Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.     

A hybrid approach based on saddlepoint and importance sampling methods for bootstrap tail probability estimation

We propose a simple hybrid method which makes use of both saddlepoint and importance sampling techniques to approximate the bootstrap tail probability of an M-estimator. The method does not rely on explicit formula of the Lugannani-Rice type, and is computationally more efficient than both uniform bootstrap sampling and importance resampling suggested in earlier literature. The method is also applied to construct confidence intervals for smooth functions of M-estimands.     

A Bayesian approach to Weibull survival models—Application to a cancer clinical trial

In this paper we outline a class of fully parametric proportional hazards models, in which the baseline hazard is assumed to be a power transform of the time scale, corresponding to assuming that survival times follow a Weibull distribution. Such a class of models allows for the possibility of time varying hazard rates, but assumes a constant hazard ratio. We outline how Bayesian inference proceeds for such a class of models using asymptotic approximations which require only the ability to maximize the joint log posterior density. We apply these models to a clinical trial to assess the efficacy of neutron therapy compared to conventional treatment for patients with tumors of the pelvic region. In this trial there was prior information about the log hazard ratio both in terms of elicited clinical beliefs and the results of previous studies. Finally, we consider a number of extensions to this class of models, in particular the use of alternative baseline functions, and the extension to multi-state data.     

Laplace approximations to means and variances with asymptotic modes

The moment-generating function method, which is proposed by Tierney et al. [1989a. Fully exponential Laplace approximations to expectations and variances of nonpositive functions. J. Amer. Statist. Assoc. 84, 710–716], is an asymptotic technique of approximating a posterior mean of a general function by approximating the moment-generating function (MGF), and then differentiating it. In this article, we give approximations to the posterior means and variances by combining the MGF method and the Laplace approximations with asymptotic modes. We prove that asymptotic errors of the approximate means and variances are of order n^-2$n^{-2}$ and of order n^-3$n^{-3}$, respectively. Our approximation is closely related to a standard-form approximation, and is given without evaluating the exact posterior mode and third derivatives of the log-likelihood function. The MGF method also improves numerical instability of the fully exponential Laplace approximation for a predictive mean in logistic regression.     

Saddlepoint approximations to sensitivities of tail probabilities of random sums and comparisons with Monte Carlo estimators

This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus, the suggested saddlepoint approximation can be conveniently used in various scientific problems.     

A bayesian approach to a reliability problem: Theory,analysis and interesting numerics

We consider approximate Bayesian inference about the quantity R = P[Y₂> Y₁] when both the random variables Y₁, Y₂ have expectations that depend on certain explanatory variables. Our interest centers on certain characteristics of the posterior of R under Jeffreys's prior, such as its mean, variance and percentiles. Since the posterior of R is not available in closed form, several approximation procedures are introduced, and their relative performance is assessed using two real datasets.     

An Elementary Introduction to Maximum Likelihood Estimation for Multinomial Models: Birch's Theorem and the Delta Method

A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second-year graduate course in categorical data analysis, can be based on Birch's theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch's theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao's (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.     

A discrete analogue of odd Weibull-G family of distributions: properties,classical and Bayesian estimation with applications to count data

In the statistical literature, several discrete distributions have been developed so far. However, in this progressive technological era, the data generated from different fields is getting complicated day by day, making it difficult to analyze this real data through the various discrete distributions available in the existing literature. In this context, we have proposed a new flexible family of discrete models named discrete odd Weibull-G (DOW-G) family. Its several impressive distributional characteristics are derived. A key feature of the proposed family is its failure rate function that can take a variety of shapes for distinct values of the unknown parameters, like decreasing, increasing, constant, J-, and bathtub-shaped. Furthermore, the presented family not only adequately captures the skewed and symmetric data sets, but it can also provide a better fit to equi-, over-, under-dispersed data. After producing the general class, two particular distributions of the DOW-G family are extensively studied. The parameters estimation of the proposed family, are explored by the method of maximum likelihood and Bayesian approach. A compact Monte Carlo simulation study is performed to assess the behavior of the estimation methods. Finally, we have explained the usefulness of the proposed family by using two different real data sets.     

A multilevel cross-classified modelling approach to peer review of grant proposals: the effects of assessor and researcher attributes on assessor ratings

Summary. The peer review of grant proposals is very important to academics from all disciplines. Although there is limited research on the reliability of assessments for grant proposals, previously reported single-rater reliabilities have been disappointingly low (between 0.17 and 0.37). We found that the single-rater reliability of the overall assessor rating for Australian Research Council grants was 0.21 for social science and humanities (2870 ratings, 1928 assessors and 687 proposals) and 0.19 for science (7153 ratings, 4295 assessors and 1644 proposals). We used a multilevel, cross-classification approach (level 1, assessor and proposal cross-classification; level 2, field of study), taking into account that 34% of the assessors evaluated more than one proposal. Researcher-nominated assessors (those chosen by the authors of the research proposal) gave higher ratings than panel-nominated assessors chosen by the Australian Research Council, and proposals from more prestigious universities received higher ratings. In the social sciences and humanities, the status of Australian universities had significantly more effect on Australian assessors than on overseas assessors. In science, ratings were higher when assessors rated fewer proposals and apparently had a more limited frame of reference for making such ratings and when researchers were professors rather than non-professors. Particularly, the methodology of this large scale study is applicable to other forms of peer review (publications, job interviews, awarding of prizes and election to prestigious societies) where peer review is employed as a selection process.