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In Bayesian statistics, one frequently encounters priors and posteriors that are product of two probability density functions. In this paper, we discuss three such priors/posteriors, provide motivation and derive expressions for their moments, median and mode. Forty seven motivating examples are discussed. We expect that this paper could serve as a useful reference for practitioners of Bayesian statistics. It could also encourage further research in this area.

  • 1

    Abdullah, M. Y., Bayesian inferences with the poly-t distribution, Ph.D. Thesis, Oklahoma State University, USA (1982).

  • 2

    Abdullah, M. Y., Bayesian analysis of normal scalar populations with a common mean, Pertanika, 9 (1986), 381386.

  • 3

    Agamennoni, G., Nieto, J. I. and Nebot, E. M., Approximate inference in statespace models with heavy-tailed noise, IEEE Transactions on Signal Processing, 60 (2012), 50245037.

    • Search Google Scholar
    • Export Citation
  • 4

    Albert, J. and Rizzo, M., R by Example: Concept to Code. Springer Verlag, New York (2012).

  • 5

    Arnold, B. C., Pareto Distributions. International Co-operative Publishing House, Burtonsville, Maryland (1983).

  • 6

    Aykac, A. and Brumat, C., New Developments in the Applications of Bayesian Methods. Proceedings of the First European Conference held at Fontainebleau, June 1976. North-Holland Publishing Company, Amsterdam (1977).

    • Search Google Scholar
    • Export Citation
  • 7

    Ball, R. J., The International Linkage of National Economic Models. North Holland Publishing Company, Holland (1975).

  • 8

    Bauwens, L., Bayesian Full Information Analysis of Simultaneous Equation Models using Integration by Monte Carlo. Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin (1984).

    • Search Google Scholar
    • Export Citation
  • 9

    Bauwens, L. and D’Alcantara, G., An export model for the Belgian industry, European Economic Review, 22 (1983), 265276.

  • 10

    Bauwens, L., Fiebig, D. G. and Steel, M. F. J., Estimating end-use demand: A Bayesian approach, Journal of Business and Economic Statistics, 12 (1994), 221231.

    • Search Google Scholar
    • Export Citation
  • 11

    Bauwens, L. and Giot, P., A Gibbs sampler approach to cointegration, Computational Statistics, 13 (1998), 339368.

  • 12

    Bauwens, L. and Lubrano, M., Identification restrictions and posterior densities in cointegrated Gaussian VAR systems, in: Advances in Econometrics, volume 11, part B, JAI Press (1996), pp. 328.

    • Search Google Scholar
    • Export Citation
  • 13

    Bauwens, L., Lubrano, M. and Richard, J. -F., Bayesian Inference in Dynamic Econometric Models. Oxford University Press, Oxford (1999).

  • 14

    Bauwens, L. and Richard, J.-F., A 1-1 poly-t random variable generator with application to Monte Carlo integration, Journal of Econometrics, 29 (1985), 1946.

    • Search Google Scholar
    • Export Citation
  • 15

    Bauwens, L. and Tompa, H., Bayesian Regression Program (BRP), CORE User’s Manual Set # A-5 (1977).

  • 16

    Bian, G., Bayesian statistical analysis with independent bivariate priors for the normal location and scale parameters, Ph.D. Thesis, University of Minnesota, USA (1989).

    • Search Google Scholar
    • Export Citation
  • 17

    Bian, G., Robust Bayesian estimators in a one-way ANOVA model, Test, 4 (1995), 115135.

  • 18

    Bian, G., Bayesian inference in location-scale distributions with independent bivariate priors, Test, 6 (1997), 137157.

  • 19

    Bian, G., Bayesian estimates in a one-way ANOVA random effects model, Australian and New Zealand Journal of Statistics, 44 (2002), 99108.

    • Search Google Scholar
    • Export Citation
  • 20

    Bian, G. and Dickey, J. M., Moments of the poly-Cauchy density with applications in estimation, Journal of the Italian Statistical Society, 1 (1996a), 111.

    • Search Google Scholar
    • Export Citation
  • 21

    Bian, G. and Dickey, J. M., Properties of multivariate Cauchy and poly-Cauchy distributions with Bayesian g-prior applications, in: Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner, editors D. A. Berry, K. M. Chaloner and J. K. Geweke, John Wiley and Sons, New York (1996b), pp. 299310.

    • Search Google Scholar
    • Export Citation
  • 22

    Bian, G. and Tiku, M. L., Bayesian inference based on robust priors and MML estimators: Part I, symmetric location-scale distributions, Statistics, 29 (1997a), 317345.

