On the Behavior of Statistical Models Used for Design

1976 ◽  
Vol 98 (2) ◽  
pp. 601-606 ◽  
Author(s):  
P. H. Wirsching
Keyword(s):  
Statistical Models ◽  
Random Variable ◽  
Upper Bounds ◽  
Probability Models ◽  
Tail Probabilities ◽  
Structural Element ◽  
Probability Estimates ◽  
Power Models ◽  
Competing Models ◽  
Two Parameter

In probabilistic design, it is common practice to use two parameter statistical models (e.g., normal, lognormal) to describe random design factors. However, given a random sample of data, it is often difficult to distinguish which of several competing models provides the best description. It is demonstrated herein that the choice of model has a profound effect on probability estimates, particularly in the tails of the distributions. Given only the mean and standard deviation of a random variable, the Tchebycheff or Camp-Meidell inequalities can be used to provide upper-bound estimates of probabilities. However, these inequalities are usually too weak for design purposes. Probability models which yield more reasonable results are proposed. The two parameter exponential and power models are proposed for quasi-upper bounds of right and left tail probabilities, respectively. The exponential and power models are used for stress and strength, respectively, to derive, from inference theory, quasi-upper bounds for probability of failure of a structural element.

STRUCTURAL RELIABILITY ANALYSIS BASED ON P-BOXES

2020 ◽  
Vol 92 (6) ◽  
pp. 51-58
Author(s):  
S.A. SOLOVYEV ◽  
Keyword(s):  
Probability Distribution ◽  
Reliability Analysis ◽  
Structural Reliability ◽  
Epistemic Uncertainty ◽  
Random Variable ◽  
Strength Criterion ◽  
Structural Elements ◽  
Structural Element ◽  
Limit States ◽  
Distribution Shape

The article describes a method for reliability (probability of non-failure) analysis of structural elements based on p-boxes. An algorithm for constructing two p-blocks is shown. First p-box is used in the absence of information about the probability distribution shape of a random variable. Second p-box is used for a certain probability distribution function but with inaccurate (interval) function parameters. The algorithm for reliability analysis is presented on a numerical example of the reliability analysis for a flexural wooden beam by wood strength criterion. The result of the reliability analysis is an interval of the non-failure probability boundaries. Recommendations are given for narrowing the reliability boundaries which can reduce epistemic uncertainty. On the basis of the proposed approach, particular methods for reliability analysis for any structural elements can be developed. Design equations are given for a comprehensive assessment of the structural element reliability as a system taking into account all the criteria of limit states.

scholarly journals Using the Econometric Models for Identification of Risk Factors for Albanian Smes (Case Study: Smes of Gjirokastra Region)

2021 ◽  
Vol 18 ◽  
pp. 163-170
Author(s):  
Lorenc Koçiu ◽  
Kledian Kodra
Keyword(s):  
Logistic Regression ◽  
Statistical Models ◽  
Logistic Regression Model ◽  
Qualitative Data ◽  
Linear Models ◽  
Binary Logistic Regression ◽  
Random Variable ◽  
Econometric Models ◽  
Binary Logistic Regression Model

Using the econometric models, this paper addresses the ability of Albanian Small and Medium-sizedEnterprises (SMEs) to identify the risks they face. To write this paper, we studied SMEs operating in theGjirokastra region. First, qualitative data gathered through a questionnaire was used. Next, the 5-level Likertscale was used to measure it. Finally, the data was processed through statistical software SPSS version 21,using the binary logistic regression model, which reveals the probability of occurrence of an event when allindependent variables are included. Logistic regression is an integral part of a category of statistical models,which are called General Linear Models. Logistic regression is used to analyze problems in which one or moreindependent variables interfere, which influences the dichotomous dependent variable. In such cases, the latteris seen as the random variable and is dependent on them. To evaluate whether Albanian SMEs can identifyrisks, we analyzed the factors that SMEs perceive as directly affecting the risks they face. At the end of thepaper, we conclude that Albanian SMEs can identify risk

