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

Bimodal and multimodal distributions are found in many continuous and discrete data sets; for example, in aggregate counts of responses to Likert scale questions, as in online ratings of movies or hotels [1]; in the durations of intervals between the eruptions of certain geysers; in the distributions of male and female body weights; in student test scores, distinguishing between those who studied for the test and those who did not; and in tourism analysis, regarding the number of nights that tourists spend at a given destination [2,3]. However, these distributions have received little attention in theoretical and empirical literature, with the exceptions of classical distributions based on continuous data, such as exponential or normal distributions [4]; discrete data frameworks, such as censored count data (where an additional mode might be used for the highest category; see [5]); latent class models for count data which account for heterogeneity using a finite mixture of unimodal Poisson distributions (i.e., the latent class truncated Poisson regression [6]); and flexible models that capture both over and underdispersion, such as the mixed Conway–Maxwell–Poisson distribution, which can reflect a wide range of truncated discrete data, and can exhibit either unimodal or bimodal behaviour [1] (the Conway–Maxwell–Poisson (CMP) distribution is a two-parameter generalisation of the Poisson distribution that allows for either over or underdispersion.). An important feature of multimodal data sets is that they can reveal when two or more types of individuals are represented in a data set (for example, consumer segments and preferences). The discrete data observed in the specific case of the length of tourist stay at a major “sun and sand” destination present not only bimodality but also overdispersion (the variance is greater than the mean). The length of stay is considered to be multimodal because tourists usually structure their trips in weekly blocks (for holidays in the Canary Islands, periods of one, two or three weeks are the most common options). However, there is also some heterogeneity of tourist preferences as regards shorter or longer stays, which may depend on socioeconomic and demographic characteristics, the time available to tourists and their prior familiarity with the destination, among other things. Reflecting on these diverse possibilities, empirical studies in this field have used several types of count data models, for example, latent class truncated Poisson models, to define different segments of tourists’ preferences [3] or count data quantile regression models to analyse the quantiles of the distribution of overall length of tourist stay [7], based on the Poisson distribution. More recently, [2] two newly proposed statistical distributions with which to explicitly incorporate bimodality, including a flexible discrete distribution and a mixture model based on the Poisson distribution (discrete choice models, have also been used, which redefine the variable by taking weekly intervals; see [8,9,10]). However, although the latter models are suitable and provide a reasonably good fit, they also present certain drawbacks. Following [2], we obtain a flexible distribution for modelling bimodal behaviour in a count data framework based on the Poisson distribution. As an anonymous reviewer pointed out, a finite mixture (non-latent) of two Poisson distributions could produce bimodality. However, these models of finite mixtures are known to present some difficulty regarding identifying and estimating the parameters. Moreover, this problem may be aggravated if covariates are included. This research contributes to the literature in two main respects. First, we present a discrete distribution characterised by overdispersion, and also by either unimodality or bimodality, according to the parameter values selected. This distribution is obtained by means of a shifted version of a classical discrete Poisson distribution. Our proposal accounts for some of the heterogeneity observed, and moreover, facilitates the inclusion of covariates; therefore, determinants to explain the bimodal variable can be considered. Second, the approach described overcomes problems that may arise in estimating the models described by [2], with respect to the existence of multiple points which maximise the logarithm function of the likelihood, thus impeding identification of the global maximum. The rest of this paper is organised as follows. Section 2 presents the model proposed for length of tourist stay, and describes its most important properties. The estimations of the parameters for the proposed distribution are discussed in Section 3, after which we present the empirical analysis performed, based on the length of stay. The results obtained are then discussed. Finally, in Section 6 we summarise the main conclusions drawn.

According to the standard literature in this field, briefly presented above, the length of tourist stay, T, like most expressions of the frequency of occurrence of an event, can be described by a Poisson distribution in which $\lambda >0$. This is the standard distribution for modelling random counts. In many cases, however, a model based on the Poisson distribution will be inadequate. Thus, in some situations counts may occur in clusters, giving rise to heterogeneity among individuals and provoking contagion (i.e., a degree of association between discrete events). When this happens, the count data may become overdispersed (i.e., the variance is greater than the mean), making the Poisson assumption very restrictive. On the other hand, if the parameter $\lambda$ fits a gamma distribution with shape $r>0$ and scale $(1-p)/p$, $0<p<1$, the unconditional distribution of T produces a negative binomial distribution with parameters r (dispersion parameter) and $1-p$. Nevertheless, although this distribution overcomes the problem of overdispersion, it still fails to reflect the bimodality observed in empirical data, such as that for the length of tourist stay (usually expressed in days). The following theorem, crucially, obtains a distribution that is appropriate for modelling the length of tourist stay.

Parameter $\theta$ controls the unimodality or bimodality of the family given in (1). Here $f_Y(y;\mu ,\sigma )$ is the parent distribution from which we can construct a distribution that can be unimodal or bimodal. From the construction established in the previous result, it is apparent that this same result can be applied to obtain generalisations of classical distributions. The first candidates for this application, which rely on just a single parameter, would be the exponential distribution, for the continuous case, and the geometric and Poisson distributions, for the discrete case. In this paper, we consider the latter case. In other words, our starting point is that of a shifted Poisson distribution with parameter $\lambda >0$. This situation is illustrated in the following result.

