Univariate and Multivariate Distributions

Abstract

In the previous chapters, the notion of risk measure has been introduced and we observe that the key point is the choice of the distributions which characterise the variables. In this chapter several methodologies to fit univariate or multivariate classes of distributions on data sets are introduced. The presentation is not exhaustive. The question of estimation is not documented here and references will be provided “au fil de l’eau”.

Appendix: Estimation

Though estimation procedures are not the focal point of this book, risk measures cannot be assessed in practice without fitting the distributions, either univariate or multivariate. Therefore, we briefly introduce in the following the main approaches to adjust the distributions presented before on data samples.

4.1.1 I Distribution’s Parameters Estimation

• Maximum likelihood estimation (MLE): the method of maximum likelihood is based on the likelihood function, $\mathcal {L}(\theta ;x)$. Assuming a distribution f(⋅;θ), where θ ∈ Θ denotes a set of parameters, and Θ represents the domain of possible parameters. The method of maximum likelihood aims at finding the parameters, θ, that maximise the likelihood function, $\mathcal {L}(\theta ;x)$.

  The method defines a maximum likelihood estimate as follows:

  $$\displaystyle \begin{aligned} \hat{\theta}\in \{{\underset {\theta \in \Theta }{\operatorname {arg\,max} }}\ {\mathcal {L}}(\theta \,;x)\}, \end{aligned} $$

  (1)

  if a maximum exists.

  In practice, it is often convenient (under specific assumptions) to work with the natural logarithm of the likelihood function, called the log-likelihood:

  $$\displaystyle \begin{aligned} \ell (\theta \,;x)=\ln {\mathcal {L}}(\theta \,;x), \end{aligned} $$

  (2)

  As the logarithm function is monotonic and strictly increasing, the MLE is the same regardless of whether the likelihood or the log-likelihood function is maximised. For the properties of the maximum likelihood, we refer to Millar 2011.

• Generalised method of moments (GMM): Let {Y _t}_t = 1, …, T, where each observation Y _t is an n-dimensional multivariate random variable. Considering a distribution driven by a parameter θ ∈ Θ, the objective is to estimate θ.

  GMM assumes that Y _t are generated by a weakly stationary stochastic process (the state independent and identically distributed is a particular case of this condition).

  In order to apply the GMM, the “moment conditions” has to be fulfilled, in other words, the distribution has to have moment. Therefore, the vector-valued function g(Y _t, θ) associated to the r.v. has to be known such that

  $$\displaystyle \begin{aligned} m(\theta _{0})\equiv \operatorname{E}[\,g(Y_t,\theta _{0})\,]=0, \end{aligned} $$

  (3)

  where E denotes expectation.

  The objective of the approach is to replace the theoretical expected value with its empirical average given as follows, assuming that we observe Y ₁, ..., Y _T:

  $$\displaystyle \begin{aligned} {\hat {m}}(\theta )\equiv {\frac {1}{T}}\sum _{{t=1}}^{T}g(Y_{t},\theta ) \end{aligned} $$

  (4)

  and then to minimise this expression with respect to θ.

  Considering the law of large numbers, ${\hat {m}}(\theta )\,\approx \;\operatorname {E}[g(Y_{t},\theta )]\,=\,m(\theta )$ for large values of T, and thus ${\hat {m}}(\theta )\;\approx \;m(\theta _{0})\;=\;0$. The generalised method of moments aims at obtaining ${\hat {\theta }}$ which makes the quantity ${\hat {m}}({\hat \theta })$ as close to zero as possible. Formally, this is equivalent to minimising a particular norm of ${\hat {m}}(\theta )$. The family of norms considered is defined as follows:

  $$\displaystyle \begin{aligned} \|{\hat {m}}(\theta )\|{}_{W}^{2}={\hat {m}}(\theta )^{\mathsf {T}}\,W{\hat {m}}(\theta ), \end{aligned} $$

  (5)

  where W is a positive-definite weighting matrix, and m ^T its transposition. In practice, the weighting matrix W is obtained using the available data, and therefore is an estimate denoted $\hat {W}$. The GMM estimator can therefore be formalised as follows:

  $$\displaystyle \begin{aligned} \hat{\theta}={\operatorname{arg}\,min}_{\theta \in \Theta}\left({\frac{1}{T}}\sum_{t=1}^{T}{g(Y_{t},\theta)}\right)^{\mathsf{T}}{\hat{W}}\left({\frac{1}{T}}\sum_{t=1}^{T}{g(Y_{t},\theta)}\right) \end{aligned} $$

  (6)

  Note that under suitable conditions this estimator is consistent, asymptotically normal, and with an appropriate weighting matrix ${\hat {W}}$, asymptotically efficient. For more details on the properties of the GMM, we refer to Hansen 1982.

• Truncated distributions: Parameter estimation—In practice considering data sets are either incomplete or truncated, therefore the following approaches might be of interest to parametrise the distributions

  • Constrained maximisation likelihood: In the constrained maximum likelihood function approach (Chernobai et al. 2007), one can obtain the parameters of the distribution by directly maximising the likelihood function. If the observed data sample X = X ₁, X ₂, …, X _n constitutes an i.i.d. left-truncated sample, then the likelihood function is expressed as:

    $$\displaystyle \begin{aligned} L(x;\theta|X \leq H) = \prod^{n}_{j=1}{\frac{f(x_{j};\theta|X \neq H)}{P(\theta;X \leq H)}}, \end{aligned} $$

    (7)

    where γ is the parameter set that defines the density.

