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The purpose of this paper is to associate a probabilistic confidence with the stock market interval forecasts obtained with the interval least squares (ILS) algorithm. The term probabilistic confidence in this paper means the probability of a point observation that will fall in the interval forecast.

Using confidence interval as input, annual ILS forecasts of the stock market were made. Then the probability of point observation that fall in the annual forecasts was examined empirically.

When using confidence interval as ILS input, the stock market annual interval forecasts may have the same level of confidence as that of the input intervals.

At the same confidence level, the ILS can produce much better quality forecasts than the traditional ordinary least squares method for the stock market. Although the algorithmic approach can be applied to analyze other datasets, one should examine implications of computational results as always.

Results of this specific paper may be interesting to executive officers, other financial decision makers and to investors.

Although the ILS algorithm has been recently developed in forecasting the variability of the stock market, this paper presents the first successful attempt in associating a probabilistic confidence with ILS interval forecasts.

The present paper aims to present the results of a simulation study on the behavior of the four 95 percent bootstrap confidence intervals for estimating C_pk when collected data are from a multiple streams process.

A computer simulation study is developed to present the behavior of four 95 percent bootstrap confidence intervals, i.e. standard bootstrap (SB), percentile bootstrap (PB), biased‐corrected percentile bootstrap (BCPB), and biased‐corrected and accelerated (BCa) bootstrap for estimating the capability index C_pk of a multiple streams process. An analysis of variance using two factorial and three‐stage nested designs is applied for experimental planning and data analysis.

For multiple process streams, the relationship between the true value of C_pk and the required sample size for effective experiment is presented. Based on the simulation study, the two‐stream process always gives a higher coverage percentage of bootstrap confidence interval than the four‐stream process. Meanwhile, BCPB and BCa intervals lead to better coverage percentage than SB and PB intervals.

Since a large number of process streams decreases the coverage percentage of the bootstrap confidence interval, it may be inappropriate to use the bootstrap method for constructing the confidence interval of a process capability index as the number of process streams is large.

The present paper is the first work to explore the behavior of bootstrap confidence intervals for estimating the capability index C_pk of a multiple streams process. It is concluded that the number of process streams definitively affects the performance of bootstrap methods.

In order to develop a feasible and efficient method to acquire the long‐run availability of a parallel system with distribution‐free up and down times, the purpose of this paper is to perform the simulation comparisons on the interval estimations of system availability using four bootstrapping methods.

By using four bootstrap methods; standard bootstrap (SB) confidence interval, percentile bootstrap (PB) confidence interval, bias‐corrected percentile bootstrap (BCPB) confidence interval, and bias‐corrected and accelerated (BCa) confidence interval. A numerical simulation study is carried out in order to demonstrate performance of these proposed bootstrap confidence intervals. Especially, we investigate the accuracy of the four bootstrap confidence intervals by calculating the coverage percentage, the average length, and the relative coverage of confidence intervals.

Among the four bootstrap confidence intervals, the PB method has the largest relative coverage in most situations. That is, the PB method is the best one made by practitioners who want to obtain an efficient interval estimation of availability.

It is the first time that the relative coverage is introduced to evaluate the performance of estimation method, which is more efficient than the existing measures.

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.

In this chapter we demonstrate the construction of inverse test confidence intervals for the turning-points in estimated nonlinear relationships by the use of the marginal or first derivative function. First, we outline the inverse test confidence interval approach. Then we examine the relationship between the traditional confidence intervals based on the Wald test for the turning-points for a cubic, a quartic, and fractional polynomials estimated via regression analysis and the inverse test intervals. We show that the confidence interval plots of the marginal function can be used to estimate confidence intervals for the turning-points that are equivalent to the inverse test. We also provide a method for the interpretation of the confidence intervals for the second derivative function to draw inferences for the characteristics of the turning-point.

This method is applied to the examination of the turning-points found when estimating a quartic and a fractional polynomial from data used for the estimation of an Environmental Kuznets Curve. The Stata do files used to generate these examples are listed in Appendix A along with the data.

In this paper, the authors propose two construction methods to estimate confidence intervals of the time-based optimal software rejuvenation policy and its associated maximum system availability via a parametric bootstrap method. Through simulation experiments the authors investigate their asymptotic behaviors and statistical properties.

