# A Bimodal Discrete Shifted Poisson Distribution. A Case Study of Tourists’ Length of Stay

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## Abstract

**:**

## 1. Introduction

## 2. The Bimodal Shifted Poisson Model

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

## 3. Model Estimation

#### Simulation Study

## 4. Including Covariates

#### Marginal Effects

## 5. Empirical Analysis

#### 5.1. Literature Review on Length of Tourist Stay

#### 5.2. Data

#### 5.3. Model Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Annex: Brief Description of Variables Used

- Length of tourist stay (days) in the Canary Islands.
- Expenditure at origin (€). Flights and accommodation for the travel party, paid in advance. This is usually the main expenditure by each tourist, and is effected in the country of origin.
- Some individual characteristics.
- 3.1
- Household income. Measured as an ordered categorical variable, not as a continuous one. This variable takes the following values: $=1$, from 12,000 to 24,000€; $=2$, from 24,001 to 36,000€; $=3$, from 36,001 to 48,000€; $=4$, from 48,001 to 60,000€; $=5$, from 60,001 to 72,000€; $=6$, from 72,001 to 84,000€; and $=7$, higher than 84,000€. Medium income is a dummy variable which takes the value one when categories are 3, 4 or 5, and 0 otherwise. High income is a dummy variable that takes the value 1 for categories 6 and 7, and 0 otherwise.
- 3.2
- Age of the survey respondent. Finally, we controlled for seasonal variables, considering three dummies: summer, autumn and winter. Summer was taken as June to September, autumn as October to December and winter as January to March. In winter, there is much less competition in the sun-and-beach tourism market than in summer, as there are few good alternatives for tourists in this demand segment (in many cases, too, tourists prefer to repeat their visits) [24].

- Some vacation characteristics.
- 4.1
- Type of accommodation. The following types of variable were considered: a dummy variable, taking the value 1 if the tourist accommodation is a 4 or 5 star hotel/aparthotel, and 0 otherwise; a second dummy variable, taking the value 1 if the tourist accommodation is a 1, 2 or 3 star hotel/aparthotel, and 0 otherwise; and a third dummy variable taking the value 1 if the tourist stays in non-hotel accommodation and the value 0 otherwise. The reference categories considered are own home or staying at the home of friends or family, or other types of accommodation.
- 4.2
- Travel party size or family size. The number of persons booking the holiday package paid for in the country of origin.
- (a)
- Repetition. The number of previous visits to the Canary Islands. A value of 0 is possible, indicating that at the moment of the interview, this is the tourist’s first visit to the Canary Islands.

- 4.3
- Transport (return flight) and accommodation booked via a tour operator.
- 4.4
- Low cost. This is a dummy variable, taking the value 1 if the travel arrangements were made with a low-cost carrier, and 0 otherwise.
- 4.5
- Sun and beach. A dummy variable, taking the value 1 if the tourist’s motive for travel is a “sun and beach” holiday, and 0 otherwise.

## References

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**Table 1.**Empirical mean, standard deviation (SD) and coverage (C) values for different values of parameters $\theta $ and $\lambda $.

n | $\mathit{\theta}$ | $\mathit{\alpha}$ | $\widehat{\mathit{\theta}}$ | $\mathit{sd}(\widehat{\mathit{\theta}})$ | $\mathit{c}(\widehat{\mathit{\theta}})$ | $\widehat{\mathit{\alpha}}$ | $\mathit{sd}(\widehat{\mathit{\alpha}})$ | $\mathit{c}(\widehat{\mathit{\alpha}})$ |
---|---|---|---|---|---|---|---|---|

