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Choice under risk has a large stochastic (unpredictable) component. This chapter examines five stochastic models for binary discrete choice under risk and how they combine with “structural” theories of choice under risk. Stochastic models are substantive theoretical hypotheses that are frequently testable in and of themselves, and also identifying restrictions for hypothesis tests, estimation and prediction. Econometric comparisons suggest that for the purpose of prediction (as opposed to explanation), choices of stochastic models may be far more consequential than choices of structures such as expected utility or rank-dependent utility.

In this chapter, we utilize the residual concept of productivity measures defined in the context of normal-gamma stochastic frontier production model with heterogeneity to differentiate productivity and inefficiency measures. In particular, three alternative two-way random effects panel estimators of normal-gamma stochastic frontier model are proposed using simulated maximum likelihood estimation techniques. For the three alternative panel estimators, we use a generalized least squares procedure involving the estimation of variance components in the first stage and estimated variance–covariance matrix to transform the data. Empirical estimates indicate difference in the parameter coefficients of gamma distribution, production function, and heterogeneity function variables between pooled and the two alternative panel estimators. The difference between pooled and panel model suggests the need to account for spatial, temporal, and within residual variations as in Swamy–Arora estimator, and within residual variation in Amemiya estimator with panel framework. Finally, results from this study indicate that short- and long-run variations in financial exposure (solvency, liquidity, and efficiency) play an important role in explaining the variance of inefficiency and productivity.

The purpose of this study is to propose a novel super-efficiency DEA model to appraise the relative efficiency of DMUs with zero data and stochastic data. Our model can work with both variable returns to scale (VRS) and constant returns to scale (CRS).

This study proposes a new stochastic super-efficiency DEA (SSDEA) model to assess the performance of airlines with stochastic and zero inputs and outputs.

This paper proposes a new analysis and contribution to the knowledge of efficiency assessment with stochastic super-efficiency DEA model by (1) using input saving and output surplus index for efficient DMUs to get the optimal solution; (2) obtaining efficiency scores from the proposed model that are equivalent to original stochastic super-efficiency model when feasible solutions exist. A case study is given to illustrate the applicability of our proposed model. Also, poor performance reasons are identified to improve the performance of inefficient airlines.

For the first time, a new SSDEA model for ranking DMUs is proposed. The introduced model produces a feasible solution when dealing with zero input or output. This paper applies the input saving and output surplus concept to rectify the infeasibility problem in the stochastic DEA model.

The purpose of this paper is to develop a stochastic detailed schedule for a preventive/scheduled/periodic maintenance program of a military aircraft, specifically a rotorcraft or helicopter.

The new model, entitled the military “periodic aviation maintenance stochastic schedule” (PAM-SS), develops a stochastic detailed schedule for a PUMA SA 330SM helicopter for the 50-h periodic inspection, using cyclic operation network (CYCLONE) and Monte Carlo simulation (MCS) techniques. The PAM-SS model identifies the different periodic inspection tasks of the maintenance schedule, allocates the resources required for each task, evaluates a stochastic duration of each inspection task, evaluates the probability of occurrence for each breakdown or repair, develops the CYCLONE model of the stochastic schedule and simulates the model using MCS.

The 50-h maintenance stochastic duration follows a normal probability distribution and has a mean value of 323 min and a standard deviation of 23.7 min. Also, the stochastic maintenance schedule lies between 299 and 306 min for a 99 per cent confidence level. Furthermore, except the pilot and the electrical team (approximately 90 per cent idle), all other teams are around 40 per cent idle. A sensitivity analysis is also performed and yielded that the PAM-SS model is not sensitive to the number of technicians in each team; however, it is highly sensitive to the probability of occurrence of the breakdowns/repairs.

The PAM-SS model is specifically developed for military rotorcrafts, to manage the different resources involved in the detailed planning and scheduling of the periodic/scheduled maintenance, mainly the 50-h inspection. It evaluates the resources utilization (idleness and queue), the stochastic maintenance duration and identifies backlogs and bottlenecks.

The PAM-SS tackles military aircraft planning and scheduling in a stochastic methodology, considering uncertainties in all inspection task durations and breakdown or repair durations. The PAM-SS, although developed for rotorcrafts can be further developed for any other type of military aircraft or any other scheduled maintenance program interval.

