There are abundant empirical illustrations of the models and techniques described, many of which could be equally applied to other financial time series, such as exchange and interest rates. The authors have taken care to make the material accessible to anyone with a basic knowledge of statistics, calculus and probability, while at the same time preserving the mathematical rigor and complexity of the original models.
This book will be an essential reference for practitioners in the finance industry, especially those responsible for managing portfolios and monitoring financial risk, but it will also be useful for mathematicians who want to know more about how their mathematical tools are applied in finance, and as a text for advanced courses in empirical finance; financial econometrics and financial derivatives. Financial Modeling Under Non-Gaussian Distributions is a very accessible textbook that covers a wide range of topics.
The authors define their target readers as specialized master and Ph. It targets practioners in the financial industry. It is suitable for use as core text for students in empirical finance, financial econometrics and financial derivatives. It is useful for mathematician who want to know more about their mathematical tools are applied in finance. Help Centre. Track My Order. My Wishlist Sign In Join. Be the first to write a review. Add to Wishlist. Ships in 15 business days.
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Link Either by signing into your account or linking your membership details before your order is placed. Description Table of Contents Product Details Click on the cover image above to read some pages of this book! Statistical Properties of Financial Market Data. Modeling Higher Moments. Modeling Correlation. Extreme Value Theory. Portfolio Allocation. Non-Structural Option Pricing. Structural Option Pricing. Martingale and Changing Measure. Characteristic Functions and Fourier Transforms. The predicted variable consists in count data of crashes in highway segments in the United States over the course of several years.
As previously stated, the Poisson regression is what usually comes first to mind when count data needs to be assessed. However, as seen for phonemic inventory size, the overdispersion is very high for the number of crashes. They thus suggest that it should be used in subsequent studies to obtain better estimates of the role of predictors. Another example is response times in psycholinguistic experiments.
While Lo and Andrews report that inverse Gaussian and Gamma distributions are equivalent good fits for response times due to theoretical reasons, analysis of experimental data reveals that the distribution of residuals is not always satisfactory, especially because of the long tail of the distribution corresponding to long response times. Relying on distributions better accounting for the skewness of the target distribution, such as the generalized Gamma distribution, leads to more satisfying results in terms of normality of the residuals.
Finally, Rigby et al. This is not a bad thing. The main package is named gamlss, but associated packages such as gamlss.
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Given the differences in the algorithms, outputs may, however, slightly differ from one model to the next. A first step in contemplating the use of GAMLSS to study phonemic inventory size is to pay a closer look at the distribution of the latter. The distribution of the dependent variable independently from any predictor is called the marginal distribution. The histDist and fitDist functions of the gamlss package come in handy to investigate what theoretical distribution comes closest to the empirical one.
The first one takes as its main inputs a vector of values and the name of a distribution, and returns how well the values fit the distribution, as expressed by the global deviance, the AIC and BIC of the fit. The second allows one to find the best fit among a list of distributions, and also returns the AIC of the different fitting attempts.
We used these two functions to compare different distributions. On the one hand, we considered distributions adapted to count data available in the gamlss. There are over 25 available distributions, among them:. We also checked all the distributions adapted to positive real numbers. However, some distributions are based on parameters that are difficult to relate to the four moments mean, variance, skewness, and kurtosis. This is the case for all previously reported discrete distributions although with a specific parametrization for the Sichel distribution , but not for all continuous distributions — some of them, however, model the median, which is easy to interpret.
Given this constraint of interpretability, we especially paid attention to:. The intuition behind testing these various distributions was that those with more parameters would better be able to account for the thick right tail of the distribution, i. Figure 5 summarizes the fits of the two most adequate discrete distributions, of the two most adequate continuous distributions, and of the Poisson and inverse-Gaussian distributions that were tested in previous models.
Fitting of several discrete and continuous theoretical distributions to the empirical distribution of Phonemic inventory size. As visible on Figure 5 , except for the Poisson distribution, all displayed theoretical distributions seem rather close to the empirical distribution.
One must be cautious here, since the marginal distribution is not the same as the conditional distribution of the dependent variable, i. The question is whether the overdispersion can be explained by one or several of these predictors, or whether it is to some extent independent of them. In the second case, overdispersion will still be manifest in the conditional distribution, and will require treatment with a distribution with the right number of parameters.
