Normal Cumulative Distribution Function Research Articles

A Bayesian Beta-Mixture Model for Nonparametric IRT (BBM-IRT)

Item response models typically assume that the item characteristic (step) curves follow a logistic or normal cumulative distribution function, which are strictly monotone functions of person test ability. Such assumptions can be overly-restrictive for real item response data. We propose a simple and more flexible Bayesian nonparametric IRT model for dichotomous items, which constructs monotone item characteristic (step) curves by a finite mixture of beta distributions, which can support the entire space of monotone curves to any desired degree of accuracy. A simple adaptive random-walk Metropolis-Hastings algorithm is proposed to estimate the posterior distribution of the model parameters. The Bayesian IRT model is illustrated through the analysis of item response data from a 2015 TIMSS test of math performance.

On estimating efficiency effects in a stochastic frontier model

This paper proposes a technical efficiency effects model within the framework of stochastic production frontier. The efficiency effects are specified by a standard normal cumulative distribution function of exogenous variables which ensures the efficiency scores to lie in a unit interval. This specification eschews one-sided error term present in almost all the existing inefficiency effects models. The efficiency scores are obtained directly once the parameters of the model are estimated. An empirical exercise based on the widely used Philippines rice farming data illustrates the simplicity and usefulness of the proposed model.

Soil health assessment for coffee farms on andosols in Colombia

Developing local soil health (SH) benchmarks for different ecosystems is important for supporting locally appropriate management decisions and correct interpretation of soil health results. This study was conducted to develop SH scoring functions as benchmarks specific to coffee production in Cauca, Colombia. A total of 223 soil samples were collected from coffee farms in six municipalities and were analyzed for 13 SH indicators including wet aggregate stability (WAS), available water capacity (AWC), respiration rate, pH, contents of active carbon (AC), organic matter (OM), protein, phosphorus (P), potassium (K), magnesium (Mg), manganese (Mn), iron (Fe) and zinc (Zn). A scoring function for each indicator was developed using the cumulative normal distribution (CND) function with parameters based on either the average local conditions for a given indicator (physical and biological indicators), or thresholds found in the literature for coffee systems (chemical indicators). Separate scoring functions by textural group (fine, medium) were necessary for AWC, OM, AC, and respiration. A best subsets regression (BSR) using the overall soil health index as the response variable was executed to determine the indicators with highest predictive power of overall soil health. AC was the best single predictor of soil health, and AC combined with protein, P and pH offer additional predictability, suggesting them for a simplified SH test.

The development of the own-race advantage in school-age children: A morphing face paradigm.

Previous studies examining the other-race effect in school-age children mostly focused on recognition memory performance. Here we investigated perceptual discriminability for Asian-like versus Caucasian-like morph faces in school-age Taiwanese children and adults. One-hundred-and-two 5- to 12-year-old children and twenty-three adults performed a sequential same/different face matching task, where they viewed an Asian- or a Caucasian-parent face followed by either the same parent face or a different morphed face (containing 15%, 30%, 45%, or 60% contribution from the other parent face) and judged if the two faces looked the same. We computed the d’ as the sensitivity index for each age groups. We also analyzed the group mean rejection rates as a function of the morph level and fitted with a cumulative normal distribution function. Results showed that the adults and the oldest 11-12-year-old children exhibited a greater sensitivity (d’) and a smaller discrimination threshold (μ) in the Asian-parent condition than those in the Caucasian-parent condition, indicating the presence of an own-race advantage. On the contrary, 5- to 10-year-old children showed an equal sensitivity and similar discrimination thresholds for both conditions, indicating an absence of the own-race advantage. Moreover, a gradual development in enhancing the discriminability for the Asian-parent condition was observed from age 5 to 12; however, the progression in the Caucasian-parent condition was less apparent. In sum, our findings suggest that expertise in face processing may take the entire childhood to develop, and supports the perceptual learning view of the other-race effect—the own-race advantage seen in adulthood likely reflects a result of prolonged learning specific to faces most commonly seen in one’s visual environment such as own-race faces.

