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To solve the problem of simulation evaluation with small samples, a fresh approach of grey estimation is presented based on classical statistical theory and grey system theory. The purpose of this paper is to make full use of the difference of data distribution and avoid the marginal data being ignored.

Based upon the grey distribution characteristics of small sample data, the definition about a new concept of grey relational similarity measure comes into being. At the same time, the concept of sample weight is proposed according to the grey relational similarity measure. Based on the new definition of grey weight, the grey point estimation and grey confidence interval are studied. Then the improved Bootstrap resampling is designed by uniform distribution and randomness as an important supplement of the grey estimation. In addition, the accuracy of grey bilateral and unilateral confidence intervals is introduced by using the new grey relational similarity measure approach.

The new small sample evaluation method can realize the effective expansion and enrichment of data and avoid the excessive concentration of data. This method is an organic fusion of grey estimation and improved Bootstrap method. Several examples are used to demonstrate the feasibility and validity of the proposed methods to illustrate the credibility of some simulation data, which has no need to know the probability distribution of small samples.

This research has completed the combination of grey estimation and improved Bootstrap, which makes more reasonable use of the value of different data than the unimproved method.

When performing system-level developmental testing, time and expenses generally warrant a small sample size for failure data. Upon failure discovery, redesigns and/or corrective actions can be implemented to improve system reliability. Current methods for estimating discrete (one-shot) reliability growth, namely the Crow (AMSAA) growth model, stipulate that parameter estimates have a great level of uncertainty when dealing with small sample sizes. The purpose of this paper is to present an application of a modified GM(1,1) model for handling system-level testing constrained by small sample sizes.

The paper presents a methodology for incorporating failure data into a modified GM(1,1) model for systems with failures following a poly-Weibull distribution. Notional failure data are generated for complex systems and characterization of reliability growth parameters is performed via both the traditional AMSAA model and the GM(1,1) model for purposes of comparing and assessing performance.

The modified GM(1,1) model requires less complex computational effort and provides a more accurate prediction of reliability growth model parameters for small sample sizes and multiple failure modes when compared to the AMSAA model. It is especially superior to the AMSAA model in later stages of testing.

This research identifies cost-effective methods for developing more accurate reliability growth parameter estimates than those currently used.

By using small reference samples, the calculation method of confidence value and prediction method of confidence interval for multi-input system are investigated. The purpose of this paper is to offer effective assessing methods of confidence value and confidence interval for the simulation models used in establishing guidance and control systems.

In this paper, first, an improved cluster estimation method is proposed to guide the selection of the small reference samples. Then, based on analytic hierarchy process method, the new calculation method of the weight of each reference sample is derived. By using the grey relation analysis method, new calculation methods of the correlation coefficient and confidence value are presented. Moreover, the confidence interval of the sample awaiting assessment is defined. A new prediction method is derived to obtain the confidence interval of the sample awaiting assessment which has no reference sample. Subsequently, by using the prediction method and original small reference samples, Bootstrap resampling method is used to obtain more correlation coefficients for the sample to reduce the probability of abandoning the true.

The grey relational analysis is used in assessing the confidence value and interval prediction. The numerical simulations are presented to demonstrate the effectiveness of the theoretical results.

Based on the selected small reference samples, new calculation methods of the correlation coefficient and confidence value are presented to assess the confidence value of model awaiting assessment. The calculation methods of maximum confidence interval, expected confidence interval and other required confidence intervals are presented, which can be used in assessing the validities of controller and guidance system obtained from the model awaiting assessment.

The purpose of this paper is to discuss the interval forecasting, prediction interval and its reliability. When the predicted interval and its reliability are construction, the general rule which must satisfy is studied, grey wrapping band forecasting method is perfect.

A forecasting method puts forward a process of prediction interval. It also elaborates on the meaning of interval (the probability of the prediction interval including the real value of predicted variable). The general rule is abstracted and summarized by many forecasting cases. The general rule is discussed by axiomatic method.

The prediction interval is categorized into three types. Three axioms that construction predicted interval must satisfy are put forward. Grey wrapping band forecasting method is improved based on the proposed axioms.

Take the Shanghai composite index as the example, according to the K-line diagram from 4 January 2013 to 9 May 2013, the reliability of predicted rebound height of subsequent two or three trading day does not exceed the upper wrapping curve is 80 per cent. It is significant to understand the forecasting range correctly, build a reasonable range forecasting method and to apply grey wrapping band forecasting method correctly.

Grey wrapping band forecasting method is improved based on the proposed axioms.

This study aims to accurately evaluate the influence of various error intervals on the performance of the wiper.

