INSTITUTE OF MATHEMATICAL STATISTICS

A semiparametric Bayesian model for multiple monotonically increasing count sequences | |

Valeria Leiva-Yamaguchi, Fernando A. Quintana | 155-170 |

Parametric Stein operators and variance bounds | |

Christophe Ley, Yvik Swan | 171-195 |

Log-symmetric distributions: Statistical properties and parameter estimation | |

Luis Hernando Vanegas, Gilberto A. Paula | 196-220 |

Local limit theorems for shock models | |

Edward Omey, Rein Vesilo | 221-247 |

A new skew logistic distribution: Properties and applications | |

D. V. S. Sastry, Deepesh Bhati | 248-271 |

Fractional absolute moments of heavy tailed distributions | |

Muneya Matsui, Zbyněk Pawlas | 272-298 |

Limiting behavior of the Jeffreys power-expected-posterior Bayes factor in Gaussian linear models | |

D. Fouskakis, I. Ntzoufras | 299-320 |

On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$ | |

Jiaojiao Ma, Shengjun Fan, Rui Fang | 321-344 |

In longitudinal clinical trials, subjects may be evaluated many times over the course of the study. This article is motivated by a medical study conducted in the U.S. Veterans Administration Cooperative Urological Research Group to assess the effectiveness of a treatment in preventing recurrence on subjects affected by bladder cancer. The data consist of the accumulated tumor counts over a sequence of regular checkups, with many missing observations. We propose a hierarchical nonparametric Bayesian model for sequences of monotonically increasing counts. Unlike some of the previous analyses for these data, we avoid interpolation by explicitly incorporating the missing observations under the assumption of these being missing completely at random. Our formulation involves a generalized linear mixed effects model, using a dependent Dirichlet process prior for the random effects, with an autoregressive component to include serial correlation along patients. This provides great flexibility in the desired inference, that is, assessing the treatment effect. We discuss posterior computations and the corresponding results obtained for the motivating dataset, including a comparison with parametric alternatives.

Aldous, D. J. (1985). *Exchangeability and Related Topics, École d’été de probabilités de Saint-Flour, XIII—1983. Lecture Notes in Math.***1117**, 1–198. Berlin: Springer.Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. *Annals of Statistics***2**, 1152–1174.Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. *Annals of Statistics***1**, 353–355.Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized in linear mixed models. *Journal of the American Statistical Association***88**, 9–25.Bush, C. A. and MacEachern, S. N. (1996). A semiparametric Bayesian model for randomised block designs. *Biometrika***83**, 275–285.Byar, D., Blackard, C. and Urological Research Group (1977). Comparisons of placebo, pyridoxine, and topical thiotepa in preventing recurrence of stage I bladder cancer. *Urology***10**, 556–561.Celeux, G., Forbes, F., Robert, C. P. and Titterington, D. M. (2006). Deviance information criteria for missing data models. *Bayesian Analysis***1**, 651–673 (electronic).Davis, C. S. and Wei, L. J. (1988). Nonparametric methods for analyzing incomplete nondecreasing repeated measurements. *Biometrics***44**, 1005–1018.Di Lucca, M., Gugliemi, A., Müller, P. and Quintana, F. A. (2013). A simple class of Bayesian nonparametric autoregression models. *Bayesian Analysis***8**, 63–88.Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. *Journal of the American Statistical Association***89**, 268–277.Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. *Annals of Statistics***1**, 209–230.Geisser, S. and Eddy, W. F. (1979). A predictive approach to model selection. *Journal of the American Statistical Association***74**, 153–160.Gelfand, A. E. and Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. *Journal of the Royal Statistical Society. Series B, Methodological***56**, 501–514.Giardina, F., Guglielmi, A., Ruggeri, F. and Quintana, F. A. (2011). Bayesian first order autoregressive latent variable models for multiple binary sequences. *Statistical Modelling International Journal***11**, 471–488.Hjort, N. L., Holmes, C., Müller, P. and Walker, S. (2010). *Bayesian Nonparametrics*. Cambridge, UK: Cambridge University Press.Ibrahim, J. G. and Kleinman, K. P. (1998). Semiparametric Bayesian methods for random effects models. In *Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist.***133**, 89–114. New York: Springer.Ishwaran, H. and James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. *Journal of the American Statistical Association***96**, 161–173.MacEachern, S. N. and Müller, P. (1998). Estimating mixture of Dirichlet process models. *Journal of Computational and Graphical Statistics***7**, 223–338.Müller, P. and Mitra, R. (2013). Bayesian nonparametric inference—Why and how. *Bayesian Analysis***8**, 269–302.Müller, P. and Quintana, F. A. (2004). Nonparametric Bayesian data analysis. *Statistical Science***19**, 95–110.Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. *Journal of Computational and Graphical Statistics***9**, 249–265.Plummer, M., Best, N., Cowles, K. and Vines, K. (2006). Coda: Convergence diagnosis and output analysis for MCMC. *R News***6**, 7–11.Roberts, G. O. and Rosenthal, J. S. (2009). Examples of adaptive MCMC. *Journal of Computational and Graphical Statistics***18**, 349–367.Rolin, J.-M. (1992). Some useful properties of the Dirichlet process. Technical Report 9207, Center for Operations Research & Econometrics, Univ. Catholique de Louvain. Sethuraman, J. (1994). A constructive definition of Dirichlet priors. *Statistica Sinica***4**, 639–650.Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. *Journal of the Royal Statistical Society. Series B, Statistical Methodology***64**, 583–639.Zeger, S. L. and Karim, M. R. (1991). Generalized linear models with random effects; a Gibbs sampling approach. *Journal of the American Statistical Association***86**, 79–86.