  • Expectation maximisation algorithm: This algorithm (Dempster et al. 1977; McLachlan and Krishnan 1997) is used to estimate unknown parameters by maximising the likelihood function expectation using available information on truncated data.

    The EM algorithm is an iterative process split up in two steps. In the initial step the set of parameters Θ are initialised at θ ₀ values. Likelihood function values are replaced by their expected values.

    In the next step, we maximise this function to obtain new parameters values. These ones are used to re-initialise θ ₀ in the first step and then re-iterate the process. The algorithm is:

Algorithm 1

• Initialisation step: We choose ς ₀ , and provide an estimation of the missing data, n ^missing.

• Expectation step: Given ς ₀ , we compute:

  $$\displaystyle \begin{aligned} Q = E_{\varsigma}\left[\log \frac{L_{\varsigma}(x^{all})}{x^{observed}}\right] = n^{missed} \times E_{\varsigma}[\log f_{\varsigma}(x^{truncated})] + \sum^{n}_{j=1}{\log f_{\varsigma}(x_{j}^{observed})} \end{aligned} $$

  (8)

  • We estimate n ^missing, $n^{missing}=n^{observed}\frac {1-P_{\varsigma _{0}}(x < u)}{P_{\varsigma _{0}}(x < u)}$

  • Random generation of n ^missing values from the theoretical distribution chosen to fit the head, with respect to the constraint x ^missing < u

  • x = (x ^observed, x ^missing)

  • Log-likelihood function computation

• Maximisation step: We compute $\varsigma _{1}=\arg _{\varsigma } \max E_{\varsigma }[\log \frac {L_{\varsigma }(x^{all})}{x^{observed}}]$

• If, we have a convergence, we have ς ^optim . If we do not $\sqrt {(\varsigma _{1})^{2} - (\varsigma _{0})^{2}} > eps_{value}$ , we repeat step 2 and 3 with ς ⁰ = ς ¹ until convergence.

4.1.2 I Copulas’ Parameters Estimation

In order to estimate the copulas’ parameters, we can carry out one of the following methods(Cherubini et al. (2004)). Regarding chosen copula characteristics, parameters enable tuning the dependence structure.

• 1. Maximum likelihood:

  The classical maximum likelihood and the two-stage inference function for margins method proposed by Joe (1997, 2005) belong to the first category. The former is based on the maximisation of the full log-likelihood

  $$\displaystyle \begin{aligned} \sum^{n}_{i=1}{\log(c_{\theta})(F_{1,\beta_{1,n}}(X_{i,1}),\ldots,F_{p,\beta_{p,n}}(X_{i,1}))}+\sum^{n}_{i=1}{\sum^{p}_{j=1}{\log f_{j,\beta_{j}}(X_{i,j})}}, \end{aligned} $$

  (9)

  where c _θ and $f_{1;\beta _{1}},\ldots ,f_{p;\beta _{p}}$ are the probability density functions obtained from C _θ and $f_{1;\beta _{1}},\ldots ,f_{p;\beta _{p}}$, respectively when we observe data sets X _i,1, ..., X _i,n, i = 1, ..., p. With the aim of decreasing the computational burden associated with maximum likelihood estimation, the inference function for margins method first estimates the parameters β ₁, …, β _p of the marginal cdfs by β _1,n, …, β _p,n, and then θ as the maximiser of

  $$\displaystyle \begin{aligned} \sum^{n}_{i=1}{\log(c_{\theta})(F_{1,\beta_{1,n}}(X_{i,1}),\ldots,F_{p,\beta_{p,n}}(X_{i,p}))} \end{aligned} $$

  (10)

• 2. Pseudo maximum likelihood:

  As it may be argued that the estimation of θ should not be affected by the choice of the marginal cdfs, many authors advocate the use of the maximum pseudo likelihood estimator studied in Genest et al. (1995) and in Shih and Louis (1995) to estimate this one. It consists in maximising the log pseudo-likelihood:

  $$\displaystyle \begin{aligned} \log L(\theta) = \sum^{n}_{i=1}{\log c_{\theta}(\hat{U}_{i})}, \end{aligned} $$

  (11)

  where the $\hat {U}_{i}=(\hat {U}_{i,1},\ldots ,\hat {U}_{i,p})$ are pseudo-observations computed from the X _i = (X _i,1, …, X _i,p) by $\hat {U}_{i,j}= \frac {R_{i,j}}{(n+1)}$, with R _i,j being the rank of X _i,j among X _1,j, …, X _n,j.

Method-of-moment approaches are based on the inversion of a consistent estimator of a moment of the copula C _θ. The two best-known moments, Spearman’s rho and Kendall’s tau, are, respectively, given by,

• 1. Spearman’s Rho:

  $$\displaystyle \begin{aligned} \rho_{\theta}=12 \int_{[0,1]^{2}}{C_{\theta}(u_{1},u_{2})d u_{1}d u_{2}}-3 \end{aligned} $$

  (12)

• 2. Kendall’s Tau:

  $$\displaystyle \begin{aligned} \tau_{\theta}=4 \int_{[0,1]^{2}}{C_{\theta}(u_{1},u_{2})d u_{1}d u_{2}}-1 \end{aligned} $$

  (13)