Objective Prior for the Number of Degrees of Freedom of a t Distribution

Villa, Cristiano; Walker, Stephen G.

doi:10.1214/13-ba854

Cited by 39 publications

(66 citation statements)

References 16 publications

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“…As ν increases, leading to t densities that are more and more similar to each other, the relative mean squared error decreases. This result is in line with previous objective studies on the number of degrees of freedom for t ‐type densities, see, for example, and . As expected, the mean squared error in higher for small sample sizes than for larger sample sizes, given that more information is carried by the observations via the likelihood function.…”

Section: Simulation Studysupporting

confidence: 91%

“…The earlier result also holds when we assume that for ν = 30,

f_{ν}^{α}

is the skewed normal distribution. These results lead to the following important considerations: The objective prior distribution for ν does not depend by the skewness parameter α ; The objective prior distribution for ν for the skewed model is exactly the prior introduced in , that is, π ( ν | μ , σ , α ) = π ( ν |0.5, μ , σ ). …”

Section: The Prior Distributions For the Parameters Of The Asymmetricmentioning

confidence: 95%

“…In this case, the density in represents a symmetrical t distribution with ν degrees of freedom, and the prior for the parameter ν has the form as in . To simplify the notation, we indicate the t ‐density with parameters ν , α , μ and σ by

f_{ν}^{α}

; therefore, the prior for ν is given by

π (ν | α, μ, σ) \propto \exp \{D (f_{ν}^{α} ∥ f_{ν + 1}^{α})\} - 1,

for ν < 29 and, for

ν ⩾ 29

π (ν | α, μ, σ) \propto \exp \{D (f_{ν}^{α} ∥ f_{ν - 1}^{α})\} - 1,

where

f_{30}^{α} \sim φ^{α}

is the skewed normal distribution with mean μ and standard deviation σ , that is,

φ^{α} (x | μ, σ) = ({, \begin{matrix} array \frac{1}{σ \sqrt{2 π}} \exp \{- {(\frac{x - μ}{2 α σ})}^{2}\} & array x ⩽ μ \\ array \frac{1}{σ \sqrt{2 π}} \exp \{- {(\frac{x - μ}{2 (1 - α) σ})}^{2}\} & array x > μ . \end{matrix})

As recalled in Section 1, the choice of α = 0.5 corresponds to the usual Student‐ t distribution, and the prior earlier is the one introduced in . The following theorem states a crucial result to set π ( ν | α , μ , σ ).…”

Section: The Prior Distributions For the Parameters Of The Asymmetricmentioning

confidence: 99%

See 2 more Smart Citations

Objective Bayesian modelling of insurance risks with the skewed Student‐t distribution

Leisen

Marín

Villa

2017

Appl Stoch Models Bus & Ind

Self Cite

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Insurance risks data typically exhibit skewed behaviour. In this paper, we propose a Bayesian approach to capture the main features of these data sets. This work extends a methodology recently introduced in the literature by considering an extra parameter that captures the skewness of the data. In particular, a skewed Student‐t distribution is considered. Two data sets are analysed: the Danish fire losses and the US indemnity loss. The analysis is carried with an objective Bayesian approach. For the discrete parameter representing the number of the degrees of freedom, we adopt a novel prior recently appeared in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

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Section: Simulation Studysupporting

confidence: 91%

“…The earlier result also holds when we assume that for ν = 30,

f_{ν}^{α}

Section: The Prior Distributions For the Parameters Of The Asymmetricmentioning

confidence: 95%

f_{ν}^{α}

; therefore, the prior for ν is given by

π (ν | α, μ, σ) \propto \exp \{D (f_{ν}^{α} ∥ f_{ν + 1}^{α})\} - 1,

for ν < 29 and, for

ν ⩾ 29

π (ν | α, μ, σ) \propto \exp \{D (f_{ν}^{α} ∥ f_{ν - 1}^{α})\} - 1,

where

f_{30}^{α} \sim φ^{α}

is the skewed normal distribution with mean μ and standard deviation σ , that is,

φ^{α} (x | μ, σ) = ({, \begin{matrix} array \frac{1}{σ \sqrt{2 π}} \exp \{- {(\frac{x - μ}{2 α σ})}^{2}\} & array x ⩽ μ \\ array \frac{1}{σ \sqrt{2 π}} \exp \{- {(\frac{x - μ}{2 (1 - α) σ})}^{2}\} & array x > μ . \end{matrix})

Section: The Prior Distributions For the Parameters Of The Asymmetricmentioning

confidence: 99%

See 1 more Smart Citation

Objective Bayesian modelling of insurance risks with the skewed Student‐t distribution

Leisen

Marín

Villa

2017

Appl Stoch Models Bus & Ind

Self Cite

View full text Add to dashboard Cite

show abstract

“…Shore and Johnson (1980) give axiomatic foundations for deriving various probabilistic rules and, more specifically, the combining mechanism for the Bayes rule in Bernardo (1979) is expected utility, in Zellner (1988) is an information processing rule, and in Zellner (1996) is a maximum entropy principle. More recently, the self information loss, together with the Kullback-Leibler divergence, has been employed in a proper Bayesian setting to derive objective prior distributions for specific discrete parameter spaces (Villa and Walker, 2015) and to estimate the number of degrees of freedom in the t-distribution (Villa and Walker, 2014).…”

Section: Introductionmentioning

confidence: 99%

Characterization of the Bayesian Posterior Distribution in Terms of Self-information

Dall’Aglio

Hill

2017

IJSP

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It is well known that the classical Bayesian posterior arises naturally as the unique solution of different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. Here it is shown that the Bayesian posterior is also the unique minimax optimizer of the loss of self-information in combining the prior and the likelihood distributions, and is the unique proportional consolidation of the same distributions. These results, direct corollaries of recent results about conflations of probability distributions, further reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.

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“…As v → ∞, Student-t distribution becomes a normal one. Thus, the estimation of v is of particular interest in its own right, whereas such a task is not straightforward within both frequentist and Bayesian frameworks; see, for example, Villa and Walker (2014). It can be seen from Theorem 5 of Fernandez and Steel (1999) that the likelihood function may have multiple local maxima and can approach infinity as v → 0.…”

Section: Introductionmentioning

confidence: 99%