Statistical inference (estimation and testing) for the stochastic volatility (SV) model Taylor (1982, 1986) is challenging, especially likelihood-based methods which are difficult to apply due to the presence of latent variables. The existing methods are either computationally costly and/or inefficient. In this paper, we propose computationally simple estimators for the SV model, which are at the same time highly efficient. The proposed class of estimators uses a small number of moment equations derived from an ARMA representation associated with the SV model, along with the possibility of using “winsorization” to improve stability and efficiency. We call these ARMA-SV estimators. Closed-form expressions for ARMA-SV estimators are obtained, and no numerical optimization procedure or choice of initial parameter values is required. The asymptotic distributional theory of the proposed estimators is studied. Due to their computational simplicity, the ARMA-SV estimators allow one to make reliable – even exact – simulation-based inference, through the application of Monte Carlo (MC) test or bootstrap methods. We compare them in a simulation experiment with a wide array of alternative estimation methods, in terms of bias, root mean square error and computation time. In addition to confirming the enormous computational advantage of the proposed estimators, the results show that ARMA-SV estimators match (or exceed) alternative estimators in terms of precision, including the widely used Bayesian estimator. The proposed methods are applied to daily observations on the returns for three major stock prices (Coca-Cola, Walmart, Ford) and the S&P Composite Price Index (2000–2017). The results confirm the presence of stochastic volatility with strong persistence.

The purpose of this paper is to investigate the effects of irregular unsettling on the smoking model in form of the stochastic model as in the deterministic model these effects are neglected for simplicity.

In this research, the authors investigate a stochastic smoking system in which the contact rate is perturbed by Lévy noise to control the trend of smoking. First, present the formulation of the stochastic model and study the dynamics of the deterministic model. Then the global positive solution of the stochastic system is discussed. Further, extinction and the persistence of the proposed system are presented on the base of the reproductive number.

The authors discuss the dynamics of the deterministic smoking model form and further present the existence and uniqueness of non-negative global solutions for the stochastic system. Some previous study’s mentioned in the Introduction can be improved with the help of obtaining results, graphically present in this manuscript. In this regard, the authors present the sufficient conditions for the extinction of smoking for reproductive number is less than 1.

In this work, the authors investigated the dynamic stochastic smoking model with non-Gaussian noise. The authors discussed the dynamics of the deterministic smoking model form and further showed for the stochastic system the existence and uniqueness of the non-negative global solution. Some previous study’s mentioned in the Introduction can be improved with the help of obtained results, clearly shown graphically in this manuscript. In this regard, the authors presented the sufficient conditions for the extinction of smoking, if <1, which can help in the control of smoking. Motivated from this research soon, the authors will extent the results to propose new mathematical models for the smoking epidemic in the form of fractional stochastic modeling. Especially, will investigate the effective strategies for control smoking throughout the world.

This study is helpful in the control of smoking throughout the world.

The purpose of this paper is to present a stochastic p-robust data envelopment analysis (DEA) model for decision-making units (DMUs) efficiency estimation under uncertainty. The main contribution of this paper consists of the development of a more robust system for the estimation of efficiency in situations of inputs uncertainty. The proposed model is used for the efficiency measurement of a commercial Iranian bank.

This paper has been arranged to launch along the following steps: the classical Charnes, Cooper, and Rhodes (CCR) DEA model was briefly reviewed. After that, the p-robust DEA model is introduced and then calculated the priority weights of each scenario for CCR DEA output oriented method. To compute the priority weights of criteria in discrete scenarios, the analytical hierarchy analysis process (AHP) is used. To tackle the uncertainty of experts’ opinion, a synthetic technique is applied based on both robust and stochastic optimizations. In the sequel, stochastic p-robust models are proposed for the estimation of efficiency, with particular attention being paid to DEA models.

The proposed method provides a more encompassing measure of efficiency in the presence of synthetic uncertainty approach. According to the results, the expected score, relative regret score and stochastic P-robust score for DMUs are obtained. The applicability of the extended model is illustrated in the context of the analysis of an Iranian commercial bank performance. Also, it is shown that the stochastic p-robust DEA model is a proper generalization of traditional DEA and gained a desired robustness level. In fact, the maximum possible efficiency score of a DMU with overall permissible uncertainties is obtained, and the minimal amount of uncertainty level under the stochastic p-robustness measure that is required to achieve this efficiency score. Finally, by an example, it is shown that the objective values of the input and output models are not inverse of each other as in classical DEA models.

This research showed that the enormous decrease in maximum possible regret makes only a small addition in the expected efficiency. In other words, improvements in regret can somewhat affect the expected efficiency. The superior issue this kind of modeling is to permit a harmful effect to the objective to better hedge against the uncertain cases that are commonly ignored.

The volatility of a financial asset is an important input for financial decision‐making in the context of asset allocation, option pricing, and risk management. The authors compare and contrast four approaches to stochastic volatility to determine which is most appropriate to each of these various needs.