In order to achieve a more elegant expression for the above probability function (pf), it is convenient to take $\lambda ={\alpha }^2$ and $\theta (1+{\alpha }^2-t)={\gamma }_{\alpha ,\theta }(t)$. The expression given in (2) can then be rewritten as where ${\omega }_{\alpha ,\theta }(t)$ with $\kappa (\theta )={(2+{\theta }^2)}^{-1}$. Figure 1 shows that the proposed distribution properly represents the unimodal or bimodal nature of empirical data. In this situation, the shifted Poisson distribution is a special case for $\theta =0$. Furthermore, it seems that as $\alpha$ tends to infinity and $\theta \to 0$, the normal distribution is an excellent approximation of the pf (4). Some tedious but simple computations then provide the probability generating function of the distribution, which is given by for $\left|z\right|\le 1$. From (5) the moments of the distribution can be obtained. In particular, the mean and the variance are given by respectively.

By determining the index of dispersion (ID) of a probability distribution, we can quantify the extent to which a set of occurrences is dispersed, compared to a standard pattern such as the Poisson distribution. The ID is defined as the variance of a distribution divided by its mean. If the ID is greater than one, the corresponding distribution is said to be overdispersed, and if it is less than one, the distribution is underdispersed. Figure 2 represents the ID plot of the proposed bimodal distribution for some supports of the two parameters of the distribution. The ID can take values greater or less than 1, and so the distribution is appropriate for fitting empirical data which present over or underdispersion. The probabilities can be computed by using the recursive formula where $f_T(1)=[2+\alpha \theta (2+\alpha \theta )]\kappa (\theta )$. Furthermore, the expression given in (6) can be used to obtain the mode or modes of the distribution, by solving, for $\left[t\right]$, the third-degree polynomial equation given by where $[·]$ represents the integer part. Equation (7) supplies either one real solution (the unimodal case) or three such solutions (two modes and the corresponding anti-mode). The cumulative distribution function, which is not reproduced here, can also be obtained in closed form.

Consider a sample with n observations $\tilde{t}=(t_1,t_2,\dots ,t_n)$, taken from the pf (2). As a first approximation, the parameters $\alpha$ and $\theta$ can be estimated by the method of moments, assuming $\widehat{\mu }=\overline{t}$, where $\overline{t}=(1/n){\sum }_{i=1}^n{t}_i$ is the sample mean. The estimation is then obtained by the maximum likelihood method. To do so, we first consider the model without covariates. Here, the log-likelihood function is proportional to The normal equations for estimating the parameters $\theta$ and $\alpha$ are given by Equations (9) and (10) can be solved numerically for $\theta$ and $\mu$ using the Newton–Raphson iteration; for example, starting from the seed point $\theta$ near to zero, with $\mu =\overline{t}+1$.

Let us now assume that covariates are to be included in the model. First, consider that where $\phi (\theta )=4+{\theta }^2(5+2{\theta }^2)$ and where $\mu >\phi (\theta )\kappa {(\theta )}^2$. Under this assumption, the mean of the pf given in (4) is just the parameter $\mu$, as is usually assumed when covariates must be included in the model. Now, let ${\mathit{x}}_{\mathit{i}}^{\mathbf{\prime }}=[x_{1i},x_{2i},\dots ,x_{ki}]$ be a vector of $k\times 1$ covariates or factors associated with the length of stay of the i-th tourist and where $x_{ji}$ is the j-th factor for the i-th observation, $j=1,2,\dots ,k$. This vector of linearly independent regressors will determine $t_i$. The model provides great simplicity and the mean is straightforwardly expressed in terms of $\mu$. Therefore, in order to introduce the covariates, we need only assume a translated one unit logit link, defined by where $\mathit{\beta }={({\beta }_1,\dots ,{\beta }_k)}^{\prime }$ denotes the corresponding vector of regression coefficients. Furthermore, this logit link ensures that ${\mu }_i=\mu ({\mathit{x}}_{\mathit{i}},\mathit{\beta })$ lies within the interval $[1,\infty )$. The log-likelihood of the model with covariates is similar to that given in (8) except that $\alpha$ is replaced by $\alpha (\theta ,\mu )$, given in (11). Thus we have The normal equations are now given by for $j=1,\dots ,k$, where, Finally, $\partial {\omega }_{\alpha (\theta ,{\mu }_i)}(t_i)/\partial \theta$ can easily be computed by means of the chain rule.

This paper proposes a count data model based on the Poisson distribution, taking into account both bimodality and overdispersion. The model proposed is the shifted Poisson distribution, in the view that it is suitable for modelling the length of tourist stay, taking bimodality into account. This model was applied to the Canary Islands, a popular holiday destination, and would also be suitable with respect to tourism in the Balearic Islands, another major tourist destination in Spain. Using data from the Canary Islands Tourist Expenditure Survey for the year 2011, and taking into account information for all tourists, our shifted Poisson model provided a reasonable fit. Only the coefficient of income did not appear to be coherent with the expenditure literature, although this outcome has also been reported by previous studies of the duration of tourist visits. Finally, several variables corresponding to vacation characteristics were found to have statistically significant effects on the length of stay, as were some individual characteristics, such as age.