Saddlepoint Approximation Technique Research Articles

The weighted log-rank tests based on stratified clustered survival data: saddle-point p-values and confidence intervals

ABSTRACT Clinical studies sometimes provide clustered data with censored failure times. A crucial factor of the randomized design that lessens selection bias is the random allocation rule. Given this, the weighted rank tests’ p-values for stratified survival clustered sampling based on the random allocation rule are approximated using the double saddle-point approximation technique. For tests of significance and confidence intervals for the treatment effect, this approximation can be utilized. Through simulation experiments, the accuracy of the saddle-point approximation is examined by comparing saddle-point and normal approximations to the exact underlying permutation distribution.

Weighted log-rank tests for left-truncated data: Saddlepoint p-values and confidence intervals

Left truncation is a feature of many clinical trials and survival analysis in which lifetimes are observed conditionally on being greater than random truncation times. The subjects of such studies are exposed to an event prior to the event of interest and both times are recorded for each subject. In other words, left-truncated data arise when we only observe these subjects whose failure event times are limited below by truncation event times. For such type of data, the most commonly used class of two-sample tests is the weighted log-rank class. The aim of this article is to use saddlepoint approximation techniques to calculate p-values from the null permutation distributions of tests from the weighted log-rank class. The saddlepoint p-values are almost closer to the exact mid-p-values. The weighted log-rank tests are inverted to determine nominal 95% confidence intervals for the treatment effect in the presence of left-truncated data. Such analytical inversions lead to permutation confidence intervals which are easily computed and virtually identical with the exact intervals that would usually require huge amounts of simulations.

Log-rank tests for censored clustered data under generalized randomized block design: Saddlepoint approximation

ABSTRACT The weighted log-rank class is the common and widely used class of two-sample tests for clustered data. Clustered data with censored failure times often arise in tumorigenicity investigations and clinical trials. The randomized block design is a significant design that reduces both unintentional bias and selection bias. Accordingly, the p-values of the null permutation distribution of weighted log-rank class for clustered data are approximated using the double saddlepoint approximation technique. Comprehensive simulation studies are carried out to appraise the accuracy of the saddlepoint approximation. This approximation exhibits a significant improvement in precision over the asymptotic approximation. This precision motivates us to determine the approximated confidence intervals for the treatment impact.

On reliability analysis method through rotational sparse grid nodes

This study aims to develop a rotational sparse grid (R-SPGR) method for statistical moment evaluation and structural reliability analysis with enhanced accuracy and efficiency. The optimal rotational angle in the proposed R-SPGR method is determined by an optimization approach, in which the objective function is constructed in terms of discrepancy between the marginal moments of each input uncertain variables calculated from these rotational sparse grid nodes and their exact values. The R-SPGR nodes allow capturing more information of exact probability distribution for the system response, which is considered critically important to the calculation of statistical moments accurately. Following this, the probability density function (PDF) and failure probability of system response can be efficiently determined by using the saddlepoint approximation technique. To demonstrate the effectiveness of the proposed method, four benchmark examples, one structural analysis example and one practical example of industrial robot are provided here, in which the results obtained from the proposed R-SPGR method are compared with those calculated from the conventional sparse grid (SPGR) method, the maximum entropy problem with fractional moments (ME-FM) method and Monte Carlo simulation (MCS) method. From these illustrative examples involving a wide range of complexity, it is demonstrated that the proposed R-SPGR method has fairly high accuracy and efficiency for structural reliability analysis.

An improved high order moment-based saddlepoint approximation method for reliability analysis

Moment-based methods use only statistical moments of random variables for reliability analysis. The cumulative distribution function (CDF) or probability density function (PDF) of a performance function can be constructed from the perspective of the first few statistical moments, and the failure probability can be evaluated accordingly. However, existing moment-based methods may lead to large errors or instability. As such, the present paper focuses on the high order moment method for higher accuracy of reliability estimation by combining the common saddlepoint approximation technique, and an improved high order moment-based saddlepoint approximation (SPA) method for reliability analysis is presented. The approximated cumulant generating function (CGF) and the CDF of the performance function in terms of its first four statistical-moments are constructed. The developed method can be used for reliability evaluation of uncertain structures follow any types of distribution. Several numerical examples are given to demonstrate the efficacy and accuracy of the proposed method. Comparisons of the new method and several existing high order moment methods are also made on the reliability assessment.