50 | 2 | 2 | 2.0678 | 0.5261 | 93.6 | 1.9961 | 0.0727 | 93.0 |

100 | 2 | 2 | 2.0364 | 0.3474 | 94.4 | 1.9974 | 0.0504 | 94.5 |

150 | 2 | 2 | 2.0270 | 0.2783 | 95.8 | 1.9988 | 0.0410 | 94.4 |

200 | 2 | 2 | 2.0177 | 0.2384 | 95.4 | 1.9987 | 0.0355 | 95.3 |

50 | 2 | 3 | 2.0634 | 0.5358 | 94.3 | 2.9946 | 0.0730 | 93.1 |

100 | 2 | 3 | 2.0377 | 0.3544 | 95.2 | 2.9974 | 0.0501 | 94.5 |

150 | 2 | 3 | 2.0254 | 0.2836 | 95.5 | 2.9984 | 0.0408 | 95.3 |

200 | 2 | 3 | 2.0169 | 0.2433 | 95.8 | 2.9981 | 0.0353 | 95.6 |

50 | 2 | 4 | 2.0548 | 0.5370 | 93.7 | 3.9928 | 0.0715 | 94.1 |

100 | 2 | 4 | 2.0359 | 0.3582 | 94.4 | 3.9957 | 0.0500 | 94.6 |

150 | 2 | 4 | 2.0233 | 0.2865 | 95.8 | 3.9967 | 0.0407 | 95.2 |

200 | 2 | 4 | 2.0140 | 0.2455 | 95.6 | 3.9965 | 0.0352 | 95.4 |

50 | 3 | 2 | 3.2750 | 1.1502 | 92.8 | 1.9999 | 0.0626 | 94.5 |

100 | 3 | 2 | 3.0860 | 0.6542 | 92.7 | 2.0007 | 0.0437 | 94.5 |

150 | 3 | 2 | 3.0693 | 0.5187 | 94.2 | 2.0008 | 0.0355 | 94.6 |

200 | 3 | 2 | 3.0428 | 0.4397 | 95.1 | 2.0004 | 0.0307 | 95.5 |

50 | 3 | 3 | 3.3158 | 1.2529 | 93.3 | 2.9985 | 0.0621 | 93.4 |

100 | 3 | 3 | 3.1165 | 0.6746 | 93.9 | 2.9996 | 0.0437 | 95.4 |

150 | 3 | 3 | 3.0896 | 0.5302 | 95.5 | 2.9996 | 0.0355 | 95.1 |

200 | 3 | 3 | 3.0594 | 0.4479 | 94.7 | 2.9996 | 0.0308 | 94.9 |

50 | 3 | 4 | 3.4002 | 2.2907 | 93.4 | 3.9967 | 0.0622 | 94.0 |

100 | 3 | 4 | 3.1205 | 0.6852 | 94.0 | 3.9974 | 0.0437 | 96.1 |

150 | 3 | 4 | 3.0886 | 0.5346 | 96.2 | 3.9977 | 0.0355 | 95.2 |

200 | 3 | 4 | 3.0594 | 0.4520 | 95.8 | 3.9976 | 0.0307 | 95.0 |

50 | −2 | 2 | −2.1168 | 0.6585 | 96.1 | 2.0050 | 0.0742 | 94.4 |

100 | −2 | 2 | −2.0606 | 0.4220 | 95.4 | 2.0004 | 0.0508 | 94.1 |

150 | −2 | 2 | −2.0405 | 0.3339 | 95.3 | 2.0009 | 0.0412 | 94.6 |

200 | −2 | 2 | −2.0265 | 0.2847 | 95.9 | 2.0002 | 0.0356 | 95.6 |

50 | −2 | 3 | −2.0988 | 0.6171 | 95.6 | 3.0047 | 0.0729 | 93.9 |

100 | −2 | 3 | −2.0468 | 0.4012 | 95.4 | 3.0010 | 0.0503 | 94.0 |

150 | −2 | 3 | −2.0335 | 0.3196 | 95.7 | 3.0011 | 0.0408 | 95.0 |

200 | −2 | 3 | −2.0209 | 0.2730 | 96.3 | 3.0005 | 0.0353 | 95.9 |

50 | −2 | 4 | −2.0935 | 0.6023 | 95.1 | 4.0026 | 0.0724 | 94.0 |

100 | −2 | 4 | −2.0480 | 0.3933 | 95.3 | 3.9986 | 0.0500 | 94.3 |

150 | −2 | 4 | −2.0367 | 0.3140 | 95.7 | 3.9987 | 0.0406 | 94.7 |

200 | −2 | 4 | −2.0255 | 0.2686 | 95.6 | 3.9982 | 0.0351 | 95.3 |

Variables | Mean | Standard Deviation |
---|---|---|

Length of stay | 8 nights | 3 nights |

Expenditure at origin | 1473 euro | 1038 euro |

Household income (categorical variable) | 3 | 2 |

Job | 84% | – |

Nationality | 17% | |

Sun & beach | 94% | |

Low cost | 36% | |

Travel party size | 2 persons | 1 persons |

Repetition | 3 times | 3 times |

4–5 star hotel or other accommodation | 47% | |

1, 2, 3 star hotel or other accommodation | 16% | |

Transport booked by tour operator | 56% | |

Accommodation booked by tour operator | 48% | |

Age of the respondent (years) | 42 | 13 |

Number of tourists after data cleansing | 17,923 |

**Table 3.**Maximum likelihood estimates for models without covariates: Zero truncated Poisson (ZTP), zero truncated negative binomial (ZTNB) and bimodal shifted Poisson. The p-values are shown in parentheses.

ZTP | ZTNB | Bimodal Shifted Poisson | |
---|---|---|---|

$\widehat{\theta}$ | 0.336 | 1.643 | |

$\left[0.00\right]$ | $\left[0.00\right]$ | ||

$\widehat{\alpha}$ | 8.454 | 8.450 | 2.978 |

$\left[0.00\right]$ | $\left[0.00\right]$ | $\left[0.00\right]$ | |

${\ell}_{max}$ | −46,519.40 | −46,003.90 | −44,386.10 |

AIC | 93,040.90 | 92,011.80 | 88,776.20 |

BIC | 93,048.70 | 92,027.40 | 88,791.80 |

CAIC | 93,049.70 | 92,029.40 | 88,793.80 |

Observations | 17,923 | 17,923 | 17,923 |

**Table 4.**Ordinary least squares (OLS) and maximum likelihood (ML) estimates for TP, TNB and the bimodal shifted Poisson models with the inclusion of covariates.