In the first case, given its degrees of freedom, this distribution will likely still provide good fitting. To this extent, the results obtained with the marginal distribution can serve as a guide in the choice of the target conditional distribution. In practice, many decisions have to be made regarding the modeling options offered by GAMLSS, from choosing the distribution to choosing the link function, the additive terms and the smoothing parameters. In our case, in the previous section, we first investigated the marginal distribution of the dependent variable to narrow down possible choices of distributions.
We also included the inverse-Gaussian distribution for the sake of comparison with previous models. Second, regarding the link function, we thought that keeping an identity link was useful to relate estimates of the models to actual number of phonemes, without the difficulties related to transforming the dependent variable — or the relationship between it and the predictors — as mentioned earlier in this article. Various link functions can actually be compared with AIC. In distributions requiring positive values, link functions such as the logarithm also prevent convergence issues that are otherwise difficult to address.
Third, which additive terms to consider was like in all previous models related to current debates in the literature, which in no way means that other predictors would not be relevant. Various methods of model selection are available, some of them mixing backward, forward, and stepwise procedures across the various parameters of the distribution Stasinopoulos et al.
However, besides the fact that some scholars disagree with the concept of model selection overall, the presence of a random effect for Family is somehow problematic. Indeed, the way this random effect is estimated in the model — a local normal approximation to likelihood, also known as penalized quasi likelihood — is different from what occurs in common LMM or GLMM — a global estimation to likelihood.
The consequence is that dropping a continuous predictor can lead to a change in the penalization of the random effect, such that a strong effect, which should be retained by the selection procedure, may be abandoned. Because of this, we chose not to rely on selection procedures, but rather compare a number of models of increasing complexity. As for smoothing finally, we considered P-splines smooth functions — cubic splines proved difficult to work with —, with a modified penalty so as to shrink toward zero when the smoothing parameter went to infinity — the pbz smooth function in gamlss Stasinopoulos et al.
The advantage of these smooth terms was that the estimation could lead to linear terms, or even to constant terms when no influence of a predictor was detected, other predictors being accounted for. Some parameter selection was thus present. Table 5 reports the deviance, the degrees of freedom used for the various parameters, the total number of used degrees of freedom, as well as the AIC and BIC of the various models tested. There were issues of convergence with Sichel models that we could not address, which explains why they are not discussed in what follows.
In terms of AIC, i.
Finally, the BIC pointed to the three Delaporte models as the most parsimonious. TABLE 5. This is not the case for the two models. Looking at the various effective degrees of freedom of the smooth terms, it appeared that many terms were actually equivalent to linear predictors, and the model could be simplified and described as follows:. Table 6 reports the outputs of this model. Several predictors appear as statistically significant, however, Stasinopoulos et al. Indeed, the values given for a smooth term correspond to its linear part, and not to its total contribution. Additionally, reminiscent of what was said for GAM, the values for non-smoothed terms do not account for the uncertainty attached to the estimation of the smoothing terms.
A partial solution to this problem is to consider likelihood-ratio tests to assess the significance of the predictors once the degrees of freedom of the smooth terms have been fixed to the values previously estimated with penalization Stasinopoulos et al. Table 7 reports the output of this function for our chosen model described in Table 6. TABLE 6. TABLE 7. Regarding the median of the distribution, the smooth term for Distance from Africa is highly significant, while Local linguistic density is barely significant and Number of speakers is not.
Finally, no predictor reaches the 0. One can observe that for P-splines smooth terms, the difference in degrees of freedom between the full model and the model without the smooth term is equal to the fixed number of degrees of this smooth term minus 1. In our case, a conclusion is that the coefficient of variation of the distribution significantly decreases as Distance from Africa increases, which means that inventories are more homogeneous in terms of size the further away from Africa, other factors being accounted for.
In order to better understand what is suggested by the model, it is necessary to look at the partial terms reproduced in Figure 8. The median of Phonemic inventory size is non-linearly related to Distance from Africa , and the two local maxima of the non-linear relation are not easy to interpret.
As for GAMM, a linear decrease is not confirmed by the observed pattern.
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A sharp decrease can, however, be observed at some distance away from Africa. Relations for Number of speakers and Local linguistic density are linear. While the first one was assessed as not significant, the second one barely is, with an increase of the median Phonemic inventory size as the local linguistic density increases.