    • Search Google Scholar
    • Export Citation
  • 23

    Bian, G. and Tiku, M. L., Bayesian inference based on robust priors and MML estimators: Part II, skew location-scale distributions, Statistics, 29 (1997b), 8199.

    • Search Google Scholar
    • Export Citation
  • 24

    Bisceglie, M. D., Galdi, C. and Griffiths, H. D., Statistical scattering model for high-resolution sonar images: Characterisation and parameter estimation, IEE Proceedings Radar, Sonar and Navigation, 146 (1999), 264272.

    • Search Google Scholar
    • Export Citation
  • 25

    Box, C. E. P. and Tiao, G. C., Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, Massachusetts (1973).

  • 26

    Broemeling, L. D., Bayesian Analysis of Linear Models. Marcel Dekker, New York (1985).

  • 27

    Broemeling, L. and Abdullah, M. Y., An approximation to the poly-t distribution, Communications in Statistics –Theory and Methods, 13 (1984), 14071422.

    • Search Google Scholar
    • Export Citation
  • 28

    Broemeling, L., Abdullah, M. Y. and Diaz, J., Some Bayesian solutions for problems of adaptive estimation in linear dynamic systems, Communications in Statistics –Theory and Methods, 14 (1985), 401418.

    • Search Google Scholar
    • Export Citation
  • 29

    Broemeling, L. and Diaz, J., Bayesian solution to the control problem for linear control systems, Communications in Statistics –Theory and Methods, 15 (1984), 18031817.

    • Search Google Scholar
    • Export Citation
  • 30

    Carriquiry, A. L. and Kliemann, W., The modes of a poly-t distribution and an application to mixed linear models, Proyecciones, 26 (2007), 281308.

    • Search Google Scholar
    • Export Citation
  • 31

    Chan, J. S. K., Choy, S. T. B. and Makov, U. E., Robust Bayesian analysis of loss reserves data using the generalized-t distribution, Astin Bulletin, 38 (2008), 207230.

    • Search Google Scholar
    • Export Citation
  • 32

    Chantas, G., Galatsanos, N., Likas, A. and Saunders, M., Variational Bayesian image restoration based on a product of t-distributions image prior, IEEE Transactions on Image Processing, 17 (2008), 17951805.

    • Search Google Scholar
    • Export Citation
  • 33

    Chebotarev, A. M., On stable Pareto laws in a hierarchical model of economy, Physica A –Statistical Mechanics and Its Applications, 373 (2007), 541559.

    • Search Google Scholar
    • Export Citation
  • 34

    Chen, D.-X., Bayesian computation methods for the poly t density, Ph.D. Thesis, Bowling Green State University, Ohio, USA (1992).

  • 35

    Chib, S., Nardari, F. and Shephard, N., Markov chain Monte Carlo methods for stochastic volatility models, Journal of Econometrics, 108 (2002), 281316.

    • Search Google Scholar
    • Export Citation
  • 36

    Choy, S. T. B. and Chan, J. S. K., Bayesian inference using Gibbs sampling for Window version (WinBUGS), software for Bayesian analysis using MCMC method and Gibbs sampler, Australian and New Zealand Journal of Statistics, 50 (2008), 135146.

    • Search Google Scholar
    • Export Citation
  • 37

    Clyde, M. and George, E. I., Flexible empirical Bayes estimation for wavelets, Journal of the Royal Statistical Society, B, 62 (2000), 681698.

    • Search Google Scholar
    • Export Citation
  • 38

    Clyde, M., Parmigiani, G. and Vidakovic, B., Multiple shrinkage and subset selection in wavelets, Biometrika, 85 (1998), 391402.

  • 39

    Corona, P., Ferrara, G. and Migliaccio, M., Generalized stochastic field model for reverberating chambers, IEEE Transactions on Electromagnetic Compatibility, 46 (2004), 655660.

    • Search Google Scholar
    • Export Citation
  • 40

    Dayal, H. H. and Dickey, J. M., Numerical evaluation of integrals involving products of t-densities, Journal of Statistical Computation and Simulation, 6 (1977), 1927.

    • Search Google Scholar
    • Export Citation
  • 41

    Dickey, J. M., Expansions of t-densities and related complete integrals, Annals of Mathematical Statistics, 38 (1967), 503510.

  • 42

    Dickey, J. M., Three multidimensional-integral identities with Bayesian applications, Annals of Statistics, 39 (1968), 16151627.

  • 43

    Dickey, J. M., Bayesian alternatives to the F-test and least-squares estimate in the normal linear model, in: Studies in Bayesian Econometrics and Statistics, editors S. E. Fienberg and A. Zellner, North-Holland, Amsterdam (1974), pp. 515554.