Heat Out of Equilibrium Driven by Potential Pulling Beyond the Overdamped Limit

We investigate heat accumulated for a long period time in nonequilibrium motion of a colloid under a potential with center pulled by a constant velocity. Heat and work have the same average value in steady state, but the distribution functions for thermal fluctuations are different. Difference comes from everlasting memory of initial high energy on total amount of accumulated heat. As a result, the positive tail of the heat distribution exhibits a distinct correction to the large deviation function, compared to a usual logarithmic correction. Mathematically, difference originates from the singularity in the generating function, the Fourier transform, of the distribution function and makes usual saddle point approximation not valid. We use the modified saddle point approximation technique to compute the heat distribution function. We find the heat distribution for the underdamped case and find the correction to large deviation function more predominant than for the overdamped case.

A method to determine kinematic accuracy reliability of gear mechanisms with truncated random variables

Gears are widely used as transmission components in various types of precision machinery, such as machine tools, vehicles and robots. The performance of these mechanical systems is directly related to the kinematic accuracy of the gear mechanism. This paper presents a novel method to evaluate the reliability of the kinematic accuracy of gear mechanisms with truncated random variables. The influence of errors originating from factors such as the gear eccentricity, profile deviation and assembly error on the transmission error are systematically analyzed and discussed. The statistical moments of the transmission error were obtained combined with the international standards for gear accuracy, tolerances, deviations and fits. A fit of the probability density function and cumulative distribution function of the transmission error was obtained using the saddlepoint approximation technique. A model was established to enable reliability analysis for gear mechanisms. Finally, a discussion of the practical applications along with a detailed analysis to evaluate the efficiency of the proposed method is presented.

An efficient third-moment saddlepoint approximation for probabilistic uncertainty analysis and reliability evaluation of structures

This paper presents an efficient third-moment saddlepoint approximation approach for probabilistic uncertainty analysis and reliability evaluation of random structures. By constructing a concise cumulant generating function (CGF) for the state variable according to its first three statistical moments, approximate probability density function and cumulative distribution function of the state variable, which may possess any types of distribution, are obtained analytically by using saddlepoint approximation technique. A convenient generalized procedure for structural reliability analysis is then presented. In the procedure, the simplicity of general moment matching method and the accuracy of saddlepoint approximation technique are integrated effectively. The main difference of the presented method from existing moment methods is that the presented method may provide more detailed information about the distribution of the state variable. The main difference of the presented method from existing saddlepoint approximation techniques is that it does not strictly require the existence of the CGFs of input random variables. With the advantages, the presented method is more convenient and can be used for reliability evaluation of uncertain structures where the concrete probability distributions of input random variables are known or unknown. It is illustrated and examined by five representative examples that the presented method is effective and feasible.

Saddlepoint Approximation Methods for Pricing Derivatives on Discrete Realized Variance

We consider the saddlepoint approximation methods for pricing derivatives whose payoffs depend on the discrete realized variance of the underlying price process of a risky asset. Most of the earlier pricing models of variance products and volatility derivatives use the quadratic variation approximation as the continuous limit of the discrete realized variance. However, the corresponding discretization gap may become significant for short-maturity derivatives. Under Levy models and stochastic volatility models with jumps, we manage to obtain the saddlepoint approximation formulas for pricing variance products and volatility derivatives using the small time asymptotic approximation of the Laplace transform of the discrete realized variance. As an alternative approach, we also develop the conditional saddlepoint approximation method based on a given simulated stochastic variance path via Monte Carlo simulation. This analytic-simulation approach reduces the dimensionality of the simulation of the discrete variance derivatives; and in some cases, the simulation procedure of the realized variance can be effectively performed using an appropriate exact simulation method. We examine numerical accuracy and reliability of various types of the saddlepoint approximation techniques when applied to pricing derivatives on discrete realized variance under different types of asset price processes. The limitations of the saddlepoint approximation methods in pricing variance products and volatility derivatives are also discussed.

Saddlepoint Approximation Methods for Pricing Derivatives on Discrete Realized Variance

We consider the saddlepoint approximation methods for pricing derivatives whose payoffs depend on the discrete realized variance of the underlying price process of a risky asset. Most of the earlier pricing models of variance products and volatility derivatives use the quadratic variation approximation as the continuous limit of the discrete realized variance. However, the corresponding discretization error may become significant for short-maturity derivatives. Under Lévy models and stochastic volatility models with jumps, we manage to obtain the saddlepoint approximation formulas for pricing variance products and volatility derivatives using the small time asymptotic approximation of the Laplace transform of the discrete realized variance. As an alternative approach, we also develop the conditional saddlepoint approximation method based on a given simulated stochastic variance path via Monte Carlo simulation. This analytic-simulation approach reduces the dimensionality of the simulation of the discrete variance derivatives; and in some cases, the simulation procedure of the realized variance can be effectively performed using an appropriate exact simulation method. We examine numerical accuracy and reliability of various types of the saddlepoint approximation techniques when applied to pricing derivatives on discrete realized variance under different types of asset price processes. The limitations of the saddlepoint approximation methods in pricing variance products and volatility derivatives are also discussed.

