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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.

Aims to test to determine whether the selection of the historical return time interval (monthly, quarterly, semiannual, or annual) used for calculating real estate investment trust (REIT) returns has a significant effect on optimal portfolio allocations.

Using a mean‐variance utility function, optimal allocations to portfolios of stocks, bonds, bills, and REITs across different levels of assumed investor risk aversion are calculated. The average historical returns, standard deviations, and correlations (assuming different time intervals) of the various asset classes are used as mean‐variance inputs. Results are also compared using more recent data, since 1988, with, data from the full REIT history, which goes back to 1972.

Using the more recent REIT datarather than the full dataset results in optimal allocations to REITs that are considerably higher. Likewise, using monthly and quarterly returns tends to understate the variability of REITs and leads to higher portfolio allocations.

The results of this study are based on the limited historical return data that are currently available for REITs. The results of future time periods may not prove to be consistent with the findings.

Numerous research papers arbitrarily decide to employ monthly or quarterly returns in their analyses to increase the number of REIT observations they have available. These shorter interval returns are generally annualized. This paper addresses the consequences of those decisions.

It has been shown that the decision to use return estimation intervals shorter than a year does have dramatic consequences on the results obtained and, therefore, must be carefully considered and justified.

Exact confidence interval estimation for the new extreme value model is often impractical. This paper seeks to evaluate the accuracy of approximate confidence intervals for the two‐parameter new extreme value model.

The confidence intervals of the parameters of the new model based on likelihood ratio, Wald and Rao statistics are evaluated and compared through the simulation study. The criteria used in evaluating the confidence intervals are the attainment of the nominal error probability and the symmetry of lower and upper error probabilities.

This study substantiates the merits of the likelihood ratio, the Wald and the Rao statistics. The results indicate that the likelihood ratio‐based intervals perform much better than the Wald and Rao intervals.

Exact interval estimates for the new model are difficult to obtain. Consequently, large sample intervals based on the asymptotic maximum likelihood estimators have gained widespread use. Intervals based on inverting likelihood ratio, Rao and Wald statistics are rarely used in commercial packages. This paper shows that the likelihood ratio intervals are superior to intervals based on the Wald and the Rao statistics.

The purpose of this paper is to focus on the influences of the uncertain dynamic responses on the reconstruction of loads.

Based on the assumption of unknown-but-bounded (UBB) noise, a time-domain approach to estimate the uncertain time-dependent external loads is presented by combining the inverse system method in modern control theory and interval analysis in interval mathematics. Inspired by the concept of set membership identification in control theory, an interval analysis model of external loads time history, which is indeed a region or feasible set containing all possible loads being consistent with the bounded structural acceleration responses is established and further solved by two interval algorithms.

Unlike traditional loads identification methods which only give a point estimation, an interval estimation of external loads time history, which is a region containing all the possible loads being consistent with the uncertain structural responses, is determined. The correlation characteristics among the responses of acceleration, velocity, and displacement are also discussed in consideration of the UBB uncertainty.

For one hand, the solution of the inverse problem in original system is transformed to the solution of the direct problem in inverse system; for another, the authors deal with the uncertainty by use of interval analysis method, and the identified interval process, which contains any possible external loads time history being consistent with the bounded structural responses can be approximately obtained.

Confidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance.

The construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation.

The GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR.

This research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance.

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.

The purpose of this paper is to present an actuator fault detection method for unmanned aerial vehicles (UAVs) based on interval observer and extended state observer.

The proposed algorithm has very little model dependency. Therefore, a six-degree-of-freedom linear equation of UAVs is first established, and then, combined with actuator failure and external disturbances in flight control, a steering gear model with actuator failure (such as stuck bias and invalidation) is designed. Meanwhile, an extended state observer is designed for fault detection. Moreover, a fault detection scheme based on interval observer is designed by combining fault and disturbances.

The method is testified on the extended state observer and the interval observer under the failure of the steering gear and bounded disturbances. The simulation results show that the two types of fault detection schemes designed can successfully detect various types of faults and have high sensitivity.

This research paper studies the failure detection scheme of the UAVs’ actuator. The fault detection scheme in this paper has better performance on actuator faults and bounded disturbances than using regular fault detection schemes.

This paper deals with the estimation of the stress–strength reliability R = P(X < Y), when X and Y follow (1) independent generalized gamma (GG) distributions with only a common shape parameter and (2) independent Weibull random variables with arbitrary scale and shape parameters and generalize the proposal from Kundu and Gupta (2006), Kundu and Raqab (2009) and Ali et al. (2012).

First, a closed form expression for R is derived under the conditions (1) and (2). Next, sufficient conditions are given for the convergence of the infinite series expansions used to calculate the value of R in case (2). The models GG and Weibull are fitted by maximum likelihood using Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton method. Confidence intervals and standard errors are calculated using bootstrap. For illustration purpose, two real data sets are analyzed and the results are compared with the existing recent results available in the literature.

The proposed approaches improve the estimation of the R by not using transformations in the data and flexibilize the modeling with Weibull distributions with arbitrary scale and shape parameters.

The proposals of the paper eliminate the misestimation of R caused by subtracting a constant value from the data (Kundu and Raqab, 2009) and treat the estimation of R in a more adequate way by using the Weibull distributions without restrictions in the parameters. The two cases covered generalize a number of distributions and unify a number of stress–strength probability P(X < Y) results available in the literature.

This study aims to propose an improved affine interval truncation algorithm to restrain interval extension for interval function.

To reduce the occurrence times of related variables in interval function, the processing method of interval operation sequence is proposed.

The interval variable is evenly divided into several subintervals based on correlation analysis of interval variables. The interval function value is modified by the interval truncation method to restrain larger estimation of interval operation results.

Through several uncertain displacement response engineering examples, the effectiveness and applicability of the proposed algorithm are verified by comparing with interval method and optimization algorithm.

Much emphasis in recent years has been placed on evaluating product quality by use of the C⊂p and C⊂pk indices. These capability indices, particularly the latter, have much appeal as they seemingly wrap up quality into the calculation of a single number. The purpose of this article is to explain the statistical essentials involved in using the C⊂p and C⊂pk indices to assess product quality. Having discussed confidence intervals, definitions and provided some quantitative results, caution is urged in using these indices in practice. Some practical suggestions provide a climax.