    • Search Google Scholar
    • Export Citation
  • 44

    Dreze, J. H., Bayesian regression analysis using poly-t densities, Journal of Econometrics, 6 (1977), 329354.

  • 45

    Dreze, J. H., Gabszewicz, J. J., Richard, J. F. and Wolsey, L. A., Economic Decision-Making: Games, Econometrics, and Optimisation: Contributions in Honor of Jacques H. Dreze. North Holland Publishing Company, Holland (1990).

    • Search Google Scholar
    • Export Citation
  • 46

    Dreze, J. H. and Richard, J.-F., Bayesian analysis of simultaneous equation system, in: Handbook of Econometrics, volume 1, editors Z. Glriliches and M. D. Intriligator, North-Holland, Amsterdam (1983), pp. 577598.

    • Search Google Scholar
    • Export Citation
  • 47

    Durbin, J. and Koopman, S., Time Series Analysis by State Space Methods. Oxford University Press, Oxford (2001).

  • 48

    Eltoft, T., Modeling the amplitude statistics of ultrasonic images, IEEE Transactions on Medical Imaging, 25 (2006), 229240.

  • 49

    Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, volumes I, II and III. Robert E. Krieger Publishing Company, Inc, Melbourne, Florida (1981).

    • Search Google Scholar
    • Export Citation
  • 50

    Esteves, L. G., Wechsler, S., Leite, J. G. and González-López, V. A., De-Finettian consensus, Theory and Decision, 49 (2000), 7996.

    • Search Google Scholar
    • Export Citation
  • 51

    Evans, R. B., Empirical Bayes inference for means using student t prior distributions, Technical Report, Department of Statistics, Iowa State University, USA (2009).

    • Search Google Scholar
    • Export Citation
  • 52

    Fan, T. H. and Berger, J. O., Exact convolutions of t distributions with applications to Bayesian inference for a normal mean with t prior distributions, Journal of Statistical Computation and Simulation, 36 (1990), 209228.

    • Search Google Scholar
    • Export Citation
  • 53

    Fan, T. H. and Berger, J. O., Behavior of the posterior distribution and inference for a normal mean with t prior distributions, Statistics and Decisions, 10 (1992), 99120.

    • Search Google Scholar
    • Export Citation
  • 54

    Fernandez, C., Osiewalski, J. and Steel, M. F. J., Classical and Bayesian inference robustness in multivariate regression models, Journal of the American Statistical Association, 92 (1997), 14341444.

    • Search Google Scholar
    • Export Citation
  • 55

    Fiorentini, G., Planas, C. and Rossi, A., The marginal likelihood of structural time series models, with application to the US and euro area NAIRU, Working Paper 21-08, The Rimini Centre for Economic Analysis (2008).

    • Search Google Scholar
    • Export Citation
  • 56

    Fiorio, C. V., Hajivassiliou, V. A. and Phillips, P. C. B., Bimodal t-ratios: The impact of thick tails on inference, Econometrics Journal, 13 (2010), 271289.

    • Search Google Scholar
    • Export Citation
  • 57

    Fisher, R. A., Dispersion on a sphere, Proceedings of the Royal Society of London, A, 217 (1953), 295305.

  • 58

    Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M., Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association, 85 (1990), 972985.

    • Search Google Scholar
    • Export Citation
  • 59

    Ghosh, A., Gangopadhyay, K. and Basu, B., Consumer expenditure distribution in India, 1983–2007: Evidence of a long Pareto tail, Physica A –Statistical Mechanics and Its Applications, 390 (2011), 8397.

    • Search Google Scholar
    • Export Citation
  • 60

    Gianola, D., Im, S. and Macedo, F. W., A framework for prediction of breeding value, in: Advances in Statistical Methods for Genetic Improvement of Livestock, editors D. Gianola and K. Hammond, Springer Verlag, Heidelberg (1990), pp. 210238.

    • Search Google Scholar
    • Export Citation
  • 61

    Gierull, C. H., Statistical analysis of multilook SAR interferograms for CFAR detection of ground moving targets, IEEE Transactions on Geoscience and Remote Sensing, 42 (2004), 691701.

    • Search Google Scholar
    • Export Citation
  • 62

    Golam Kibria, B. M., Sun, L., Zidek, J. V. and Le, N. D., Bayesian spatial prediction of random space-time fields with application to mapping PM2.5 exposure, Journal of the American Statistical Association, 97 (2002), 112124.

    • Search Google Scholar
    • Export Citation
  • 63

    Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, sixth edition. Academic Press, San Diego (2000).