New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function

In this paper, we develop a new matrix-based implementation of the Mendell and Elston (ME) analytic approximation to evaluate the multivariate normal cumulative distribution (MVNCD) function, using an LDLT decomposition method followed by a rank 1 update of the LDLT factorization. Our implementation is easy to code for individuals familiar with matrix-based coding. Further, our new matrix-based implementation for the ME algorithm allows us to efficiently write the analytic matrix-based gradients of the approximated MVNCD function with respect to the abscissae and correlation parameters, an issue that is important in econometric model estimation. In addition, we propose four new analytic methods for approximating the MVNCD function. The paper then evaluates the ability of the multiple approximations for individual MVNCD evaluations as well as multinomial probit model estimation. As expected, in our tests for evaluating individual MVNCD functions, we found that the traditional GHK approach degrades rapidly as the dimensionality of integration increases. Concomitant with this degradation in accuracy is a rapid increase in computational time. The analytic approximation methods are also much more stable across different numbers of dimensions of integration, and even the simplest of these methods is superior to the GHK-500 beyond seven dimensions of integration. Based on all the evaluation results in this paper, we recommend the new Two-Variate Bivariate Screening (TVBS) method proposed in this paper as the evaluation approach for MVNCD function evaluation.

A threshold stresses-based permeability variation model for microcracked porous rocks

The dependence of permeability on stress is an important property of rocks related to their microstructural alterations. Justified by the distinct regions on the stress–strain curves under mechanical loading, this study proposed a threshold stresses-based permeability variation model for microcracked porous rocks by separating voids of rocks into a more-rounded pore system and a flat crack system. In the proposed model, the permeability reduction caused by progressive closure of pre-existing cracks is characterised by a negative exponential function of the effective mean stress, while the enhancement of permeability induced by the initiation and growth of new cracks is correlated to the cumulative acoustic emission counts described with a normal cumulative distribution function. The permeability induced by the elastic compaction of the less-compressible pores is assumed to decrease linearly with effective mean stress, typically with a rather small slope, which lower bounds the permeability of rocks when the pre-existing cracks are sufficiently closed while new cracks are not yet initiated. The proposed model has advantages of clear physical meaning, easy laboratory parameterisation and low computational cost and reproduces well the main features of permeability variation observed in the laboratory for sedimentary rocks under hydrostatic loading and granitic rocks under triaxial loading.

Modelling Efficiency Effects in a True Fixed Effects Stochastic Frontier

This paper proposes a stochastic frontier panel data model which includes time-invariant unobserved heterogeneity along with the efficiency effects. Following Paul and Shankar (2018), the efficiency effects are specified by a standard normal cumulative distribution function of exogenous variables which ensures the efficiency scores to lie in a unit interval. This specification eschews one-sided error term present in almost all the existing inefficiency effects models. The model parameters can be estimated by non-linear least squares after removing the individual effects by the usual within transformation or using non-linear least squares dummy variables (NLLSDV) estimator. The efficiency scores are directly calculated once the model is estimated. An empirical illustration based on widely used panel data on Indian farmers is presented.

A sharp Pólya-based approximation to the normal cumulative distribution function

We study an expansion of the cumulative distribution function of the standard normal random variable that results in a family of closed form approximations that converge at 0. One member of the family that has only five explicit constants offers the absolute error of 5.79·10−6 across the entire range of real numbers. With its simple form and applicability for all real numbers, our approximation surpasses either in computational efficiency or in relative error, and most often in both, other approximation formulas based on numerical algorithms or ad-hoc approximations. An extensive overview and classification of the existing approximations from the literature is included.

Computation of the quadrivariate and pentavariate normal cumulative distribution functions

ABSTRACTThis article provides explicit integration rules for the quadrivariate and the pentavariate normal distribution. By analytically reducing the dimension of the problem and simplifying the functions to be integrated, these rules form the basis for a numerical evaluation scheme yielding an observed maximum error in the order of 10− 7 and a computational time of less than 10− 6 s. The implementation is very straightforward as it is based on a classical Gauss–Legendre quadrature. Order statistics are also dealt with.