Saddlepoint Approximation to Cumulative Distribution Function for Poisson–Exponential Distribution

The random sum distribution is a key role in probability theory and its applications as well, these applications could be used in different sciences such as insurance system, biotechnology, allied health science, etc. The statistical significance in random sum distribution initiates when using the applications of probability theory in the real life, where the total quantity X can be only observed, which is included of an unknown random number X of random contributions. Saddlepoint approximation techniques overcome this problem. Saddlepoint approximations are effective tools in getting exact expressions for distribution functions that are not known in closed form. Saddlepoint approximations usually better than the other methods in which calculation costs, but not necessarily about accuracy. This paper introduces the saddlepoint approximations to the cumulative distribution function for random sum Poisson- Exponential distributions in continuous settings. We discuss approximations to random sum random variable with dependent components assuming existence of the moment generating function. A numerical example of continuous distributions from the Poisson- Exponential distribution is presented.

Reliability analysis using a 'Taylor expansion-expected value' saddlepoint approximation

The problem of reliability analysis of dynamical systems is considered. Failure event in the problem is formulated as the exceedance of a performance variable over a prescribed threshold level. Saddlepoint approximation technique provides a choice to estimate the Cumulative Distribution Function (CDF) of the performance variable. The failure probability is obtained as the value of the complement of the CDF at a specified threshold. The method requires finding the saddlepoint from a simple algebraic equation that depends on the Cumulant Generating Function (CGF) of the performance variable. Two different methods, which respectively use Taylor series expansion and statistical averaging, are investigated for estimating the CGF of the performance variable based on its random samples. A ten-storey shear building model subjected to white noise excitation is used to show the preference of a combination of these two methods in terms of accuracy and efficiency.

Improved performance evaluation for DC-coupled burst mode reception in the presence of amplified spontaneous emission noise and interchannel crosstalk

Performance evaluations, for bit error rate (BER) and power penalty, are performed for DC coupled burst mode (BM) receivers in the presence of both amplified spontaneous emission (ASE) noise and interchannel crosstalk. Such BM receivers acquire the threshold from preamble bits, so the threshold is a random variable, because of contamination by noise and crosstalk. Here such a crosstalk contaminated threshold is described using a moment generating function for the first time. This facilitates the first use of Chernoff bound, modified Chernoff bound and saddle-point approximation techniques for BM reception under these conditions. A Gaussian approximation (GA) is also calculated. Results for an optically preamplified receiver indicate that the overall penalty (threshold acquisition penalty in the presence of crosstalk (and ASE)) exceeds the straightforward summation of the separate crosstalk and threshold acquisition penalties (in dB) and while this difference is particularly pronounced for the GA it remains significant for the other techniques. Furthermore, the GA is seen to be more pessimistic in BER compared to the other approaches and less accurate (as it exceeds an upper bound). Relative to the other techniques, the penalty predicted by the GA is optimistic for low crosstalk but pessimistic for high crosstalk.

Conditional probability Markov chain simulation based reliability analysis method for nonnormal variables

Based on fast Markov chain simulation for generating the samples distributed in failure region and saddlepoint approximation (SA) technique, an efficient reliability analysis method is presented to evaluate the small failure probability of non-linear limit state function (LSF) with non-normal variables. In the presented method, the failure probability of the non-linear LSF is transformed into a product of the failure probability of the introduced linear LSF and a feature ratio factor. The introduced linear LSF which approximately has the same maximum likelihood points as the non-linear LSF is constructed and its failure probability can be calculated by SA technique. The feature ratio factor, which can be evaluated on the basis of multiplicative rule of probability, exhibits the relation between the failure probability of the non-linear LSF and that of the linear LSF, and it can be fast computed by utilizing the Markov chain algorithm to directly simulate the samples distributed in the failure regions of the non-linear LSF and those of the linear LSF. Moreover, the expectation and variance of the failure probability estimate are derived. The results of several examples demonstrate that the presented method has wide applicability, can be easily implemented, and possesses high precision and high efficiency.

