# A Comparison of Forecasting Mortality Models Using Resampling Methods

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## Abstract

**:**

## 1. Introduction

## 2. Fitting and Prediction of the Lee–Carter Models

## 3. Resampling Methods for Evaluating the Forecasting Abilities of the Models

- Hold-out;
- Repeated hold-out;
- Leave-one-out-CV (Cross Validation); and
- K-fold CV.

#### 3.1. Hold-Out or Out-Of-Sample

#### 3.2. Repeated Hold-Out

#### 3.3. Leave-One-Out Cross-Validation

#### 3.4. K-Fold Cross-Validation

## 4. Choosing the Optimal Mortality Model

## 5. Analysis of the Mortality Data from the Human Mortality Database

#### 5.1. Description of the Data

#### 5.2. Forecasting Abilities of the Models

#### 5.3. Hold-Out

- The sample is subdivided into two subsets: the training set contains 75% of the data that correspond to the 1990–2009 period, and the validation set comprises the remaining 25% of the data that cover the 2010–2016 period. The validation set that includes the last years of the sample period is employed to evaluate the forecasting ability of the models.
- The three mortality models are fitted by using the training set, and the corresponding estimations of parameters ${a}_{x}$, ${b}_{x}^{\left(i\right)}$ and ${k}_{t}^{\left(i\right)}$ are obtained for each model.
- Once the ${k}_{t}^{\left(i\right)}$ values are estimated with the training set data, we proceed to fit an ARIMA model to forecast the ${k}_{t}$ values of the validation set (2010–2016). The particular ARIMA model is selected according to the AIC, as explained in Section 2.
- Forecasted life tables are obtained for each model, and then, the forecasting ability of each model is obtained by using the goodness-of-fit measures described in Section 4 that are calculated with the validation dataset.

#### 5.4. Repeated Hold-Out

- We randomly subdivide the sample into two subsets. Of the total data, 75% are used as the training subset, and the remaining 25% are the validation subset. Here, the data that correspond to the years 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1998, 1999, 2000, 2001, 2002, 2003, 2005, 2006, 2009, 2011, 2013, 2014 and 2016 are used as the training set, and the data that correspond to the years 1997, 2004, 2007, 2008, 2010, 2012 and 2015 form the validation set.
- The three models are fitted with the training data set that obtains the corresponding estimates ${a}_{x}$, ${b}_{x}^{\left(i\right)}$ and ${k}_{t}^{\left(i\right)}$.
- Since the training set does not contain serialized data, we use the na.kalman function from the imputeTS library of [70] to estimate the missing values by using ARIMA time series models and obtain the ${k}_{t}^{\left(i\right)}$ values that corresponds to the years included in the validation set.
- Finally, we obtain the forecasted life tables of the years included in the validation set; then, the forecasting ability of the model is obtained by using the goodness-of-fit measures described in Section 4 that are applied to the validation dataset.

#### 5.5. Leave-One-Out CV

- We use the first three years of the sample (1990, 1991 and 1992) as the training set. According to the tsCV function of the forecast library developed by [50], three is the minimum number of years necessary to fit the mortality models used in this study.
- We obtain the estimations of ${a}_{x}$, ${b}_{x}^{\left(i\right)}$ and ${k}_{t}^{\left(i\right)}$.
- By using the ARIMA model that best fits the ${k}_{t}^{\left(i\right)}$ values, a single forecast is obtained for the ${k}_{t}^{\left(i\right)}$ that correspond to the year 1993.
- Once these data are projected for 1993, we obtain the corresponding forecasted probabilities of death for all ages (from zero to 109 years), countries and populations, and we then proceed to calculate the forecasting ability measures with the 1993 data as the validation set.