  • 64

    Griffiths, W. E., Bayesian inference in the seemingly unrelated regressions model, in: Computer-Aided Econometrics, editor David E. A. Giles, Chapter 9 (2003).

    • Search Google Scholar
    • Export Citation
  • 65

    Griffiths, W. E. and Rebecca Valenzuela, Ma., Gibbs samplers for a set of seemingly unrelated regressions, Australian and New Zealand Journal of Statistics, 48 (2006), 335351.

    • Search Google Scholar
    • Export Citation
  • 66

    Guida, M. and Maria, F., Topology of the Italian Airport Network: A scale-free small-world network with a fractal structure? Chaos Solitons and Fractals, 31 (2007), 527536.

    • Search Google Scholar
    • Export Citation
  • 67

    Guidolin, M., International asset prices and portfolio choices under Bayesian learning, Research in Economics, 57 (2003), 383437.

  • 68

    Guidolin, M., Home bias and high turnover in an overlapping-generations model with learning, Review of International Economics, 13 (2005), 725756.

    • Search Google Scholar
    • Export Citation
  • 69

    Guthrie, R. H. and Evans, S. G., Magnitude and frequency of landslides triggered by a storm event, Loughborough Inlet, British Columbia, Natural Hazards and Earth System Sciences, 4 (2004), 475483.

    • Search Google Scholar
    • Export Citation
  • 70

    Han, X. P. and Hu, C. D., Power law distributions in the experiment for adjustment of the ion source of the NBI system, Plasma Science and Technology, 7 (2005), 31023104.

    • Search Google Scholar
    • Export Citation
  • 71

    Harville, D. A. and Zimmermann, A. G., The posterior distribution of the fixed and random effects in a mixed effects linear model, Journal of Statistical Computation and Simulation, 54 (1996), 211229.

    • Search Google Scholar
    • Export Citation
  • 72

    Hill, D. J., Data mining approaches to complex environmental problems, Ph.D. Thesis, Department of Environmental Engineering in Civil Engineering, University of Illinois at Urbana-Champaign, USA (2007).

    • Search Google Scholar
    • Export Citation
  • 73

    Hiroki, T. and Neil, S., Bayesian tests of a parameter shift under heteroscedasticity: Weighted-t vs. double-t approaches, Communications in Statistics –Theory and Methods, 13 (1984), 10031013.

    • Search Google Scholar
    • Export Citation
  • 74

    Hoogerheide, L. F., Essays on Neural Network Sampling Methods and Instrumental Variables. Rozenberg Publishers (2006).

  • 75

    Jackman, S., Bayesian Analysis for the Social Sciences. John Wiley and Sons, New York (2009).

  • 76

    Jara, A., Quintana, F. A. and Martín, E. S., Linear effects mixed models with skew-elliptical distributions: A Bayesian approach, Computational Statistics and Data Analysis, 52 (2008), 50335045.

    • Search Google Scholar
    • Export Citation
  • 77

    Jeffreys, H. and Zellner, A., Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. North Holland Publishing Company, Holland (1980).

    • Search Google Scholar
    • Export Citation
  • 78

    Johnson, N. L., Kotz, S. and Balakrishnan, N., Continuous Univariate Distributions, volume 1, second edition. John Wiley and Sons, New York (1994).

    • Search Google Scholar
    • Export Citation
  • 79

    Johnson, N. L., Kotz, S. and Balakrishnan, N., Continuous Univariate Distributions, volume 2, second edition. John Wiley and Sons, New York (1995).

    • Search Google Scholar
    • Export Citation
  • 80

    Johnson, R. W., Fitting a sum of exponentials to lattice correlation functions using a non-uniform prior, The European Physical Journal C –Particles and Fields, 70 (2010), 233241.

    • Search Google Scholar
    • Export Citation
  • 81

    Judge, G. G., The Theory and Practice of Econometrics. John Wiley and Sons, New York (1985).

  • 82

    Jupp, P. E. and Mardia, K. V., Maximum likelihood estimation for the matrix von Mises-Fisher and Bingham distributions, Annals of Statistics, 7 (1979), 599606.

    • Search Google Scholar
    • Export Citation
  • 83

    Karlsson, L. S., Bayesian analysis of vector autoregressions, Ph.D. Thesis, Department of Economics, Purdue University, USA (1989).

  • 84

    Kaufman, G. M., Posterior inference for structural parameters using cross-section and time series data, Operations Research Center Working Paper OR 007-71, Sloan School of Management, Massachusetts Institute of Technology, USA (1971).