Stein operators are (differential/difference) operators which arise within the so-called Stein’s method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric distributions with continuous dependence on the parameter. We provide explicit general expressions for location, scale and skewness families. We also provide a general expression for discrete distributions. We use properties of our operators to provide upper and lower variance bounds (only lower bounds in the discrete case) on functionals $h(X)$ of random variables $X$ following parametric distributions. These bounds are expressed in terms of the first two moments of the derivatives (or differences) of $h$. We provide general variance bounds for location, scale and skewness families and apply our bounds to specific examples (namely the Gaussian, exponential, gamma and Poisson distributions). The results obtained via our techniques are systematically competitive with, and sometimes improve on, the best bounds available in the literature.

Afendras, G., Papadatos, N. and Papathanasiou, V. (2011). An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds. *Bernoulli***17**, 507–529.Bakry, D., Gentil, I. and Ledoux, M. (2014). *Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]***348**. Cham: Springer.Barbour, A. D. (1990). Stein’s method for diffusion approximations. *Probability Theory and Related Fields***84**, 297–322.Barbour, A. D. and Chen, L. H. (2014). Steins (magic) method. Preprint. Available at arXiv:1411.1179. Barbour, A. D. and Chen, L. H. Y. (2005). *An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap.***4**. Singapore: Singapore Univ. Press.Barbour, A. D., Holst, L. and Janson, S. (1992). *Poisson Approximation. Oxford Studies in Probability***2**. New York: The Clarendon Press, Oxford Univ. Press.Borovkov, A. and Utev, S. (1984). On an inequality and a related characterization of the normal distribution. *Theory of Probability and Its Applications***28**, 219–228.Brown, T. C. and Xia, A. (1995). On Stein–Chen factors for Poisson approximation. *Statistics and Probability Letters***23**, 327–332.Cacoullos, T. (1982). On upper and lower bounds for the variance of a function of a random variable. *The Annals of Probability***10**, 799–809.Cacoullos, T. and Papathanasiou, V. (1995). A generalization of covariance identity and related characterizations. *Mathematical Methods of Statistics***4**, 106–113.Cacoullos, T., Papathanasiou, V. and Utev, S. A. (1994). Variational inequalities with examples and an application to the central limit theorem. *The Annals of Probability***22**, 1607–1618.Chatterjee, S., Fulman, J. and Röllin, A. (2011). Exponential approximation by exchangeable pairs and spectral graph theory. *ALEA Latin American Journal of Probability and Mathematical Statistics***8**, 1–27.Chen, L. H. Y. (1975). Poisson approximation for dependent trials. *The Annals of Probability***3**, 534–545.Chen, L. H. Y. (1982). An inequality for the multivariate normal distribution. *Journal of Multivariate Analysis***12**, 306–315.Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). *Normal Approximation by Stein’s Method. Probability and Its Applications (New York)*. Heidelberg: Springer.Chernoff, H. (1980). The identification of an element of a large population in the presence of noise. *The Annals of Statistics***8**, 1179–1197.Chernoff, H. (1981). A note on an inequality involving the normal distribution. *The Annals of Probability***9**, 533–535.Diaconis, P. and Zabell, S. (1991). Closed form summation for classical distributions: Variations on a theme of de Moivre. *Statistical Science***6**, 284–302.Döbler, C. (2012). Stein’s method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model. Preprint. Available at arXiv:1207.0533. Döbler, C., Gaunt, R. E., Ley, C., Reinert, G. and Swan, Y. (2015). *A Handbook of Stein Operators*. In progress.Goldstein, L. and Reinert, G. (2005). Distributional transformations, orthogonal polynomials, and Stein characterizations. *Journal of Theoretical Probability***18**, 237–260.Goldstein, L. and Reinert, G. (2013). Stein’s method for the Beta distribution and the Pólya–Eggenberger urn. *Journal of Applied Probability***50**, 1187–1205.Götze, F. (1991). On the rate of convergence in the multivariate CLT. *The Annals of Probability***19**, 724–739.Götze, F. and Tikhomirov, A. N. (2003). Rate of convergence to the semi-circular law. *Probability Theory and Related Fields***127**, 228–276.Holmes, S. (2004). Stein’s method for birth and death chains. In *Stein’s Method: Expository Lectures and Applications. IMS Lecture Notes Monogr. Ser.***46**, 42–65. Beachwood, OH: Institute of Mathematical Statistics.Houdré, C. and Kagan, A. (1995). Variance inequalities for functions of Gaussian variables. *Journal of Theoretical Probability***8**, 23–30.Hudson, H. M. (1978). A natural identity for exponential families with applications in multiparameter estimation. *The Annals of Statistics***6**, 473–484.Hwang, J. T. (1982). Improving upon standard estimators in discrete exponential families with applications to Poisson and negative binomial cases. *The Annals of Statistics***10**, 857–867.Jones, M. C. and Pewsey, A. (2009). Sinh–arcsinh distributions. *Biometrika***96**, 761–780.Klaassen, C. A. J. (1985). On an inequality of Chernoff. *The Annals of Probability***13**, 966–974.Landsman, Z., Vanduffel, S. and Yao, J. (2015). Some Stein-type inequalities for multivariate elliptical distributions and applications. *Statistics and Probability Letters***97**, 54–62.Ledoux, M. (2001). *The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs***89**. Providence, RI: American Mathematical Society.Ledoux, M., Nourdin, I. and Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. *Geometric and Functional Analysis***25**, 256–306.Lehmann, E. L. and Casella, G. (1998). *Theory of Point Estimation*, 2nd ed. New York: Springer.Ley, C. and Paindaveine, D. (2010). Multivariate skewing mechanisms: A unified perspective based on the transformation approach. *Statistics and Probability Letters***80**, 1685–1694.Ley, C., Reinert, G. and Swan, Y. (2014). Approximate computation of expectations: A canonical Stein operator. Preprint. Available at arXiv:1408.2998v2. Ley, C. and Swan, Y. (2013a). Local Pinsker inequalities via Stein’s discrete density approach. *IEEE Transactions on Information Theory***59**, 5584–4491.Ley, C. and Swan, Y. (2013b). Stein’s density approach and information inequalities. *Electronic Communications in Probability***18**, 1–14.Liu, J. S. (1994). Siegel’s formula via Stein’s identities. *Statistics and Probability Letters***21**, 247–251.Luk, H. M. (1994). Stein’s method for the gamma distribution and related statistical applications. Ph.D. thesis, Univ. Southern California. Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. *Probability Theory and Related Fields***145**, 75–118.Nourdin, I. and Peccati, G. (2012). *Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics*. Cambridge: Cambridge Univ. Press.Papadatos, N. and Papathanasiou, V. (2001). Unified variance bounds and a Stein-type identity. In *Probability and Statistical Models with Applications*87–100. Boca Raton, FL: Chapman & Hall/CRC.Peköz, E. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. *The Annals of Probability***39**, 587–608.Peköz, E., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. *The Annals of Applied Probability***23**, 1188–1218.Picket, A. (2004). Rates of convergence of $\chi^{2}$ approximations via Stein’s method. Ph.D. thesis, Lincoln College, Univ. Oxford. Reinert, G. (2004). Three general approaches to Stein’s method. In *An Introduction to Stein’s Method. Lecture Notes Series, Institute for Mathematical Sciences, National Univ. Singapore***4**. Singapore: World Scientific.Röllin, A. (2012). On the optimality of Stein factors. In *Probability Approximations and Beyond*61–72. New York: Springer.Stein, C. (1970/1971). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In *Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability*,*Vol. II: Probability Theory*583–602. Berkeley, CA: Univ. California, Berkeley.Stein, C. (1986). *Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series***7**. Hayward, CA: Institute of Mathematical Statistics.Stein, C., Diaconis, P., Holmes, S. and Reinert, G. (2004). Use of exchangeable pairs in the analysis of simulations. In *Stein’s Method: Expository Lectures and Applications. IMS Lecture Notes Monogr. Ser.***46**, 1–26. Beachwood, OH: Institute of Mathematical Statistics.