This could be due to less satisfying statistical approaches, but should also serve as a warning of the limited trust one should put in this result. Three aspects can be put forward in discussing the previous results and observations. The first aspect concerns the specific nature of our target dependent variable, i. The very large inventories of some languages, and the overdispersion of the connected variable, can be in good part explained by how features are combined into phonemes. Multiplicative processes are therefore at the origin of at least some the variance and overdispersion of phonemic inventory size.
From this observation, one could argue that applying a transformation to the dependent variable makes sense, even if it is not an easy question to answer which transformation is respectful of the specific multiplicative processes at play. However, this transformation may run counter to the nature of the mechanisms hypothesized with the inclusion of a predictor. For example, referring to the impact of the number of speakers, does one conceive this impact at the level of phonemes, or at the level of features? In the latter case, the transformation would perhaps be justified.
In the former, some situations could appear as less convincing. Although this hypothesis is far-fetched and is only put forward to the sake of argumentation, one could argue that having a larger number of speakers does not increase the number of features at the basis of the phonemic inventory, but rather influences the way speakers combine these features, in such a way that the system tends to display greater feature economy.
Along the same line of thought, with respect to linguistic contact and the putative effect of the local linguistic density, the meaningful question would be whether speakers mostly borrow phonemes or features from other languages. To move further in this direction, future work will consist in extracting the features of each phonemic inventory used in the test case of this article. It will then become possible to study the distribution of feature inventory size, much in the way phonemic inventory size was scrutinized in the previous sections. There are no multiplicative processes at the level of features, and it will therefore be relevant to evaluate the overdispersion of the marginal and conditional distributions.
If overdispersion is still present and high, a possible conclusion will be that the overdispersion of phonemic inventory size derives from multiplicative processes when combining features, but also from the properties of the systems of features themselves. A second point is the issue of weak effects in regression modeling. As it appears from our various analyses, Distance from Africa appears as a very significant effect in all models. One can assume that very strong and significant effects will be observed even with imperfect models.
Another predictors of our models, Local linguistic density , has p -values well above 0. Luke, On the other hand, a conclusion is that weak signals are at the mercy of the chosen model, and thus this model should be chosen and assessed with care. For example, in the case of phonemic inventories, in addition to the assumptions we tested for residuals, potential spatial autocorrelation should be accounted for in order to minimize related type I errors. All in all, with respect to our test case, whether language contact significantly affects phonemic inventory size through borrowing remains to us an open question.
What geo-linguistic measures best capture language borrowing is a connected question that requires further investigation. Finally, we argue that linguistics and psycholinguistics could benefit from the use of GAMLSS when regression models are envisaged to explore a phenomenon. The adequacy of the Delaporte distribution to model phonemic inventory size in no way means that this distribution in particular is the solution to a large number of problems.
In other contexts, similar investigations would lead to another distribution or narrow choice of distributions. Relating the mean of response times as dependent variable to a number of factors such as number of phonological neighbors, frequency, number of letters etc. Besides psycholinguistics, work in preparation suggests that another variable which can benefit from GAMLSS is speech rate.
Indeed, speech rate — the number of syllables uttered by second — presents interesting variations between speakers and languages Pellegrino et al. More generally, we have little doubt that many other variables, either continuous, discrete or count data, can benefit from both the smooth functions and distributions of GAMLSS. Various statistical tools are available to linguists willing to explain how a given linguistic variable varies across its domain.
This seems especially relevant when non-linguistic causes of linguistic diversity such as climatic or sociodemographic factors are considered, since their study can often be conducted with regression models. The distributions offered by GAMLSS can be more appropriate from a methodological point of view, and both the possibility to include additive terms and the possibility to model the scale and shape of the distribution in addition to its location can be put to use to better understand the behavior of a system. The raw data supporting the conclusions of this manuscript will be made available by the author, without undue reservation, to any qualified researcher.
The author designed the work, assembled the studied dataset from other sources of data, conducted the different analyses, and wrote the article. The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Atkinson, Q. Linking spatial patterns of language variation to ancient demography and population migrations. Phonemic diversity supports a serial founder effect model of language expansion from Africa.
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Zha, L. The Poisson inverse Gaussian PIG generalized linear regression model for analyzing motor vehicle crash data. Zuur, A. A protocol for data exploration to avoid common statistical problems. Keywords : mixed-effects models, generalized linear models, generalized additive models, smooth terms, phonemic inventory size, Delaporte distribution, Box-Cox t distribution, GAMLSS. The use, distribution or reproduction in other forums is permitted, provided the original author s and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice.
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