    • Search Google Scholar
    • Export Citation
  • 85

    Kiefer, N. M., Limited information analysis of a small underidentified macroeconomic model, International Economic Review, 22 (1981), 429442.

    • Search Google Scholar
    • Export Citation
  • 86

    Kirman, A. and Gerard-Varet, L. A., Economics Beyond the Millennium. Oxford University Press, Oxford (1999).

  • 87

    Kleiber, C. and Kotz, S., Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley and Sons, New York (2003).

  • 88

    Kleibergen, F. and Van Dijk, H. K., On the shape of the likelihood/posterior in cointegration models, Econometric Theory, 10 (1994), 514551.

    • Search Google Scholar
    • Export Citation
  • 89

    Klemelä, J., Visualization of the spread of multivariate distributions. Department of Statistics, Economics Faculty University of Mannheim, Germany (2005).

    • Search Google Scholar
    • Export Citation
  • 90

    Koop, G., Recent progress in applied Bayesian econometrics, Journal of Economic Surveys, 8 (1994), 134.

  • 91

    Koop, G., Osiewalski, J. and Steel, M. F. J., Bayesian long-run prediction in time series models, Journal of Econometrics, 69 (1995), 6180.

    • Search Google Scholar
    • Export Citation
  • 92

    Koop, G., Strachan, R., van Dijk, H. K. and Villani, M., Bayesian approaches to cointegration, Econometric Institute Report EI 2005-13. (2005).

    • Search Google Scholar
    • Export Citation
  • 93

    Kotz, S., Kozubowski, T. J. and Podgórski, K., The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhäuser Boston, Boston, Massachusetts (2001).

    • Search Google Scholar
    • Export Citation
  • 94

    Le, N. D., Sun, L. and Zidek, J. V., Spatial prediction and temporal backcasting for environmental fields having monotone data, Canadian Journal of Statistics, 29 (2001), 529554.

    • Search Google Scholar
    • Export Citation
  • 95

    Le, N. D. and Zidek, J. V., Statistical Analysis of Environmental Space-Time Processes. Springer Verlag, New York (2006).

  • 96

    Lebedev, N. N., Special Functions and Their Applications. Dover Publications, Inc, New York (1972).

  • 97

    Li, W. and Cai, X., Statistical analysis of airport network of China, Physical Review E, 69 (2004), Article Number: 046106.

  • 98

    Li, X., Testing on the common mean of normal distributions using Bayesian method, Ph.D. Thesis, Department of Statistics, Case Western Reserve University, USA (2011).

    • Search Google Scholar
    • Export Citation
  • 99

    Lindley, D. V., Reconciliation of probability distributions, Operations Research, 31 (1983), 866880.

  • 100

    Lindley, D. V and Smith, A. F. M., Bayes estimates for the linear model (with discussion), Journal of the Royal Statistical Society, B, 34 (1972), 141.

    • Search Google Scholar
    • Export Citation
  • 101

    Loxam, J. and Drummond, T., Efficient parametric non-Gaussian dynamical filtering, in: Proceedings of the IEEE Nonlinear Statistical Signal Workshop (2006).

    • Search Google Scholar
    • Export Citation
  • 102

    Loxam, J. and Drummond, T., Student t mixture filter for robust, real-time visual tracking, in: Proceedings of the 10th European Conference on Computer Vision, part 3, volume 3, editors D. Forsyth, P. H. S. Torr, and A. Zisserman, Springer Verlag, Berlin (2008), pp. 372385.

    • Search Google Scholar
    • Export Citation
  • 103

    Luoma, A. and Luoto, J., Is there support for the sticky information models in the Michigan inflation expectation data? Working paper, School of Business and Economics, University of Jyvaskyla, Finland (2000).

    • Search Google Scholar
    • Export Citation
  • 104

    Maplesoft, a division of Waterloo Maple, Inc., Maple, Version 12.0, Waterloo, Ontario (2008).

  • 105

    Mardia, K. V., Statistics of directional data (with discussion), Journal of the Royal Statistical Society, B, 37 (1975a), 349393.

  • 106

    Mardia, K. V., Characterizations of directional distributions, in: Statistical Distributions in Scientific Work, volume 3, editors G. P. Patil, S. Kotz and J. K. Ord, Reidel, Dordrecht (1975b), pp. 365385.

    • Search Google Scholar
    • Export Citation
  • 107

    MathWorks, Inc., Matlab, Version 7.10.0, Natick, Massachusetts (2010).

  • 108

    McDonald, J. B. and Newey, W. K., Partially adaptive estimation of regression models via the generalized t distribution, Econometric Theory, 4 (1988), 428457.

