NON-CENTRAL MOMENTS OF THE TRUNCATED NORMAL VARIABLE IN FINANCE

This paper computes the Non-central Moments of the Truncated Normal variable, i.e. a Normal constrained to assume values in the interval with bounds that may be finite or infinite. We define two recursive expressions where one can be expressed in closed form. Another closed form is defined using the Lower Incomplete Gamma Function. Moreover, an upper bound for the absolute value of the Non-central Moments is determined. The numerical results of the expressions are compared and the different behavior for high value of the order of the moments is shown. The limitations to the use of Truncated Normal distributions with a lower negative limit regarding financial products are considered. Limitations in the application of Truncated Normal distributions also arise when considering a CRRA utility function.

Stochastic aerodynamic analysis for compressor blades with manufacturing variability based on a mathematical dimensionality reduction method

It is essential for engineering manufacture and robust design to evaluate the impact of manufacturing variability on the aerodynamics of compressor blades efficiently and accurately. In the paper, a novel quadratic curve approximation method based on the scanning points of blade design profiles was introduced and combined with Karhunen–Loève expansion, a mathematical dimensionality reduction method for modeling manufacturing variability as truncated Normal process was proposed. Subsequently, Sparse Approximation of Moment-based Arbitrary Polynomial Chaos (SAMBA PC) and computational fluid dynamics (CFD) were applied to build a computational framework for stochastic aerodynamic analysis considering manufacturing variability. Finally, the framework was adopted to evaluate the aerodynamic variations of a high subsonic compressor cascade under the design incidence. The results illustrate that the SAMBA PC method is more efficient than the traditional methods such as Monte Carlo simulation (MCS) for stochastic aerodynamic analysis. Through uncertainty quantification, the impact of manufacturing variability on the global aerodynamic performance is primarily reflected in the fluctuation of aerodynamic losses, and the fluctuation of the total losses is mainly contributed by the fluctuation of the separation loss after the suction peak (a negative pressure spike near the leading edge (LE)) and the boundary-layer loss on the suction surface (SS). With sensitivity analysis, the most important geometric modes to aerodynamics can be revealed, which provides a useful reference for manufacturing inspection process and helps reduce computational cost in robust design.

Reasoning Method Based on Intervals with Symmetric Truncated Normal Density

Error parameters are inevitable in systems. In formal verification, previous reasoning methods seldom considered the probability information of errors. In this article, errors are described as symmetric truncated normal intervals consisting of the intervals and symmetric truncated normal probability density. Furthermore, we also rigorously prove lemmas and a theorem to partially simplify the calculation process of truncated normal intervals and independently verify the formulas of variance and expectation of symmetric truncated interval given by some scholars. The mathematical derivation process or verification codes are provided for most of the key formulas in this article. Hence, we propose a new reasoning method that combines the probability information of errors with the previous statistical reasoning methods. Finally, an engineering example of the reasoning verification of train acceleration is provided. After simulating the large-scale cases, it is shown that the simulation results are consistent with the theoretical reasoning results. This method needs more calculation, while it is more effective in detecting non-error’s fault factors than other error reasoning methods.

Impact of trade liberalization on technical efficiency of mining sector: A case of selected SADC countries

Productive inefficiency and lagging technology progress are major reasons behind the Southern Africa Development Community’s (SADC) continued exportation of unprocessed minerals to the world markets. The study seeks to uncover the impact of trade openness on the technical efficiency of the mining sector in selected SADC countries (Botswana, DRC, Namibia, South Africa, Zambia, and Zimbabwe). Technical efficiency is the ability of any production process to produce maximum output from minimum quantities of inputs. A Cobb Douglas Stochastic Frontier Approach in a single-stage maximum likelihood estimation of Green’s true fixed effects was used to compute technical efficiency (scores) and the technological progress in the mining sector of SADC. Results indicate that there is no technical efficiency gains from trade liberalization during the period under study together with positive and significant technological progress. A coefficient of 0.72 suggests that a 1% increase in trade openness increases technical inefficiency in the mining sector by 0.72%. The parameter coefficient from the truncated normal distribution of the true fixed effects model indicated that technological progress from one year to the next year would lead to a 2.6% increase in the output index of the mining. Technological progress in the mining sector should target upstream mineral value chains instead of only upgrading technology in one dimension of extraction. In addition, countries should collectively and gradually put across laws that force new investments in the extraction of minerals to erect processing plants in mining value addition of host countries to re-direct economies into a growth path. AcknowledgmentsThe authors are grateful to the North West University (RSA) for financing this study.  