The wiper structural system is decomposed into classical four-link planar for kinematics analysis, and it was modeled respectively by using interval method, universal grey number theory and enumeration approach depending on the nature of uncertainty.

The universal grey number theory is a viable methodology for the accurate analysis of uncertain structural system.

(1) The model of uncertain wiper structural system is established. (2) Universal grey number theory and new parameters are adopted to analyze the presence of uncertain wiper structural system. (3) Comparative analysis of response quantities is obtained by interval method, universal grey number theory and enumeration method.

In contrast to point forecasts, interval forecasts provide information on future variability. This research thus aimed to develop interval prediction models by addressing two significant issues: (1) a simple average with an additive property is commonly used to derive combined forecasts, but this unreasonably ignores the interaction among sequences used as sources of information, and (2) the time series often does not conform to any statistical assumptions.

To develop an interval prediction model, the fuzzy integral was applied to nonlinearly combine forecasts generated by a set of grey prediction models, and a sequence including the combined forecasts was then used to construct a neural network. All required parameters relevant to the construction of an interval model were optimally determined by the genetic algorithm.

The empirical results for tourism demand showed that the proposed non-additive interval model outperformed the other interval prediction models considered.

The private and public sectors in economies with high tourism dependency can benefit from the proposed model by using the forecasts to help them formulate tourism strategies.

In light of the usefulness of combined point forecasts and interval model forecasting, this research contributed to the development of non-additive interval prediction models on the basis of combined forecasts generated by grey prediction models.

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems.

After presenting the concepts of grey systems and grey numbers, this paper surveys existing approaches for solving grey linear programming problems and applications. Also, methods and approaches for solving grey linear programming are classified, and its advantages and disadvantages are expressed.

The progress of grey programming has been expressed from past to present. The main methods for solving the grey linear programming problem can be categorized as Best-Worst model, Confidence degree, Whitening parameters, Prediction model, Positioned solution, Genetic algorithm, Covered solution, Multi-objective, Simplex and dual theory methods. This survey investigates the developments of various solving grey programming methods and its applications.

Different methods for solving grey linear programming problems are presented, where each of them has disadvantages and advantages in providing results of grey linear programming problems. This study attempted to review papers published during 35 years (1985–2020) about grey linear programming solving and applications. The review also helps clarify the important advantages, disadvantages and distinctions between different approaches and algorithms such as weakness of solving linear programming with grey numbers in constraints, inappropriate results with the lower bound is greater than upper bound, out of feasible region solutions and so on.

In the context of Dynamic Factor Models, we compare point and interval estimates of the underlying unobserved factors extracted using small- and big-data procedures. Our paper differs from previous works in the related literature in several ways. First, we focus on factor extraction rather than on prediction of a given variable in the system. Second, the comparisons are carried out by implementing the procedures considered to the same data. Third, we are interested not only on point estimates but also on confidence intervals for the factors. Based on a simulated system and the macroeconomic data set popularized by Stock and Watson (2012), we show that, for a given procedure, factor estimates based on different cross-sectional dimensions are highly correlated. On the other hand, given the cross-sectional dimension, the maximum likelihood Kalman filter and smoother factor estimates are highly correlated with those obtained using hybrid procedures. The PC estimates are somehow less correlated. Finally, the PC intervals based on asymptotic approximations are unrealistically tiny.

This chapter develops a novel bootstrap procedure to obtain robust bias-corrected confidence intervals in regression discontinuity (RD) designs. The procedure uses a wild bootstrap from a second-order local polynomial to estimate the bias of the local linear RD estimator; the bias is then subtracted from the original estimator. The bias-corrected estimator is then bootstrapped itself to generate valid confidence intervals (CIs). The CIs generated by this procedure are valid under conditions similar to Calonico, Cattaneo, and Titiunik’s (2014) analytical correction – that is, when the bias of the naive RD estimator would otherwise prevent valid inference. This chapter also provides simulation evidence that our method is as accurate as the analytical corrections and we demonstrate its use through a reanalysis of Ludwig and Miller’s (2007) Head Start dataset.

This study aims to use gray models to predict abnormal stock returns.

Data are collected from listed companies in the Tehran Stock Exchange during 2005-2015. The analyses portray three models, namely, the gray model, the nonlinear gray Bernoulli model and the Nash nonlinear gray Bernoulli model.

Results show that the Nash nonlinear gray Bernoulli model can predict abnormal stock returns that are defined by conditions other than gray models which predict increases, and then after checking regression models, the Bernoulli regression model is defined, which gives higher accuracy and fewer errors than the other two models.

The stock market is one of the most important markets, which is influenced by several factors. Thus, accurate and reliable techniques are necessary to help investors and consumers find detailed and exact ways to predict the stock market.