Asymptotics for the time of ruin in the war of attrition

AbstractWe consider two players, starting withmandnunits, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probabilityp(m,n) that the first player wins. Whenm~Nx0,n~Ny0, we prove the fluid limit asN→ ∞. Whenx0=y0,z→p(N,N+z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τNis established as (T- τN) ~N-βW1/β, β = ¼,T=x0+y0. Modulo a constant,W~ χ21(z02/T2).

System reliability analysis of tunnels reinforced by rockbolts

System reliability analysis of tunnelling problems is illustrated by using an iterative closed-form formulation for a circular tunnel reinforced by end-anchored rockbolts. Three performance functions, namely, the mobilized tensile force in the rockbolt, the maximum displacement and the plastic zone size of the tunnel, are considered. The bimodal bounds method and the multivariate normal cumulative distribution function (mvncdf) method are applied to the solution. Results show that the system probability of failure (Pf) estimated from the mvncdf method lies within the lower bound and upper bound set by the bimodal bounds method for the cases studied. The second-order reliability method (SORM) is used to refine the reliability indices and to improve the accuracy of the estimated system Pf. The influence of the correlation coefficient between the cohesion and the friction angle and the influence of the rockbolt installation position on the system Pf are discussed. A modified hybrid approach using the linear response surface method (RSM) to locate the design point and the artificial neural network (ANN) to approximate the actual limit state surface (LSS) is suggested to be used in the system reliability analysis for problems where closed-form solutions are not available. Traditional second-order RSM without cross terms may be inaccurate and may suffer from the problem of low convergence rate and the second-order RSM with cross terms may encounter the “false branch” problem. It is also shown in this paper that the “false branch” problem can be overcome by the importance sampling technique. Comparison with the second-order RSM shows that the proposed approach is efficient, accurate and robust for the system reliability analysis.

A New Proof for Winitzki's Approximation of Normal Cumulative Distribution Function

Among all statistical distributions, standard normal distribution has been the most important and practical distribution in which calculation of area under probability density function and cumulative distribution function are required. Unfortunately, the cumulative distribution function of this is, in general, expressed as a definite integral with no closed form or analytical solution. Consequently, it has to be approximated. In this paper, attempts have been made for Winitzki's approximation to be proved by a new approach. Then, the approximation is improved with some modifications and shown that the maximum error resulted from this is less than 0.0000584. Finally, an inverse function for computation of normal distribution quantiles has been derived.

An Alternative Specification for Technical Efficiency Effects in a Stochastic Frontier Production Function

This paper proposes an alternative specification for technical efficiency effects in a stochastic production frontier model. The proposed specification is distribution free and thus eschews one-sided error term present in almost all the existing inefficiency effects models. The efficiency effects are represented by the standard normal cumulative distribution function of exogenous variables which ensures the efficiency scores to lie in a unit interval. An empirical exercise based on widely used Philippines rice farming data set illustrates the simplicity and usefulness of the proposed model.

Statistical Analysis of Similarity Measures for Solving Cell Formation Problems

Hierarchical clustering (HC) has been used to solve cell formation (CF) problem, which has advantages of solution convergence and not requiring the number of cells as the input. While evolutionary algorithms (EA) are more common in recent years due to their capability in solution searching, they also imply more time and complexity involved in problem solving. At the same time, some studies have shown that HC can give comparable results if the CF structures are not very “messy”. The term messiness is used to indicate the potential of a given CF problem to yield nearly independent machine groups and part families. Then, the purpose of this research is to distinguish whether a CF problem is “messy” or not so that the users can decide whether using economical HC or relatively demanding EA for problem solving. In the methodology, the similarity values of machine pairs are first determined for a given incidence matrix for CF. Then, histogram is used to report the distribution of similarity values. By analyzing this histogram, it is found that the distribution of similarity values of a less messy problem often presents a U-shape. To further quantify this observation, the Kolmogorov–Smirnov test (K-S test) is applied to examine how well the distribution of similarity values fits the normal cumulative distribution function (CDF). The result indicates that the similarity values of messy CF problems tend to demonstrate normal distribution in view of the K-S test.