    • Search Google Scholar
    • Export Citation
  • 109

    Meinhold, R. J. and Singpurwalla, N. D., Robustification of Kalman filter models, Journal of the American Statistical Association, 84 (1989), 479486.

    • Search Google Scholar
    • Export Citation
  • 110

    Meng, X.-L., Posterior predictive p-values, Annals of Statistics, 22 (1994), 11421160.

  • 111

    Minka, T., Expectation maximization as lower bound maximization (1998).

  • 112

    Mitzenmacher, M., Dynamic models for file sizes and double Pareto distributions, Internet Mathematics, 1 (2004), 305333.

  • 113

    Muller-Plantenberg, N. A., Long swings in Japan’s current account and in the Yen, Unpublished Working Paper, London School of Economics, UK (2006).

    • Search Google Scholar
    • Export Citation
  • 114

    Nadarajah, S., Product Bessel distributions of the first and second kinds, International Journal of Mathematics and Mathematical Sciences, Article ID 57956 (2007).

    • Search Google Scholar
    • Export Citation
  • 115

    Nadarajah, S., A Pareto model for classical systems, Mathematical Methods in the Applied Sciences, 31 (2008a), 3544.

  • 116

    Nadarajah, S., Some product Bessel density distributions, Taiwanese Journal of Mathematics, 12 (2008b), 191211.

  • 117

    Nadarajah, S., The product t density distribution arising from the product of two Student’s t PDFs, Statistical Papers, 50 (2009), 605615.

    • Search Google Scholar
    • Export Citation
  • 118

    Nadarajah, S. and Gupta, A. K., A product Pareto distribution, Metrika, 68 (2008), 199208.

  • 119

    Nadarajah, S. and Kotz, S., Moments of the product F distribution, Applied Mathematics E-Notes, 6 (2006a), 4148.

  • 120

    Nadarajah, S. and Kotz, S., The product Cauchy distribution, The Mathematical Scientist, 31 (2006b), 5357.

  • 121

    Nadarajah, S. and Kotz, S., Moments of a product Pearson-type VII density distribution, AStA Advances in Statistical Analysis, 91 (2007a), 441448.

    • Search Google Scholar
    • Export Citation
  • 122

    Nadarajah, S. and Kotz, S., The compound F distribution, Missouri Journal of Mathematical Sciences, 19 (2007b), 200212.

  • 123

    Nadarajah, S. and Kotz, S., A product Pearson-type VII density distribution, Journal of Computational and Applied Mathematics, 211 (2008), 103113.

    • Search Google Scholar
    • Export Citation
  • 124

    Neuringer, J. L., Derivation of an analytic symmetric bi-modal probability density function, Chaos, Solitons and Fractals, 14 (2002), 543545.

    • Search Google Scholar
    • Export Citation
  • 125

    Oh, M.-S. and Berger, J. O., Adaptive importance sampling in Monte Carlo integration, Journal of Statistical Computation and Simulation, 41 (1992), 143168.

    • Search Google Scholar
    • Export Citation
  • 126

    Oh, M.-S. and Berger, J. O., Integration of multimodal functions by Monte Carlo importance sampling, Journal of the American Statistical Association, 88 (1993), 450456.

    • Search Google Scholar
    • Export Citation
  • 127

    O’Hagan, A., Outliers and credence for location parameter inference, Journal of the American Statistical Association, 85 (1990), 172176.

    • Search Google Scholar
    • Export Citation
  • 128

    Osiewalski, J. and Steel, M. F. J., Robust Bayesian inference in empirical regression models. Madrid University Carlos III, Spain (1991).

    • Search Google Scholar
    • Export Citation
  • 129

    Osiewalski, J. and Steel, M. F. J., Regression models under competing covariance structures: A Bayesian perspective, Annals of Economics and Statistics (1993a), 6579.

    • Search Google Scholar
    • Export Citation
  • 130

    Osiewalski, J. and Steel, M. F. J., Robust Bayesian inference in elliptical regression models, Journal of Econometrics, 57 (1993b), 345363.

    • Search Google Scholar
    • Export Citation
  • 131

    Pilz, J., Bayesian Estimation and Experimental Design in Linear Regression Models. Teubner (1983).

  • 132

    Poirier, D. J., Intermediate Statistics and Econometrics: A Comparative Approach. Massachusetts Institute of Technology Press, Massachusetts Institute of Technology, Massachusetts (1995).

    • Search Google Scholar
    • Export Citation
  • 133

    Pollice, A. and Lasinio, G. J., Spatiotemporal analysis of the PM10 concentration over the Taranto area, Environmental Monitoring and Assessment, 162 (2010), 177190.