#### 5.6. The 5-Fold CV

- We proceed to subdivide the sample into six equally sized subsets, that include subset data from four consecutive years. The first subset consists of data from 1990 to 1994 and is used only as a training set. The second subset contains data from 1995 to 1998, the third subset contains data from 1999 to 2002, the fourth subset contains data from 2003 to 2006, the fifth subset contains data from 2007 to 2011 and the sixth subset contains data from 2012 to 2016.
- With the data that correspond to the period from 1990 to 1994, we obtain the estimations of ${a}_{x}$, ${b}_{x}^{\left(i\right)}$ and ${k}_{t}^{\left(i\right)}$.
- We fit the ARIMA model to the values of ${k}_{t}^{\left(i\right)}$ by obtaining projections for the ${k}_{t}^{\left(i\right)}$ values that correspond to the second subset (from 1995 to 1998) that is used as the validation set.
- Finally, we forecast the life tables for each country according to sex and age from 1995 to 1998, and we can then proceed to determine the different measures of the forecasting ability of the mortality models employed in this study.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Plot summarizing the sum of squares errors (SSE) of each European country that applies the unifactorial Lee–Carter model to males and females, according to every employed resampling method.

**Figure A2.**Plot summarizing the mean absolute measure (MAE) of each European country that applies the unifactorial Lee–Carter model to males and females, according to every employed resampling method.

**Figure A3.**Plot that summarizing the sum of squares errors (SSE) of each European country that applies bifactorial Lee–Carter model to males and females, according to every employed resampling method.

**Figure A4.**Plot summarizing the mean absolute measure (MAE) of each European country that applies bifactorial Lee–Carter model to males and females, according to every employed resampling method.

**Figure A5.**Plot summarizing the sum of squares errors (SSE) of each European country that applies bifactorial Lee–Carter model with orthogonalized parameters to males and females, according to every employed resampling method.

**Figure A6.**Plot summarizing the mean absolute measure (MAE) of each European country that applies bifactorial Lee–Carter model with orthogonalized parameters to males and females, according to every employed resampling method.

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**Figure 1.**A schematic display of the employed resampling methods for an embedded time series. The training set, validation set and omitted set are shown in gray, white and black, respectively.

**Figure 2.**A schematic display of the employed resampling methods for nonembedded time series. The training sets and validation sets are shown in gray and white, respectively.

**Figure 3.**Radar graph summarizing the number of countries for which each model (LC, LC2, and LC2-O) achieves the best forecasting ability according to the hold-out method for males (

**a**) and females (

**b**).

**Figure 4.**Radar graph summarizing the number of countries for which each model (LC, LC2, and LC2-O) achieves the best forecasting ability according to the repeated hold-out method for males (

**a**) and females (

**b**).

**Figure 5.**Radar graphs summarizing the number of countries for which each model (LC, LC2, and LC2-O) achieves the best forecasting ability according to the leave-one-out method for males (

**a**) and females (

**b**).

**Figure 6.**Radar graph that summarizing the number of countries for which each model (LC, LC2, and LC2-O) achieves the best forecasting ability according to the 5-fold CV method for males (

**a**) and females (

**b**).

Paper | Measure of Goodness Fit | Mortality Models | Selected Model |
---|---|---|---|