The present paper is the first challenge to derive the confidence intervals of the optimal software rejuvenation schedule, which maximizes the system availability in the sense of long run. In other words, the authors concern the statistical software fault management by employing an idea of process control in quality engineering and a parametric bootstrap.

As a remarkably different point from the existing work, the authors carefully take account of a special case where the two-sided confidence interval of the optimal software rejuvenation time does not exist due to that fact that the estimator distribution of the optimal software rejuvenation time is defective. Here the authors propose two useful construction methods of the two-sided confidence interval: conditional confidence interval and heuristic confidence interval.

Although the authors applied a simulation-based bootstrap confidence method in this paper, another re-sampling-based approach can be also applied to the same problem. In addition, the authors just focused on a parametric bootstrap, but a non-parametric bootstrap method can be also applied to the confidence interval estimation of the optimal software rejuvenation time interval, when the complete knowledge on the distribution form is not available.

The statistical software fault management techniques proposed in this paper are useful to control the system availability of operational software systems, by means of the control chart.

Through the online monitoring in operational software systems, it would be possible to estimate the optimal software rejuvenation time and its associated system availability, without applying any approximation. By implementing this function on application programming interface (API), it is possible to realize the low-cost fault-tolerance for software systems with aging.

In the past literature, almost all authors employed parametric and non-parametric inference techniques to estimate the optimal software rejuvenation time but just focused on the point estimation. This may often lead to the miss-judgment based on over-estimation or under-estimation under uncertainty. The authors overcome the problem by introducing the two-sided confidence interval approach.

The purpose of this study is to investigate the impacts of interval measured data, rather than traditional point data, on economic variability studies.

The study uses interval measured data to forecast the variability of future stock market changes. The variability (interval) forecasts are then compared with point data‐based confidence interval forecasts.

Using interval measured data in stock market variability forecasting can significantly increase forecasting accuracy, compared with using traditional point data.

An interval forecast for stock prices essentially consists of predicted levels and a predicted variability which can reduce perceived uncertainty or risk embedded in future investments, and therefore, may influence required returns and capital asset prices.

Since the early 1980s, currency options have become a popular means for hedging foreign currency positions or speculating on anticipated movements in exchange rates. Yet, they can also be used to enhance the forecasting of exchange rates. Corporate forecasts of exchange rates involve two tasks: (1) a point estimate of a currency's exchange rate, and (2) a confidence interval that suggests the degree of uncertainty associated with the point estimate forecast. A currency forward or futures price is often used as the point estimate required. The confidence interval is commonly developed by using the historical volatility of exchange rate movements. However, an alternative method is to use the market's anticipated volatility in developing the confidence interval. Scott and Tucker (1990) have shown that the volatility implied from contemporaneous currency option prices is a better forecast of future volatility than historical measures. Therefore, a confidence interval implied by currency options should also be more reliable. Our objective is to illustrate how confidence intervals can be developed from currency option information. Given the degree of difficulty in forecasting exchange rates, more reliable confidence intervals could greatly improve managerial decisions.

Distribution. The purpose of this study is to obtain the modified maximum likelihood estimator of stress–strength model using the ranked set sampling, to obtain the asymptotic and bootstrap confidence interval of P[Y < X], to compare the performance of author’s estimates with the estimates under simple random sampling and to apply author’s estimates on head and neck cancer.

The maximum likelihood estimator of R = P[Y < X], where X and Y are two independent inverse Weibull random variables common shape parameter that affect the shape of the distribution, and different scale parameters that have an effect on the distribution dispersion are given under ranked set sampling. Together with the asymptotic and bootstrap confidence interval, Monte Carlo simulation shows that this estimator performs better than the estimator under simple random sampling. Also, the asymptotic and bootstrap confidence interval under ranked set sampling is better than these interval estimators under simple random sampling. The application to head and neck cancer disease data shows that the estimator of R = P[Y < X] that shows the treatment with radiotherapy is more efficient than the treatment with a combined radiotherapy and chemotherapy under ranked set sampling that is better than these estimators under simple random sampling.

The ranked set sampling is more effective than the simple random sampling for the inference of stress-strength model based on inverse Weibull distribution.

This study sheds light on the author’s estimates on head and neck cancer.

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.