In this paper, we study the main statistical properties of the class of log-symmetric distributions, which includes as special cases bimodal distributions as well as distributions that have heavier/lighter tails than those of the log-normal distribution. This family includes distributions such as the log-normal, log-Student-$t$, harmonic law, Birnbaum–Saunders, Birnbaum–Saunders-$t$ and generalized Birnbaum–Saunders. We derive quantile-based measures of location, dispersion, skewness, relative dispersion and kurtosis for the log-symmetric class that are appropriate in the context of asymmetric and heavy-tailed distributions. Additionally, we discuss parameter estimation based on both classical and Bayesian approaches. The usefulness of the log-symmetric class is illustrated through a statistical analysis of a real dataset, in which the performance of the log-symmetric class is compared with that of some competitive and very flexible distributions.

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In *International Symposium on Information Theory*(B. N. Petrov and F. Csaki, eds.) 267–281. Budapest, Hungary: Akademiai Kiado.Andrews, D. R. and Mallows, C. L. (1974). Scale mixtures of normal distributions. *Journal of the Royal Statistical Society. Series B***36**, 99–102.Azzalini, A., dal Cappello, T. and Kotz, S. (2003). Log-skew-normal and log-skew-$t$ distributions as model for family income data. *Journal of Income Distribution***11**, 12–20.Balakrishnan, N., Leiva, V., Sanhueza, A. and Vilca, F. (2009). Estimation in the Birnbaum–Saunders distributions based on scale-mixture of normals and the EM-algorithm. *Statistics and Operations Research Transactions (SORT)***33**, 171–192.Barndoff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. *Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences***353**, 401–419.Barros, M., Paula, G. A. and Leiva, V. (2008). A new class of survival regression models with heavy-tailed errors: Robustness and diagnostics. *Lifetime Data Analysis***14**, 316–332.Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. *Journal of Applied Probability***6**, 319–327.Bonett, D. G. (2006). Confidence interval for a coefficient of quartile variation. *Computational Statistics & Data Analysis***50**, 2953–2957.Box, G. E. P. and Tiao, G. C. (1973). *Bayesian Inference in Statistical Analysis*. Reading, MA: Addison-Wesely Publishing Company.Carrasco, J. M. F., Ortega, E. M. M. and Cordeiro, G. M. (2008). A generalized modified Weibull distribution for lifetime modeling. *Computational Statistics & Data Analysis***53**, 450–462.Chib, S. and Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. *The American Statistician***49**, 327–335.Cordeiro, G. M., Ferrari, S. L. P., Uribe-Opazo, M. A. and Vasconcellos, K. L. P. (2000). Corrected maximum-likelihood estimation in a class of symmetric nonlinear regression models. *Statistics & Probability Letters***46**, 317–328.Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. *Journal of the Royal Statistical Society, Series B***39**, 1–38.Díaz-García, J. A. and Leiva, V. (2005). A new family of life distributions based on elliptically contoured distributions. *Journal of Statistical Planning and Inference***128**, 445–457.Fang, K. T., Kotz, S. and Ng, K. W. (1990). *Symmetric Multivariate and Related Distributions*. London: Chapman & Hall.Fonseca, T. C. O., Migon Helio, S. and Ferreira Marco, A. R. (2012). Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution. *Brazilian Journal of Probability and Statistics***26**, 327–343.Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. *Journal of the American Statistical Association***85**, 398–409.Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). *Bayesian Data Analysis*. Boca Raton: Chapman & Hall.Glaser, R. E. (1980). Bathtub and related failure rate characterizations. *Journal of the American Statistical Association***75**, 667–672.Groeneveld, R. A. and Meeden, G. (1984). Measuring skewness and kurtosis. *The Statistician***33**, 391–399.Hinkley, D. V. (1975). On power transformations to symmetry. *Biometrika***62**, 101–111.Hörmann, W. and Leydold, J. (2015). Generating generalized inverse Gaussian random variates. *Statistics and Computing***24**, 547–557.Jørgensen, B. (1982). *Statistical Properties of the Generalized Inverse Gaussian Distribution*. New York: Springer.Kano, Y., Berkane, M. and Bentler, P. (1993). Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations. *Journal of the American Statistical Association***88**, 135–143.Leiva, V., Riquelme, M., Balakrishnan, N. and Sanhueza, A. (2008). Lifetime analysis based on the generalized Birnbaum–Saunders distribution. *Computational Statistics & Data Analysis***52**, 2079–2097.Limpert, E., Stahel, W. A. and Abbt, M. (2001). Log-normal distributions across the sciences: Key and clues. *BioScience***51**, 341–352.Lucas, A. (1997). Robustness of the Student-$t$ based $M$-estimator. *Communications in Statistics, Theory and Methods***26**, 1165–1182.Marchenko, Y. V. and Genton, M. G. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. *Environmetrics***21**, 318–340.Marshall, A. W. and Olkin, I. (2007). *Life Distributions*. New York: Springer.Moors, J. J. A. (1988). A quantile alternative for kurtosis. *The Statistician***37**, 25–32.Nadarajah, S. and Kotz, S. (2006). The exponentiated type distributions. *Acta Applicandae Mathematicae***92**, 97–111.Paula, G. A., Leiva, V., Barros, M. and Liu, S. (2012). Robust statistical modeling using the Birnbaum–Saunders-$t$ distribution applied to insurance distribution. *Applied Stochastic Model in Business and Industry***28**, 16–34.Puig, P. (2008). A note on the harmonic law: A two-parameter family of distributions for ratios. *Statistics & Probability Letters***78**, 320–326.R Core Team (2013). *R: A Language and Environment for Statistical Computing*. Vienna, Austria: R Foundation for Statistical Computing. Available at http://www.r-project.org.Rieck, J. R. (1999). A moment-generating function with application to the Birnbaum–Saunders distribution. *Communications in Statistics—Theory and Methods***28**, 2213–2222.Rieck, J. R. and Nedelman, J. R. (1991). A log-linear model for the Birnbaum–Saunders distribution. *Technometrics***33**, 51–60.Rigby, R. A. and Stasinopoulos, D. M. (2006). Using the box-cox $t$ distribution in GAMLSS to model skewness and kurtosis. *Statistical Modelling***6**, 209–229.Rogers, W. H. and Tukey, J. W. (1972). Understanding some long-tailed symmetrical distributions. *Statistica Neerlandica***26**, 211–226.Schwarz, G. E. (1978). Estimating the dimension of a model. *Annals of Statistics***6**, 461–464.Stacy, E. W. (1962). A generalization of the gamma distribution. *The Annals of Mathematical Statistics***33**, 1187–1192.Villegas, C., Paula, G. A., Cysneiros, F. J. A. and Galea, M. (2013). Influence diagnostics in generalized symmetric linear models. *Computational Statistics & Data Analysis***59**, 161–170.West, M. (1987). On scale mixtures of normal distributions. *Biometrika***74**, 646–648.Zwillinger, D. and Kokoska, S. (2000). *Standard Probability and Statistical Tables and Formula*. Boca Raton: Chapman & Hall.

In many physical systems, failure occurs when the stress after shock $n$ first exceed a critical level $x$. We consider the number of shocks $\tau(x)$ to failure and obtain more detailed information that is usually obtained about asymptotic distribution by using local limit theorems. We consider extreme and cumulative shock models with both univariate and multivariate shock types. We derive the limiting distribution of $\tau(x)$ and the rate of convergence to that. For the extreme shock model, rate of convergence for regularly varying shock distributions is found using the weighted Kolmorogov probability metric. For the cumulative shock model, we examine the rate of convergence to Gaussian densities.