Probabilistic Forecast of Low Level Wind Shear of Gimpo, Gimhae, Incheon and Jeju International Airports Using Ensemble Model Output Statistics

Wind shear can occur at all flight levels; however, it is particularly dangerous at low levels, from the ground up to approximately 2000 feet. If this phenomenon can occur during the take-off and landing of an aircraft, it may interfere with the normal altitude change of the aircraft, causing delay and cancellation of the aircraft, as well as economic damage. In this paper, to estimate the probabilistic forecasts of low-level wind shear at Gimpo, Gimhae, Incheon and Jeju International Airports, an Ensemble Model Output Statistics (EMOS) model based on a left-truncated normal distribution with a cutoff zero was applied. Observations were obtained from Gimpo, Gimhae, Incheon and Jeju International Airports and 13 ensemble member forecasts generated from the Limited-Area Ensemble Prediction System (LENS), for the period December 2018 to February 2020. Prior to applying to EMOS models, statistical consistency was analyzed by using a rank histogram and kernel density estimation to identify the uniformity of ensembles with corresponding observations. Performances were evaluated by mean absolute error, continuous ranked probability score and probability integral transform. The results showed that probabilistic forecasts obtained from the EMOS model exhibited better prediction skills when compared to the raw ensembles.

A MATHEMATICAL APPROXIMATION TO LEFT-SIDED TRUNCATED NORMAL DISTRIBUTION BASED ON HART’S MODEL

Left-sided truncated distributions (LSTD) have been found in different situations in the industry. For example, the life distribution of used devices is left-sided truncated distribution. Moreover, if a lower specification exists without the upper specification limit, the product distribution is truncated from the left side. Left-sided truncated normal distributions (LSTND) is a special case where the original distribution is normal. LSTND characteristics, as well as cumulative densities and probabilities can be difficult to employ manually, with most practitioners relying largely on specialized (and expensive) software. In many cases, practitioners are against purchasing software, as they are often limited in the number of estimations. The paper will provide an accurate and straightforward approximation to the cumulative density of LSTND. Hart’s normal distribution is simplified and used as a foundation of this model. The maximum absolute error for the curve at different truncation points (i.e., ZL) over the definition range (i.e., [zL: ∞]) is as follows: 0.004303 for ZL=-4, 0.00432 for ZL=-3, 0.00449 for ZL=-2, 0.005727 for ZL=-1, and 0.0106 for ZL=0. Even the maximum errors are very ignorable in probability applications. Further, it is rare to find a truncation point of higher than -2 in the industry.

On the Generalized Power Transformation of Left Truncated Normal Distribution

In this study, we considered various transformation problems for a left-truncated normal distribution recently announced by several researchers and then possibly seek to establish a unified approach to such transformation problems for certain type of random variable and their associated probability density functions in the generalized setting. The results presented in this research, actually unify, improve and as well trivialized the results recently announced by these researchers in the literature, particularly for a random variable that follows a left-truncated normal distribution. Furthermore, we employed the concept of approximation theory to establish the existence of the optimal value y_max in the interval denoted by (σ_a,σ_b) ((σ_p,σ_q)) corresponding to the so-called interval of normality estimated by these authors in the literature using the Monte carol simulation method.

Deterministic and Probabilistic Analysis of Dasdhunga Soil Slope along Narayangarh-Mugling Road Section

Instability of slopes is usually governed by a combination of intrinsic and extrinsic factors. The inherent variability of parameters make the problem probabilistic rather than a deterministic one. This research deals with evaluation of stability of slopes with the calculation of the factor of safety of Dasdhunga soil slope along Narayangarh- Mugling road section under different rainfall conditions through the use of coupled finite element and limit equilibrium method in GeoStudio and the determination of probability of failure by sliding, modeled as infinite slopes by using Monte Carlo simulation in R-Studio. Mean, standard deviation, minimum and maximum values of the parameters like- friction angle, cohesion and unit weight were computed from eight samples of the slope. The pore water pressure developed and its corresponding statistical data for different rainfall conditions were computed from FEM based SEEP/W simulation. The above parameters are assumed to follow truncated normal probability distribution function and the geometric parameters like height and slope angle are regarded as constant parameters. It was observed that the safety factors for theslopeis low in high intensity-low duration rainfalls and the probability of failure is high. The tendency to fail increases as the return period of rainfall increases and viceversa. Sensitivity analysis performed in both deterministic and probabilistic methods showed that friction angle is the most sensitive.