Theoretical estimation of the probability of weight rank reversal in pairwise comparisons

Pairwise comparison is a key component in multi-criteria decision making. The probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome. In the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons. We provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function. The model is applied to two alternative weight extraction methods frequently used in the literature: the geometric mean and the eigenvalue method. We introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross-correlation of the average weights which are required in the application of the model. The theoretical results are compared against numerical simulations and a very good agreement is observed. We further show how our model can be extended in applications of a full multi-criteria decision making analysis, such as the analytic hierarchy process. We also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty-induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts. The methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome.

One-term Approximation for Normal Distribution Function

This paper presents a one-term approximation to the cumulative normal distribution functions. The absolute maximum error of the proposed approximation is 0.0018 less than 0.003 of Polya's approximation. Comparisons between the proposed approximation and the different approximations with one-term that stated in the literature are given.

Mapping conduction velocity of early embryonic hearts with a robust fitting algorithm.

Cardiac conduction maturation is an important and integral component of heart development. Optical mapping with voltage-sensitive dyes allows sensitive measurements of electrophysiological signals over the entire heart. However, accurate measurements of conduction velocity during early cardiac development is typically hindered by low signal-to-noise ratio (SNR) measurements of action potentials. Here, we present a novel image processing approach based on least squares optimizations, which enables high-resolution, low-noise conduction velocity mapping of smaller tubular hearts. First, the action potential trace measured at each pixel is fit to a curve consisting of two cumulative normal distribution functions. Then, the activation time at each pixel is determined based on the fit, and the spatial gradient of activation time is determined with a two-dimensional (2D) linear fit over a square-shaped window. The size of the window is adaptively enlarged until the gradients can be determined within a preset precision. Finally, the conduction velocity is calculated based on the activation time gradient, and further corrected for three-dimensional (3D) geometry that can be obtained by optical coherence tomography (OCT). We validated the approach using published activation potential traces based on computer simulations. We further validated the method by adding artificially generated noise to the signal to simulate various SNR conditions using a curved simulated image (digital phantom) that resembles a tubular heart. This method proved to be robust, even at very low SNR conditions (SNR = 2-5). We also established an empirical equation to estimate the maximum conduction velocity that can be accurately measured under different conditions (e.g. sampling rate, SNR, and pixel size). Finally, we demonstrated high-resolution conduction velocity maps of the quail embryonic heart at a looping stage of development.

Damage to Oil Storage Tanks from the 2011 Mw 9.0 Tohoku-Oki Tsunami

The Mw 9.0 2011 Tohoku-oki tsunami damaged 418 oil storage tanks and moved 157 of them. Using data on the severity of damage and the maximum inundation depth of the tsunami, a fragility curve representing the damage to oil storage tank plumbing is presented in this paper: P( η)= Φ ((ln η − 1.02)/0.574), where P is the damage rate, η is the maximum inundation depth in meters, and Φ is the standard normal cumulative distribution function. The existing method of predicting the movement of tanks exposed to a tsunami is validated by comparing the predicted damage with actual damage data from the 2011 tsunami. The accuracy (hit rate) is 76%.

A Taylor series approach to pricing and implied volatility for local–stochastic volatility models

Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local–stochastic volatility setting. Our price approximations require only a normal cumulative distribution function and our implied volatility approximations are fully explicit (ie, they require no special functions, no infinite series and no numerical integration). As such, approximate prices can be computed as efficiently as Black– Scholes prices, and approximate implied volatilities can be computed nearly instantaneously.

On a Logistic Approximation to the Normal and Other Cumulative Distribution Functions

This paper proposes an approach that maintains the simplicity of variousapproximations as well as having a closed- form algebraic inverse yet easily generalizable to the approximation of other cumulative distribution functions. The proposed logistic approximation to the normal cumulative distribution function has a maximum error of 0.000560, lower than the maximum error reported by Aludaat et al (2007). Furthermore, the logistic approximation is more flexible in that it can be used to approximate the cumulative distribution functions of other distribution. A Chi-square approximation was also investigated and reported a maximum error of 0.00494.