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). *Regular Variation. Encyclopedia of Mathematics and Its Applications***27**. Cambridge: Cambridge Univ. Press.Feller, W. (1971). *An Introduction to Probability Theory and Its Applications. Vol. II*, 2nd ed. New York: Wiley.Gut, A. (1988). *Stopped Random Walks*. New York: Springer.Gut, A. (1990). Cumulative shock models. *Advances in Applied Probability***22**, 504–507.Gut, A. (2001). Mixed shock models. *Bernoulli***7**, 541–555.Gut, A. and Hüsler, J. (1999). Extreme shock models. *Extremes***2**, 295–307.de Haan, L. (1970). *On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Math. Centre Tract***32**. Amsterdam: Mathematisch Centrum.Landau, L. D. and Lifshits, E. M. (1976). *Teoreticheskaya fizika. Tom V. Statisticheskaya fizika. Chast 1*. Moscow: Izdat. “Nauka”.Mallor, F. and Omey, E. (2001). Shocks, runs and random sums. *Journal of Applied Probability***38**, 438–448.Mallor, F. and Omey, E. (2006). *Univariate and Multivariate Weighted Renewal Theory*. Collection of Monographies from the Department of Statistics and Operations Research, No. 2, Public Univ. Navarre.Mallor, F., Omey, E. and Santos, J. (2006). Asymptotic results for a run and cumulative mixed shock model. *Journal of Mathematical Sciences***138**, 5410–5414.Omey, E. and Rachev, S. (1988). On the rate of convergence in extreme value theory. *Theory of Probability and Its Applications***33**, 560–566.Rachev, S. (1991). *Probability Metrics and the Stability of Stochastic Models*. Chichester: Wiley.Shanthikumar, J. G. and Sumita, U. (1983). General shock models associated with correlated renewal sequences. *Journal of Applied Probability***20**, 600–614.Shorack, G. R. (2000). *Probability for Statisticians*. New York: Springer.Smith, R. L. (1982). Uniform rates of convergence in extreme-value theory. *Advances in Applied Probability***14**, 600–622.Sumita, U. and Shanthikumar, J. G. (1985). A class of correlated cumulative shock models. *Advances in Applied Probability***17**, 347–366.Virchenko, Y. P. (1998). Percolation mechanism of material ageing and distribution of the destruction time. *Functional Materials***5**, 7–13.Virchenko, Y. P. and Sheremet, O. I. (1999). The formation of destruction time distribution of material ageing by statistically independent perturbations. *Functional Materials***6**, 5–12.

Following the methodology of Azzalini, researchers have developed skew logistic distribution and studied its properties. The cumulative distribution function in their case is not explicit and therefore numerical methods are employed for estimation of parameters. In this paper, we develop a new skew logistic distribution based on the methodology of Fernández and Steel and derive its cumulative distribution function and also the characteristic function. For estimating the parameters, Method of Moments, Modified Method of Moment and Maximum likelihood estimation are used. With the help of simulation study, for different sample sizes, the parameters are estimated and their consistency was verified through Box Plot. We also proposed a regression model in which probability of occurrence of an event is derived from our proposed new skew logistic distribution. Further, proposed model fitted to a well studied lean body mass of Australian athlete data and compared with other available competing distributions.