[31] | ${R}^{2}$ | Gompertz–Makeham $(0,11)$ | Logit–Gompertz–Makeham $(0,11)$ |

MAPE | Logit–Gompertz–Makeham $(0,11)$ | ||

Heligman–Pollard–2law | |||

[32] | MAE | LC1/LM | BMS with small differences |

ME | BMS/HU | ||

LC-smooth | |||

[33] | AIC | LC1-Negative Binomial | LC1-Negative Binomial |

BIC | LC1-Poisson | ||

[24] | MAPE | LC1-Logit/LC2-Logit | MP |

MSE | Median Polish (MP) | ||

[34] | MAPE | LC1-SVC/LC1-GLM | LC2 |

MSE | LC1-ML/LC2-SVD | ||

LC2-GLM/LC2-ML | |||

[6] | BIC | LC1/LC2-Cohort | LC2-Cohort |

APC/M5 | M8 | ||

[35] | |||

M6/M7/M8 | |||

[36] | SSE | M5-Logit | M5-Logit |

M5-Log/M5-Probit | |||

[37] | BIC | [37] model | [37] model |

LC1/M7/M5 | |||

LC2-Cohort | |||

[35] model | |||

[38] | Deviance | LC1/LC1-res | LC-APC |

MSE | LC2/LC2-res | ||

MAPE | LC-APC/LC-APC-res | ||

MP/MP-res | |||

MP-APC/MP-APC-res | |||

[39] | MAPE | LC1/APC1 | [39] model |

BIC | APC2/CBD | ||

[39] model | |||

[40] | AIC | LC1/${H}_{1}$ | M7/M8/${H}_{1}$ |

BIC | M/LC2 | ||

HQC | M5/M6 | ||

M7/M8 | |||

[41] | RSSE | [41] model | RH |

$U{V}_{x}$ | LC1/RH | [41] | |

BIC | ${H}_{1}$/${M}_{3}$ | ||

[37] model | |||

[42] | MSE | LC1/CBD | [42] model |

MAPE | [42] model | ||

[43] | MAPE | P-Double-LC2/M-Double-LC2 | P-Common-LC2 |

AIC | P-Common-LC2/M-Common-LC1 | ||

BIC | P-Simple-LC1/M-Simple-LC1 | ||

MAPE | P-Division-LC1/M-Division-LC1 | ||

P-One-LC1/M-One-LC1 | |||

[44] | BIC | PCFC | PCFC |

MAPE | PCFM | ||

[45] | AIC | GAS Poisson/GAS Binomial | GAS Negative Binomial |

MAPE | GAS Negative Binomial | ||

GAS Gaussian/GAS Beta |

**Table 2.**List of the Lee–Carter models used in this study, with the acronyms, parameter constraints and equations.

Label Model | Parameter Constraints | Formula |
---|---|---|

LC | ${\sum}_{x}{b}_{x}^{\left(1\right)}=1$ | $ln\left(\right)open="("\; close=")">\frac{{q}_{x,t}}{1-{q}_{x,t}}$ |

${\sum}_{t}{k}_{t}^{\left(1\right)}=0$ | ||

LC2 | ${\sum}_{x}{b}_{x}^{\left(1\right)}={\sum}_{x}{b}_{x}^{\left(2\right)}=1$ | $ln\left(\right)open="("\; close=")">\frac{{q}_{x,t}}{1-{q}_{x,t}}$ |

${\sum}_{t}{k}_{t}^{\left(1\right)}={\sum}_{t}{k}_{t}^{\left(2\right)}=0$ | ||

LC2-O | ${\sum}_{x}|{b}_{x}^{\left(1\right)}|={\sum}_{x}\left|{b}_{x}^{\left(2\right)}\right|=1$ | $ln\left(\right)open="("\; close=")">\frac{{q}_{x,t}}{1-{q}_{x,t}}$ |

${\sum}_{t}{k}_{t}^{\left(1\right)}={\sum}_{t}{k}_{t}^{\left(2\right)}=0$ | ||

${\sum}_{t}{b}_{x}^{\left(1\right)}\xb7{b}_{x}^{\left(2\right)}=0$ | ||

${\sum}_{t}{k}_{t}^{\left(1\right)}\xb7{k}_{t}^{\left(2\right)}=0$ |

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Atance, D.; Debón, A.; Navarro, E.
A Comparison of Forecasting Mortality Models Using Resampling Methods. *Mathematics* **2020**, *8*, 1550.
https://doi.org/10.3390/math8091550

**AMA Style**

Atance D, Debón A, Navarro E.
A Comparison of Forecasting Mortality Models Using Resampling Methods. *Mathematics*. 2020; 8(9):1550.
https://doi.org/10.3390/math8091550

**Chicago/Turabian Style**

Atance, David, Ana Debón, and Eliseo Navarro.
2020. "A Comparison of Forecasting Mortality Models Using Resampling Methods" *Mathematics* 8, no. 9: 1550.
https://doi.org/10.3390/math8091550