    • Search Google Scholar
    • Export Citation
  • 134

    Potzelberger, K., Regions of highest posterior densities for univariate poly-t distributions, in: Proceedings of the First International Conference on Statistical Computing, editors Ozturk, E. C. van der Meulen, E. J. Dudewicz and P. R. Nelson, volume II, American Sciences Press, Columbus, Ohio (1991), pp. 383389.

    • Search Google Scholar
    • Export Citation
  • 135

    Press, S. J., Applied Multivariate Analysis. Robert E. Kreiger Publishing Company, Melbourne, Florida (1982).

  • 136

    Press, S. J. and Shigemasu, K., Bayesian MANOVA and MANOCOVA under exchangeability, Communications in Statistics –Theory and Methods, 14 (1985), 10531078.

    • Search Google Scholar
    • Export Citation
  • 137

    Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I., Integrals and Series, volumes 1, 2 and 3. Gordon and Breach Science Publishers, Amsterdam (1986).

    • Search Google Scholar
    • Export Citation
  • 138

    Quartieri, J., Guida, M., Guarnaccia, C., D’Ambrosio, S. and Guadagnuolo, D., Complex network applications to the infrastructure systems: The Italian airport network case, in: New Aspects of Urban Planning and Transportation, editors T. Panagopoulos, J. B. Burley and S. Celikyay (2008), pp. 96100.

    • Search Google Scholar
    • Export Citation
  • 139

    R Development Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria (2012).

  • 140

    Rajagopalan, M. and Broemeling, L. D., Bayesian inference for the variance components in general mixed linear models, Communications in Statistics –Theory and Methods, 12 (1983), 701723.

    • Search Google Scholar
    • Export Citation
  • 141

    Ramirez-Cobo, P., Lillo, R. E., Wilson, S. and Wiper, M. P., Bayesian inference for double Pareto lognormal queues, Annals of Applied Statistics, 4 (2010), 15331557.

    • Search Google Scholar
    • Export Citation
  • 142

    Richard, J.-F. and Steel, M., Bayesian analysis of systems of seemingly unrelated regression equations under a recursive extended natural conjugate prior density, Journal of Econometrics, 38 (1998), 737.

    • Search Google Scholar
    • Export Citation
  • 143

    Richard, J.-E. and Tompa, H., On the evaluation of poly-t density function, Journal of Econometrics, 12 (1980), 335351.

  • 144

    Robert, C. P., Bayesian computational methods, in: Handbook of Computational Statistics, editors J. Gentle, W. Hardle and Y. Mori (2003).

  • 145

    Robert, C. P., The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer Verlag, New York (2007).

    • Search Google Scholar
    • Export Citation
  • 146

    Robert, C. P., Chopin, N. and Rousseau, J., Harold Jeffreys’ theory of probability revisited, Statistical Science, 24 (2009), 141172.

    • Search Google Scholar
    • Export Citation
  • 147

    Sanso, B., Simple approximations for location and ANOVA models with nonconjugate priors, Test, 6 (1997), 119126.

  • 148

    SAS Institute, Inc., SAS OnlineDoc, Version 9.1, SAS Institute, Inc., Cary, North Carolina (2010).

  • 149

    Savage, L. J., Fienberg, S. E. and Zellner, A., Studies in Bayesian Econometrics and Statistics: In Honor of Leonard J. Savage. North Holland Publishing Company, Holland (1975).

    • Export Citation
  • 150

    Sendra, M., Distribución final de referencia para el problema de Fieller–Creasy, Trabajos de Estadística e Investigación Operativa, 33 (1982), 5572.

    • Search Google Scholar
    • Export Citation
  • 151

    Seshadri, V., The Inverse Gaussian Distribution: Statistical Theory and Applications. Springer Verlag, New York (1999).

  • 152

    Silvestrini, A., Testing fiscal sustainability in Poland: A Bayesian analysis of cointegration, Empirical Economics, 39 (2010), 241274.

    • Search Google Scholar
    • Export Citation
  • 153

    Smith, M. and Kohn, R., Nonparametric seemingly unrelated regression, Journal of Econometrics, 98 (2000), 257282.

  • 154

    Sorensen, D. and Gianola, D., Likelihood, Bayesian and MCMC Methods in Quantitative Genetics. Springer Verlag, New York (2002).

  • 155

    Srivastava, H. M. and Nadarajah, S., Some families of Bessel distributions and their applications, Integral Transforms and Special Functions, 17 (2006), 6573.

    • Search Google Scholar
    • Export Citation
  • 156

    Steel, M. F. J., A Bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches, Journal of Econometrics, 48 (1991), 83117.