Arellano-Valle, R. B. and Azzalani, A. (2013). The centred parameterization and related quantities of the skew-$t$ distribution. *Journal of Multivariate Analysis***113**, 73–90.Arellano-Valle, R. B., Gómez, H. W. and Quintana, F. A. (2003). Statistical inference for a general class of asymmetric distributions. *Journal of Statistical Planning and Inference***128**, 427–443.Arnold, B. C. and Beaver, R. J. (2000). The skew-Cauchy distribution. *Statistics & Probability Letters***49**, 285–290.Asgharzadeh, A., Esmaily, L., Nadarajah, S. and Shih, S. H. (2013). A generalized skew logistic. *REVSTAT Statistical Journal***11**, 317–338.Ayebo, A. and Kozubowski, T. J. (2003). An asymmetric generalization of Gaussian and Laplace laws. *Journal of Probability and Statistical Science***2**, 187–210.Azzalini, A. (1985). A class of distributions which includes the normal ones. *Scandinavian Journal of Statistics***12**, 171–178.Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew $t$ distribution. *Journal of the Royal Statistical Society, B***65**, 367–389.Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. *Biometrika***83**, 715–726.Bazan, J. L., Branco, M. and Bolfarine, H. (2006). A skew item response model. *Bayesian Analysis***1**, 861–892.Bolfarine, H. and Bazan, J. L. (2010). Bayesian estimation of the logistic positive exponent IRT model. *Journal of Educational and Behavioral Statistics***35**, 693–713.Chakraborty, S., Hazarika, P. J. and Ali, M. M. (2012). A new skew logistic distribution and its properties. *Pakistan Journal of Statistics***28**, 513–524.Chen, M. H., Dey, D. K. and Shao, Q.-M. (1999). A new skewed link model for dichotomous quantal response data. *Journal of the American Statistical Association***94**, 1172–1186.Cook, R. D. and Weisberg, S. (1994). *Regression Graphics*. New York: Wiley.Fernández, C., Osiewalski, J. and Steel, M. F. J. (1995). Modeling and inference with $v$-spherical distributions. *Journal of the American Statistical Association***90**, 1331–1340.Fernández, C. and Steel, M. F. J. (1998). On Bayesian modeling of fat tails and skewness. *Journal of the American Statistical Association***93**, 359–371.Gupta, A. K., Chang, F. C. and Huang, W. J. (2002). Some skew-symmetric models. *Random Operators and Stochastic Equations***10**, 133–140.Gupta, R. D. and Kundu, D. (2010). Generalized logistic distributions. *Journal of Applied Statistical Science***18**, 51–66.Huang, W. J. and Chen, Y. H. (2007). Generalized skew-Cauchy distribution. *Statistics & Probability Letters***77**, 1137–1147.Jones, M. C. and Faddy, M. J. (2003). A skew extension of the $t$-distribution, with applications. *Journal of Royal Statistical Society, Series B, Statistical Methodology***65**, 159–174.Lane, M. N. (2004). Pricing risk transfer transaction. *ASTIN Bulletin***30**, 259–293.Ma, Y. and Genton, M. G. (2004). A flexible class of skew symmetric distribution. *Scandinavian Journal of Statistics***31**, 259–468.Nadarajah, S. and Kotz, S. (2003). Skewed distributions generated by the normal kernel. *Statistics & Probability Letters***65**, 269–277.Pewsey, A. (2009). Problems of inference for Azzalini’s skew normal distribution. *Journal of Applied Statistics***27**, 859–870.Samejima, F. (1997). Departure from normal assumptions: A promise for future psychometrics with substantive mathematical modelling. *Psychometrika***62**, 471–493.Samejima, F. (2000). Logistic positive exponent family of models: Virtue of asymmetric item characteristic curves. *Psychometrika***65**, 319–335.Simon, H. A. (1955). On a class of skew distribution funtions. *Biometrika***42**, 425–440.Tse, Y.-K. (2009). *Non-life Actuarial Models: Theory, Methods and Evaluation*. Cambridge: Cambridge University Press.Wahed, A. S. and Ali, M. M. (2001). The skew logistic distribution. *Journal of Statistical Research***2001**, 71–80. 2.Wang, J., Boyer, J. and Genton, M. G. (2004). A skew symmetric representation of multivariate distribution. *Statistica Sinica***14**, 1259–1270.

Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. Our main focus is on an infinite variance case with finite mean, that is, we are interested in formulae for $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$ with $1<\gamma<2$ and $\mu\in\mathbb{R}$. We review techniques of fractional differentiation of Laplace transforms and characteristic functions. Several examples are given with analytical expressions of $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$. We also evaluate the fractional moment errors for both prediction and parameter estimation problems.