Application of saddlepoint approximation in reliability analysis of dynamic systems

The application of the saddlepoint approximation to reliability analysis of dynamic systems is investigated. The failure event in reliability problems is formulated as the exceedance of a single performance variable over a prescribed threshold level. The saddlepoint approximation technique provides a choice to estimate the cumulative distribution function (CDF) of the performance variable. The failure probability is obtained as the value of the complement CDF at a specified threshold. The method requires computing the saddlepoint from a simple algebraic equation that depends on the cumulant generating function (CGF) of the performance variable. A method for calculating the saddlepoint using random samples of the performance variable is presented. The applicable region of the saddlepoint approximation is discussed in detail. A 10-story shear building model with white noise excitation illustrates the accuracy and efficiency of the proposed methodology.

Higher-order saddlepoint approximations in the Vasicek portfolio credit loss model

This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value-atrisk (VaR), the risk measure chosen in the Basel II Accord for the evaluation of capital requirement, can then be found by inverting the loss distribution. The VaR contribution (VaRC), expected shortfall (ES) and ES contribution (ESC) can all be calculated accurately. Saddlepoint approximation is well known to provide good approximations to very small tail probabilities, which makes it a very suitable technique in the context of portfolio credit loss. The portfolio credit model we employ is the Vasicek one-factor model, which has an analytical solution if the portfolio is well diversified. The Vasicek asymptotic formula fails, however, when the portfolio is dominated by a few loans much larger than the rest. We show that saddlepoint approximation is able to handle such exposure concentration. We also point out that the saddlepoint approximation technique can be readily applied to more general Bernoulli mixture models (possibly multi-factor). It can further handle portfolios with random loss given default (LGD).

Tail behavior of random products and stochastic exponentials

In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven by infinite variance α -stable Lévy motion. We show that the solution is regularly varying with index α . An important step in the proof is the study of a Poisson number of products of independent random variables with regularly varying tail. The study of these products merits its own interest because it involves interesting saddle-point approximation techniques.

Statistical analysis of coherent ultrashort light pulse CDMA with multiple optical amplifiers using additive noise model

This paper describes a study of the performance of various configurations for placing multiple optical amplifiers in a typical coherent ultrashort light pulse code-division multiple access (CULP-CDMA) communication system using the additive noise model. For this study, a comprehensive performance analysis was developed that takes into account multiple-access noise, noise due to optical amplifiers, and thermal noise using the saddle-point approximation technique. Prior to obtaining the overall system performance, the input/output statistical models for different elements of the system such as encoders/decoders, star coupler, and optical amplifiers were obtained. Performance comparisons between an ideal and lossless quantum-limited case and a typical CULP-CDMA with various losses exhibit more than 30 dB more power requirement to obtain the same bit-error rate (BER). Considering the saturation effect of optical amplifiers, this paper discusses an algorithm for amplifiers' gain setting in various stages of the network in order to overcome the nonlinear effects on signal modulation in optical amplifiers. Finally, using this algorithm, various configurations of multiple optical amplifiers in CULP-CDMA are discussed and the rules for the required optimum number of amplifiers are shown with their corresponding optimum locations to be implemented along the CULP-CDMA system.

A lower bound on compression of unknown alphabets

Many applications call for universal compression of strings over large, possibly infinite, alphabets. However, it has long been known that the resulting redundancy is infinite even for i.i.d. distributions. It was recently shown that the redundancy of the strings’ patterns, which abstract the values of the symbols, retaining only their relative precedence, is sublinear in the blocklength n, hence the per-symbol redundancy diminishes to zero. In this paper we show that pattern redundancy is at least ( 1.5 log 2 e ) n 1 / 3 bits. To do so, we construct a generating function whose coefficients lower bound the redundancy, and use Hayman's saddle-point approximation technique to determine the coefficients’ asymptotic behavior.

Explaining the Saddlepoint Approximation

Saddlepoint approximations are powerful tools for obtaining accurate expressions for densities and distribution functions. \Ve give an elementary motivation and explanation of saddlepoint approximation techniques, stressing the connection with the familiar Taylor series expansions and the Laplace approximation of integrals. Saddlepoint methods are applied to the convolution of simple densities and, using the Fourier inversion formula, the saddlepoint approximation to the density of a random variable is derived. \Ve then apply the method to densities of sample means of iid random variables, and also demonstrate the technique for approximating the density of a maximum likelihood estimator in exponential families.