Geometric bounds on certain sublinear functionals of geometric Brownian motion

2003 ◽  
Vol 40 (4) ◽  
pp. 893-905 ◽  
Author(s):  
Per Hörfelt
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Borel Measure ◽  
Random Variable ◽  
Geometric Brownian Motion ◽  
Upper And Lower Bounds ◽  
Tail Probabilities ◽  
Absolutely Continuous ◽  
Basket Options ◽  
Sublinear Functionals

Suppose that {Xs, 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝm → [0,∞) is a (weighted) lq(ℝm)-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the Lp(μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(Xs), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

scholarly journals Hölder-Type Inequalities for Norms of Wick Products

2008 ◽  
Vol 2008 ◽  
pp. 1-22 ◽  
Author(s):  
Alberto Lanconelli ◽  
Aurel I. Stan
Keyword(s):  
Random Variable ◽  
Upper Bounds ◽  
Minimal Condition ◽  
Wick Product ◽  
Identity Operator ◽  
Second Quantization ◽  
Measurable Functions ◽  
Quantization Operator ◽  
Wick Products ◽  
Finite Moments

Various upper bounds for the L2-norm of the Wick product of two measurable functions of a random variable X, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Alen Alexanderian ◽  
Liang Zhu ◽  
Maher Salloum ◽  
Ronghui Ma ◽  
Meilin Yu
Keyword(s):  
Uncertainty Quantification ◽  
Material Properties ◽  
Statistical Models ◽  
Gaussian Random Field ◽  
Concentration Field ◽  
Random Variable ◽  
Spectral Projection ◽  
Velocity Fields ◽  
Efficient Computation ◽  
Uncertain Material Properties

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen–Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

scholarly journals Modelling the COVID-19 Mortality Rate with a New Versatile Modification of the Log-Logistic Distribution

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Abdisalam Hassan Muse ◽  
Ahlam H. Tolba ◽  
Eman Fayad ◽  
Ola A. Abu Ali ◽  
M. Nagy ◽  
...  
Keyword(s):  
Mortality Rate ◽  
Goodness Of Fit ◽  
Likelihood Estimation ◽  
Cumulative Distribution ◽  
Logistic Distribution ◽  
Parameter Distribution ◽  
Unknown Parameters ◽  
Failure Rate Function ◽  
Competing Models ◽  
Two Parameter

The goal of this paper is to develop an optimal statistical model to analyze COVID-19 data in order to model and analyze the COVID-19 mortality rates in Somalia. Combining the log-logistic distribution and the tangent function yields the flexible extension log-logistic tangent (LLT) distribution, a new two-parameter distribution. This new distribution has a number of excellent statistical and mathematical properties, including a simple failure rate function, reliability function, and cumulative distribution function. Maximum likelihood estimation (MLE) is used to estimate the unknown parameters of the proposed distribution. A numerical and visual result of the Monte Carlo simulation is obtained to evaluate the use of the MLE method. In addition, the LLT model is compared to the well-known two-parameter, three-parameter, and four-parameter competitors. Gompertz, log-logistic, kappa, exponentiated log-logistic, Marshall–Olkin log-logistic, Kumaraswamy log-logistic, and beta log-logistic are among the competing models. Different goodness-of-fit measures are used to determine whether the LLT distribution is more useful than the competing models in COVID-19 data of mortality rate analysis.

scholarly journals From theories to models to predictions: A Bayesian model comparison approach

2017 ◽  
Author(s):  
Jeffrey Rouder ◽  
Julia M. Haaf ◽  
Frederik Aust
Keyword(s):  
Statistical Models ◽  
Bayesian Model ◽  
Model Comparison ◽  
Bayes Factor ◽  
Significance Testing ◽  
Bayesian Model Comparison ◽  
Comparison Approach ◽  
Competing Models ◽  
Competing Hypotheses

A key goal in research is to use data to assess competing hypotheses or theories. Analternative to the conventional significance testing is Bayesian model comparison. The mainidea is that competing theories are represented by statistical models. In the Bayesianframework, these models then yield predictions about data even before the data are seen.How well the data match the predictions under competing models may be calculated, andthe ratio of these matches—the Bayes factor—is used to assess the evidence for one modelcompared to another. We illustrate the process of going from theories to models and topredictions in the context of two hypothetical examples about how exposure to media affectsattitudes toward refugees.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

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