Adler, R. J., Feldman, R. E. and Taqqu, M. S., eds. (1998). *A Practical Guide to Heavy Tails: Statistical Techniques and Applications*. New York: Birkhäuser.Blattberg, R. and Sargent, T. (1971). Regression with non-Gaussian stable disturbances: Some sampling results. *Econometrica***39**, 501–510.Brown, B. M. (1970). Characteristic functions, moments, and the central limit theorem. *The Annals of Mathematical Statistics***41**, 658–664.Brown, B. M. (1972). Formulae for absolute moments. *Journal of the Australian Mathematical Society***13**, 104–106.Cline, D. B. H. and Brockwell, P. J. (1985). Linear prediction of ARMA processes with infinite variance. *Stochastic Processes and Their Applications***19**, 281–296.Cressie, N. and Borkent, M. (1986). The moment generating function has its moments. *Journal of Statistical Planning and Inference***13**, 337–344.Cressie, N., Davis, A. S., Folks, J. L. and Policello, G. E. (1981). The moment-generating function and negative integer moments. *American Statistician***35**, 148–150.Devroye, L. (1990). A note on Linnik’s distribution. *Statistics and Probability Letters***9**, 305–306.Gradshteyn, I. S. and Ryzhik, I. M. (2007). *Table of Integrals, Series, and Products*, 7th ed. San Diego, CA: Academic Press.Hardin, C. D., Jr., Samorodnitsky, G. and Taqqu, M. S. (1991). Nonlinear regression of stable random variables. *The Annals of Applied Probability***1**, 582–612.Hsu, P. L. (1951). Absolute moments and characteristic functions. *Journal Chinese Mathematical Society (New Series)***1**, 257–280.Kawata, T. (1972). *Fourier Analysis in Probability Theory*. New York: Academic Press.Kokoszka, P. S. (1996). Prediction of inifinite variance fractional ARIMA. *Probability and Mathematical Statistics***16**, 65–83.Kozubowski, T. J. (2001). Fractional moment estimation of Linnik and Mittag–Leffler parameters. *Mathematical and Computer Modelling***34**, 1023–1035.Kozubowski, T. J. and Meerschaert, M. M. (2009). A bivariate infinitely divisible distribution with exponential and Mittag–Leffler marginals. *Statistics and Probability Letters***79**, 1596–1601.Kozubowski, T. J., Podgórski, K. and Samorodnitsky, G. (1999). Tails of Lévy measure of geometric stable random variables. *Extremes***1**, 367–378.Laue, G. (1980). Remarks on the relation between fractional moments and fractional derivatives of characteristic functions. *Journal of Applied Probability***17**, 456–466.Laue, G. (1986). Results on moments of non-negative random variables. *Sankhyā Series A***48**, 299–314.Lim, S. C. and Teo, L. P. (2010). Analytic and asymptotic properties of multivariate generalized Linnik’s probability densities. *The Journal of Fourier Analysis and Applications***16**, 715–747.Lin, G. D. (1998). On the Mittag–Leffler distributions. *Journal of Statistical Planning and Inference***74**, 1–9.Linnik, Y. V. (1953). Linear forms and statistical criteria. I, II. *Ukrainskij Matematicheskij Zhurnal***5**, 207–243, 247–290 (in Russian). English translation in*Selected Translation in Mathematical Statistics and Probability***3**(1962/1963), 1–90.Matsui, M. and Mikosch, T. (2010). Prediction in a Poisson cluster model. *Journal of Applied Probability***47**, 350–366.Matsui, M. and Takemura, A. (2006). Some improvements in numerical evaluation of symmetric stable density and its derivatives. *Communications in Statistics. Theory and Methods***35**, 149–172.Mikosch, T., Samorodnitsky, G. and Tafakori, L. (2013). Fractional moments of solutions to stochastic recurrence equations. *Journal of Applied Probability***50**, 969–982.Nguyen, T. T. (1995). Conditional distributions and characterizations of multivariate stable distribution. *Journal of Multivariate Analysis***53**, 181–193.Nolan, J. P. (2013). Multivariate elliptically contoured stable distributions: Theory and estimation. *Computational Statistics***28**, 2067–2089.Paolella, M. S. (2007). *Intermediate Probability: A Computational Approach*. Chichester: Wiley.Pinelis, I. (2011). Positive-part moments via the Fourier–Laplace transform. *Journal of Theoretical Probability***24**, 409–421.Podlubny, I. (1999). *Fractional Differential Equations*. San Diego, CA: Academic Press.Ramachandran, B. (1969). On characteristic functions and moments. *Sankhyā Series A***31**, 1–12.Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). *Fractional Integrals and Derivatives: Theory and Applications*. Yverdon: Gordon and Breach Science Publishers.Samorodnitsky, G. and Taqqu, M. S. (1991). Conditional moments and linear regression for stable random variables. *Stochastic Processes and Their Applications***39**, 183–199.Samorodnitsky, G. and Taqqu, M. S. (1994). *Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance*. Boca Raton, FL: Chapman and Hall/CRC.Sato, K.