OLS | ZTP | ZTNB | Bimodal Shifted Poisson | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Coeff | p-Value | Coeff | p-Value | ME | Coeff | p-Value | ME | Coeff | p-Value | ME | |

Expenditure at origin (in logs) | 2.255 | <0.01 | 0.165 | <0.01 | 1.26 | 0.168 | <0.01 | 1.28 | 0.200 | <0.01 | 1.53 |

Medium income | −0.398 | <0.01 | −0.042 | <0.01 | −0.32 | −0.041 | <0.01 | −0.32 | −0.043 | <0.01 | −0.33 |

High income | −0.870 | <0.01 | −0.093 | <0.01 | −0.71 | −0.092 | <0.01 | −0.70 | −0.102 | <0.01 | −0.78 |

Repetition | 0.128 | <0.01 | 0.017 | <0.01 | 0.13 | 0.017 | <0.01 | 0.13 | 0.016 | <0.01 | 0.13 |

Sun and beach | −0.014 | 0.876 | −0.005 | 0.592 | −0.04 | −0.002 | 0.855 | −0.02 | 0.019 | 0.102 | 0.15 |

Age | 0.036 | <0.01 | 0.004 | <0.01 | 0.04 | 0.004 | <0.01 | 0.04 | 0.004 | <0.01 | 0.04 |

Nationality | −0.861 | <0.01 | −0.199 | <0.01 | −1.51 | −0.198 | <0.01 | −1.51 | −0.193 | <0.01 | −1.47 |

Pre-booked transport and accommodation | −0.475 | <0.01 | −0.071 | <0.01 | −0.54 | −0.069 | <0.01 | −0.53 | −0.065 | <0.01 | −0.50 |

Low cost | −0.177 | <0.01 | −0.030 | <0.01 | −0.23 | −0.030 | <0.01 | −0.23 | −0.033 | <0.01 | −0.26 |

4–5 star hotels | −2.785 | <0.01 | 0.165 | <0.01 | 1.26 | 0.168 | <0.01 | 1.28 | 0.200 | <0.01 | 1.53 |

1, 2 or 3 star hotels | −2.017 | <0.01 | 0.194 | <0.01 | 1.47 | 0.197 | <0.01 | 1.50 | 0.237 | <0.01 | 1.80 |

Non-hotel accommodation | −1.269 | <0.01 | 0.265 | <0.01 | 2.01 | 0.268 | <0.01 | 2.04 | 0.312 | <0.01 | 2.37 |

Travel party size | −0.665 | <0.01 | −0.043 | <0.01 | −0.33 | −0.043 | <0.01 | −0.33 | −0.050 | <0.01 | −0.38 |

Q1 | −0.205 | 0.002 | −0.019 | 0.004 | −0.15 | −0.020 | 0.035 | −0.16 | −0.032 | <0.01 | −0.25 |

Q3 | 0.918 | <0.01 | 0.129 | <0.01 | 0.98 | 0.128 | <0.01 | 0.98 | 0.138 | <0.01 | 1.06 |

Q4 | −0.150 | 0.023 | −0.017 | 0.009 | −0.13 | −0.017 | 0.052 | −0.13 | −0.019 | 0.019 | −0.14 |

Constant | −4.982 | <0.01 | 0.700 | <0.01 | 5.32 | 0.679 | <0.01 | 5.16 | 0.295 | <0.01 | 2.24 |

$\theta $ | 0.091 | <0.01 | 0.890 | 0.000 | |||||||

${\ell}_{max}$ | −45,212.191 | −43,962.17 | −43,914.45 | −42,937.100 | |||||||

AIC | 87,958.30 | 87,864.90 | 85,910.20 | ||||||||

BIC | 88,090.80 | 88,005.20 | 86,050.50 | ||||||||

CAIC | 88,107.80 | 88,023.20 | 86,068.50 | ||||||||

Observations | 17,923 | 17,923 | 17,923 | 17,923 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gómez-Déniz, E.; Pérez-Rodríguez, J.V.; Reyes, J.; Gómez, H.W.
A Bimodal Discrete Shifted Poisson Distribution. A Case Study of Tourists’ Length of Stay. *Symmetry* **2020**, *12*, 442.
https://doi.org/10.3390/sym12030442

**AMA Style**

Gómez-Déniz E, Pérez-Rodríguez JV, Reyes J, Gómez HW.
A Bimodal Discrete Shifted Poisson Distribution. A Case Study of Tourists’ Length of Stay. *Symmetry*. 2020; 12(3):442.
https://doi.org/10.3390/sym12030442

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Jorge Vicente Pérez-Rodríguez, Jimmy Reyes, and Héctor W. Gómez.
2020. "A Bimodal Discrete Shifted Poisson Distribution. A Case Study of Tourists’ Length of Stay" *Symmetry* 12, no. 3: 442.
https://doi.org/10.3390/sym12030442