1.0735 | 0.7308 | 0.2507 | 2.8913 |

$\lambda $ | 3.0298 | 1.0019 | 2.0155 | 5.6100 |

$\varphi $ | 1.2187 | 1.8100 | 0.1268 | 5.2827 |

${\beta}_{0}$ | −1.7046 | 0.3675 | −2.4282 | −1.0042 |

${\beta}_{1}$ | 0.3640 | 0.0724 | 0.2164 | 0.5122 |

${\beta}_{2}$ | 0.0103 | 0.0060 | −0.0012 | 0.0225 |

${p}_{1}$ | 0.6490 | 0.0464 | 0.5486 | 0.7349 |

${p}_{2}$ | 0.5384 | 0.0412 | 0.4506 | 0.6141 |

${p}_{3}$ | 0.4110 | 0.0424 | 0.3269 | 0.4906 |

${p}_{4}$ | 0.2796 | 0.0525 | 0.1798 | 0.3827 |

Parameters | Estimates | Std. Dev. | Inf 95% CI | Sup 95% CI |
---|---|---|---|---|

$\alpha $ | 0.7298 | 0.5598 | 0.1084 | 2.1395 |

$\lambda $ | 2.8893 | 0.8340 | 2.0228 | 4.7584 |

$\varphi $ | 2.4622 | 4.8243 | 0.1135 | 16.4430 |

${\beta}_{0}$ | −1.7930 | 0.4827 | −2.7416 | −0.8808 |

${\beta}_{1}$ | 0.5083 | 0.0902 | 0.3292 | 0.6860 |

${\beta}_{2}$ | 0.0144 | 0.0079 | −0.0001 | 0.0300 |

${p}_{1}$ | 0.6421 | 0.0543 | 0.5212 | 0.7303 |

${p}_{2}$ | 0.5207 | 0.0486 | 0.4147 | 0.5995 |

${p}_{3}$ | 0.3963 | 0.0457 | 0.2976 | 0.4788 |

${p}_{4}$ | 0.2846 | 0.0462 | 0.1905 | 0.3772 |

Parameters | Estimates | Std. Dev. | Inf 95% CI | Sup 95% CI |
---|---|---|---|---|

$\alpha $ | 0.3499 | 0.3798 | 0.0533 | 1.3675 |

$\lambda $ | 2.8630 | 0.5271 | 2.1450 | 4.1202 |

$\varphi $ | 9.7127 | 15.0785 | 0.0946 | 56.0397 |

$\eta $ | 3.1508 | 1.6134 | 0.7643 | 7.0171 |

${\beta}_{0}$ | −1.4374 | 1.1867 | −3.0628 | 1.6385 |

${\beta}_{1}$ | 0.7673 | 0.2003 | 0.4468 | 1.2376 |

${\beta}_{2}$ | 0.0211 | 0.0123 | −0.0014 | 0.0480 |

${p}_{1}$ | 0.6073 | 0.0865 | 0.3520 | 0.7217 |

${p}_{2}$ | 0.4956 | 0.0703 | 0.2981 | 0.5906 |

${p}_{3}$ | 0.3931 | 0.0577 | 0.2470 | 0.4778 |

${p}_{4}$ | 0.3065 | 0.0533 | 0.1883 | 0.3937 |

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**MDPI and ACS Style**

D’Andrea, A.; Rocha, R.; Tomazella, V.; Louzada, F. Negative Binomial Kumaraswamy-G Cure Rate Regression Model. *J. Risk Financial Manag.* **2018**, *11*, 6.
https://doi.org/10.3390/jrfm11010006

**AMA Style**

D’Andrea A, Rocha R, Tomazella V, Louzada F. Negative Binomial Kumaraswamy-G Cure Rate Regression Model. *Journal of Risk and Financial Management*. 2018; 11(1):6.
https://doi.org/10.3390/jrfm11010006

**Chicago/Turabian Style**

D’Andrea, Amanda, Ricardo Rocha, Vera Tomazella, and Francisco Louzada. 2018. "Negative Binomial Kumaraswamy-G Cure Rate Regression Model" *Journal of Risk and Financial Management* 11, no. 1: 6.
https://doi.org/10.3390/jrfm11010006