-i. (1999). *Lévy Processes and Infinitely Divisible Distributions*. Cambridge: Cambridge Univ. Press.Shanbhag, D. N. and Sreehari, M. (1977). On certain self-decomposable distributions. *Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete***38**, 217–222.von Bahr, B. (1965). On the convergence of moments in the central limit theorem. *The Annals of Mathematical Statistics***36**, 808–818.Wolfe, S. J. (1971). On moments of infinitely divisible distribution functions. *The Annals of Mathematical Statistics***42**, 2036–2043.Wolfe, S. J. (1973). On the local behavior of characteristic functions. *Annals of Probability***1**, 862–866.Wolfe, S. J. (1975a). On moments of probability distribution functions. In *Fractional Calculus and Its Applications*(B. Ross, ed.).*Lecture Notes in Mathematics***457**, 306–316. Berlin: Springer.Wolfe, S. J. (1975b). On derivatives of characteristic functions. *Annals of Probability***3**, 737–738.Wolfe, S. J. (1978). On the behavior of characteristic functions on the real line. *Annals of Probability***6**, 554–562.Zolotarev, V. M. (1986). *One-Dimensional Stable Distributions. Translations of Mathematical Monographs***65**. Providence, RI: Amer. Math. Soc.

Expected-posterior priors (EPPs) have been proved to be extremely useful for testing hypotheses on the regression coefficients of normal linear models. One of the advantages of using EPPs is that impropriety of baseline priors causes no indeterminacy in the computation of Bayes factors. However, in regression problems, they are based on one or more training samples, that could influence the resulting posterior distribution. On the other hand, the power-expected-posterior priors are minimally-informative priors that reduce the effect of training samples on the EPP approach, by combining ideas from the power-prior and unit-information-prior methodologies. In this paper, we prove the consistency of the Bayes factors when using the power-expected-posterior priors, with the independence Jeffreys as a baseline prior, for normal linear models, under very mild conditions on the design matrix.

Berger, J. O. and Pericchi, L. R. (1996). The intrinsic Bayes factor for linear models. In *Bayesian Statistics*(J. Bernardo, J. Berger, A. Dawid and A. Smith, eds.)**5**, 25–44. New York: Oxford Univ. Press.Casella, G., Girón, F. J., Martínez, M. L. and Moreno, E. (2009). Consistency of Bayesian procedures for variable selection. *The Annals of Statistics***37**, 1207–1228.Fouskakis, D., Ntzoufras, I. and Draper, D. (2015). Power-expected-posterior priors for variable selection in Gaussian linear models. *Bayesian Analysis***10**, 75–107.Good, I. J. (2004). *Probability and the Weighting of Evidence*. New York: Haffner.Ibrahim, J. G. and Chen, M. H. (2000). Power prior distributions for regression models. *Statistical Science***15**, 46–60.Iwaki, K. (1997). Posterior expected marginal likelihood for testing hypotheses. *Journal of Economics, Asia University***21**, 105–134.Kass, R. E. and Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. *Journal of the American Statistical Association***90**, 928–934.Liang, F., Paulo, R., Molina, G., Clyde, M. A. and Berger, J. O. (2008). Mixtures of *g*priors for Bayesian variable selection.*Journal of the American Statistical Association***103**, 410–423.O’Hagan, A. (1995). Fractional Bayes factors for model comparison. *Journal of the Royal Statistical Society, Series B***57**, 99–138.Pérez, J. M. and Berger, J. O. (2002). Expected-posterior prior distributions for model selection. *Biometrika***89**, 491–511.Schwarz, G. (1978). Estimating the dimension of a model. *The Annals of Statistics***6**, 461–464.Spiegelhalter, D. J., Abrams, K. R. and Myles, J. P. (2004). *Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Statistics in Practice*. Chichester: Wiley.Spiegelhalter, D. J. and Smith, A. (1988). Bayes factors for linear and log-linear models with vague prior information. *Journal of the Royal Statistical Society, Series B***44**, 377–387.Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-$t$ error terms. *Journal of the American Statistical Association***71**, 400–405.Zellner, A. and Siow, A. (1980). Posterior odds ratios for selected regression hypothesis (with discussion). In *Bayesian Statistics*(J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.)**1**, 585–606 & 618–647 (discussion). Oxford: Oxford Univ. Press.