    • Search Google Scholar
    • Export Citation
  • 157

    Steel, M. and Richard, J.-F., Bayesian multivariate exogeneity analysis: An application to a UK money demand equation, Journal of Econometrics, 49 (1991), 239274.

    • Search Google Scholar
    • Export Citation
  • 158

    Stone, M., Comments on a posterior distribution of Geisser and Cornfield, Journal of the Royal Statistical Society, B, 26 (1964), 274276.

    • Search Google Scholar
    • Export Citation
  • 159

    Strachan, R. W., Valid Bayesian estimation of the cointegrating error correction model, Journal of Business and Economic Statistics, 21 (2003), 185195.

    • Search Google Scholar
    • Export Citation
  • 160

    Sugita, K., Bayesian cointegration analysis, Warwick Economic Research Paper No 591, Department of Economics, University of Warwick, UK (2001).

    • Search Google Scholar
    • Export Citation
  • 161

    Takano, K., On infinite divisibility of normed product of Cauchy densities, Journal of Computational and Applied Mathematics, 150 (2003), 253263.

    • Search Google Scholar
    • Export Citation
  • 162

    Takano, K., On Laplace transforms of some probability densities, Applied Mathematics and Computation, 187 (2007), 501506.

  • 163

    Tiao, G. C. and Zellner, A., On the Bayesian estimation of multivariate regression, Journal of the Royal Statistical Society, B, 26 (1964), 277285.

    • Search Google Scholar
    • Export Citation
  • 164

    Tompa, H., Poly-t distributions (PTD). CORE User’s Manual Set # C-9 (1977).

  • 165

    Tsionas, E. G., Bayesian inference in generalized error and generalized studentt regression models, Communications in Statistics –Theory and Methods, 37 (2008), 388407.

    • Search Google Scholar
    • Export Citation
  • 166

    Villani, M., Bayesian reference analysis of cointegration, Econometric Theory, 21 (2005), 326357.

  • 167

    Von Mises, R., Probability Statistics and Truth, second edition. Springer Verlag, New York (1981).

  • 168

    Wang, H.-L., Bayesian and non-Bayesian estimators in the simultaneous equation model, Ph.D. Thesis, Department of Mathematics, National Cheng Kung University, Taiwan (2007).

    • Search Google Scholar
    • Export Citation
  • 169

    Warne, A., Bayesian inference in cointegrated VAR models with applications to the demand for euro area M3, Working Paper 692, European Central Bank (2006).

    • Search Google Scholar
    • Export Citation
  • 170

    Welling, M., Hinton, G. and Osindero, S., Learning sparse topographic representations with products of student t distributions, Advances in Neural Information Processing Systems, 15 (2003), 13591366.

    • Search Google Scholar
    • Export Citation
  • 171

    Wolfram Research, Inc., Mathematica, Version 8.0, Champaign, Illinois (2010).

  • 172

    Wong, C. Y., Asymptotic normality of poly-t densities with Bayesian applications, Communications in Statistics –Theory and Methods, 17 (1988), 16131627.

    • Search Google Scholar
    • Export Citation
  • 173

    Wong, F., Carter, C. and Kohn, R., Efficient estimation of covariance selection models, Biometrika, 90 (2003), 809830.

  • 174

    Wong, W.-K. and Bian, G., Robust estimation in capital asset pricing model, Journal of Applied Mathematics and Decision Sciences, 4 (2000), 6582.

    • Search Google Scholar
    • Export Citation
  • 175

    Yang, R. and Berger, J., Estimation of a covariance matrix using the reference prior, Annals of Statistics, 22 (1994), 11951211.

  • 176

    Yen, T.-Y., A majorization-minimization approach to variable selection using spike and slab priors. Submitted to Annals of Statistics (2010).

    • Search Google Scholar
    • Export Citation
  • 177

    Zellner, A., An Introduction to Bayesian Inference in Econometrics. John Wiley and Sons, New York (1971).

  • 178

    Zellner, A., Ando, T., Basturk, N., Hoogerheide, L. and van Dijk, H. K., Bayesian analysis of instrumental variable models: The potential of direct Monte Carlo. Tinbergen Institute Discussion Paper TI 2012-095/III (2012a).

    • Search Google Scholar
    • Export Citation
  • 179

    Zellner, A., Ando, T., Basturk, N. and van Dijk, H. K., Bayesian analysis of instrumental variable models: Acceptance-rejection within direct Monte Carlo. Tinbergen Institute Discussion Paper TI 2012-098/III (2012b).

    • Search Google Scholar
    • Export Citation

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