In this paper, we first establish an existence and uniqueness result of $L^{p}$ ($p>1$) solutions for multidimensional backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a certain one-sided Osgood condition with a general growth in $y$ as well as a uniform continuity condition in $z$, and the $i$th component ${}^{i}g$ of $g$ depends only on the $i$th row ${}^{i}z$ of matrix $z$ besides $(\omega,t,y)$. Then we put forward and prove a stability theorem for $L^{p}$ solutions of this kind of multidimensional BSDEs. This generalizes some known results.

Bahlali, K., Essaky, E. and Hassani, M. (2010). Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs. *Comptes Rendus de l’Académie des Sciences—Series I***348**, 677–682.Briand, P., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L. (2003). $L^{p}$ solutions of backward stochastic differential equations. *Stochastic Processes and Their Applications***108**, 109–129.Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. *Probability Theory and Related Fields***141**, 543–567.Briand, P., Lepeltier, J.-P. and San Martin, J. (2007). One-dimensional BSDEs whose coefficient is monotonic in $y$ and non-Lipschitz in $z$. *Bernoulli***13**, 80–91.Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. *Probability Theory and Related Fields***150**, 145–192.El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. *Mathematical Finance***7**, 1–72.Fan, S. (2014). $L^{p}$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators. Available at arXiv:1403.5005v1 [math.PR]. Fan, S. and Jiang, L. (2013a). $L^{p}$ solutions to multidimensional backward stochastic differential equations with uniformly continuous generators. *Chinese Annals of Mathematics, Series A***34A**, 761–770 (in Chinese).Fan, S. and Jiang, L. (2013b). Multidimensional BSDEs with weak monotonicity and general growth generators. *Acta Mathematica Sinica. English Series***23**, 1885–1906.Fan, S., Jiang, L. and Davison, M. (2010). Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generator. *Comptes Rendus de l’Académie des Sciences—Series I***348**, 683–686.Fan, S., Jiang, L. and Davison, M. (2013). Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type. *Frontiers of Mathematics in China***8**, 811–824.Hamadène, S. (1996). Équations différentielles stochastiques rétrogrades: Le cas localement lipschitzien. *Annales de L’Institut Henri Poincaré***32**, 645–659.Hamadène, S. (2003). Multidimensional backward stochastic differential equations with uniformly continuous coefficients. *Bernoulli***9**, 517–534.Hu, Y. and Tang, S. (2014). Multidimensional backward stochastic differential equations of diagonally quadratic generators. Available at arXiv:1408.4579v1 [math.PR]. Jia, G. (2010). Backward stochastic differential equations with a uniformly continuous generator and related $g$-expectation. *Stochastic Processes and Their Applications***120**, 2241–2257.Kobylanski, M. (2000). Backward stochastic differential equations and partial equations with quadratic growth. *Annals of Probability***28**, 259–276.Lepeltier, J.-P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient. *Statistics and Probability Letters***32**, 425–430.Ma, M., Fan, S. and Song, X. (2013). $L^{p}$ ($p>1$) solutions of backward stochastic differential equations with monotonic and uniformly continuous generators. *Bulletin des Sciences Mathématiques***137**, 97–106.Mao, X. (1995). Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficient. *Stochastic Processes and Their Applications***58**, 281–292.Pardoux, E. (1999). BSDEs, weak convergence and homogenization of semilinear PDEs. In *Nonlinear Analysis, Differential Equations and Control*(*Montreal, QC*,*1998*), 503–549. Dordrecht: Kluwer Academic Publishers.Pardoux, E. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. *Systems and Control Letters***14**, 55–61.Peng, S. (1997). Backward SDE and related g-expectation. In *Backward Stochastic Differential Equations*(N. El Karoui and L. Mazliak, eds.).*Pitman Research Notes Mathematical Series***364**, 141–159. Harlow: Longman.Richou, A. (2012). Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition. *Stochastic Processes and Their Applications***122**, 3173–3208.Xing, H. (2012). On backward stochastic differential equations and strict local martingales. *Stochastic Processes and Their Applications***122**, 2265–2291.Yamada, T. (1981). On the successive approximation of solutions of stochastic differential equations. *Journal of Mathematics of Kyoto University***21**, 501–515.