Weibull Distributed Research Articles

Non-Linearity in MCP with Weibull Distributed Wind Speeds

Conventional MCP (Measure Correlate Predict) techniques often employ linear regression to model the relationship between wind speeds at two sites and use this model to predict the long term wind regime at one site from the long term wind regime at the other. In considering sites whose wind speeds are Weibull distributed it can be demonstrated analytically that linear relationships do not, in fact, hold. The general relationship is found to be of the form y = m xα + c where α is the ratio of the shape parameters of the Weibull distributions. Linearity prevails only when the shape parameters are equal. This is seen to have implications for MCP analysis.

On the cascaded Weibull fading channel model

A new stochastic fading channel model called cascaded Weibull fading is introduced and the associated capacity is derived in closed form. This model is generated by the product of independent, but not necessarily identically distributed, Weibull random variables (RVs). By quantifying the convergence rate of the central limit theorem as pertaining to the multiplication of Weibull distributed RVs, the statistical basis of the lognormal distribution is investigated. By performing Kolmogorov–Smirnov tests, the null hypothesis for this product to be approximated by the lognormal distribution is studied. Another null hypothesis is also examined for this product to be approximated by a Weibull distribution with properly adjusted statistical parameters.

A discussion of statistical methods used to estimate extreme wind speeds

Wind speeds in extra-tropical latitudes are known to be approximately Weibull distributed. Hence a Weibull distribution fitted to all available data is often used to predict extreme winds. The most extreme values then, however, have little influence on the estimated parent Weibull distribution, and the accuracy of the extreme value predictions obtained in this manner may be questioned. In the present paper such a "Weibull method" is compared to a method based on statistical extreme value theory, "the annual maxima method". The comparison is based on 30 years of 10 minute wind speed averages measured hourly at 12 meteorological stations located at airports in Sweden. Results show that the Weibull method generates incorrect estimates of the tails of the distributions of wind speeds and of the distribution of yearly maximum wind speed, and that serial dependence of individual measurements has to be taken into account. In addition, it is inherent in the Weibull method that it does not provide any confidence bounds for the estimates. The annual maxima method avoids these problems. The measurements were rounded, first to entire knots, and then to m/s. A further, "technical", result is that if this rounding were disregarded in the estimation procedure, then the computed standard errors of the parameter estimates would be erroneously low. Hence, if rounding is done, it should be taken into account in the estimation procedure. We also believe this to be a clear indication that rounding of the data decreases estimation accuracy.

Specification Diagnostics for Econometric Models of Duration

This paper deals with diagnostics for evaluating the appropriateness of distributional assumptions in econometric models of duration. The purpose is to point out certain extensions, equivalences and interpretations. The exposition is in terms of the Weibull distribution, but the framework is applicable more generally. The diagnostics developed are score tests of the null hypothesis of Weibull distributed durations (conditional on covariates) against alternatives which are based on expansions of various orders in a system of orthogonal functions. The unified framework put forward By Tauchen (1985) and Newey (1985) is used to develop robust conditional moment tests which generalize the score tests. The paper also shows that recently developed diagnostics for uncorrected heterogeneity essentially amount to testing whether a particular moment relationship implied by the null is satisfied. Lastly, some Monte Carlo results are presented.

Evaluating project completion times when activity times are Weibull distributed

Activity networks have been used to model complex projects with wide applications in the field of production, planning and control. Stochastic activity networks (SANs) require a priori distribution function (d.f.) such as normal, exponential, Erlang and others. In this paper, a development of the moments method based on Weibull distribution of activity time is presented. The method provides an accurate estimate of the project completion time compared with other d.f.'s estimates.

Optimal replenishment scheduling for inventory items with Weibull distributed deterioration and time-varying demand

This paper proposes a dynamic programming model for solving replenishment scheduling problems. The model is developed based on the following assumptions: the inventory items are subject to deterioration of two-parameter Weibull distribution, the demand rate is exponentially distributed over time, shortages are allowed, and the effects of inflation and time-value of money are taken into consideration. The solutions of the model determine the optimal replenishment schedule over a finite horizon so that the present worth of the total cost is minimized. The solution procedure is illustrated by two numerical examples. The strength of the proposed model is supported by carrying out a comparison study with an existing model. The proposed model permits variation in both the replenishment intervals and the service levels between order cycles. As a result, it generates a better solution than other optimization models having fixed order intervals and fixed service levels. In addition, the Weibull deterioration used in the model is a generalized form of exponentially decaying functions, and hence it is applicable to the management of a wider range of deteriorating items in practice, like agricultural products, blood, drugs, fashion goods, high-tech and electronic components, volatile liquids, and radioactive substances

The Recent Rainfall Climatology of the Mediterranean Californias

In this work, recent (1948‐2001) rainfall data in a southwestern California station (San Diego) and a northwestern Baja California station (Ensenada) within a region called Mediterranean California, around 33 8N, 1178W, are studied. Cumulative annual means are used as indicators of climatological variability; but the entire datasets are analyzed by modeling the histogram of each set as a Weibull distribution probability density function, f . The climatology of both stations, defined simply as the arithmetic average, is compared with their theoretical mean; that is, the first moment of f . It is assumed that this comparison would be indicative of the reliability of the available rainfall climatologies. If these assumptions hold, in particular if the data is indeed Weibull distributed, it can be concluded that the climatological annual mean precipitation in this region is slightly overestimated at this time.

Properties of a multivariate survival distribution generated by a Weibull and inverse-Gaussian mixture

The authors consider the influence of the work environment on a system of nonrenewable components. The failure times for the components are Weibull distributed and the work environment has an inverse Gaussian distribution. A multivariate Weibull and inverse Gaussian mixture distribution is derived. Several pertinent properties for this multivariate distribution are discussed that shed some light on the nature of the distribution. The authors account for the operating environment and its changing nature by averaging over a parameter corresponding to the environment. The distribution is applied to find the mean number of components working at some mission time and the reliability for k-out-of-n components.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimating Charpy transition temperature shift using Weibull analysis

A primary consideration for operation of commercial nuclear light-water reactors is the fracture toughness of the reactor pressure vessel. One of the tests conducted for toughness determinations of the pressure vessel materials is the Charpy V-notch impact test. The tests are conducted over a range of test temperatures and the total impact energy required to break each specimen is determined and plotted vs test temperature. The impact energy values for a fixed temperature and radiation exposure are usually assumed to be Weibull distributed. This assumption was validated by a statistical selection process. An important parameter in the evaluation of reactor vessel materials is termed the ductile-to-brittle transition temperature shift (ΔT 4) and is the temperature shift between the unirradiated and irradiated Charpy V-notch curves at the 41-J (30-ft-lb) energy level. The purpose of this paper is to introduce a methodology for estimating the ΔT 41 value and to place limits on the error of estimation. This has been accomplished by assuming that the scale parameter in the Weibull distribution is a function of temperature (and neutron fluence for the irradiated Charpy V-notch data). An inverse regression technique is then used to solve for the temperature corresponding to an impact energy of 41 J. To obtain error bounds on this estimate, the parameters of the equation describing the scale parameter are resampled assuming a normal distribution and the inverse regression is repeated several times. This leads to several estimates of the temperature, and the variance of these values can be used to give bounds on the error of estimation.

An approximation for large consecutive-k-out-of-n:F systems

The authors consider a consecutive-k-out-n:F system consisting of identically distributed and statistically independent components, where the life distribution of an individual component is Weibull distributed with scale parameter 1/ lambda and shape parameter B. Let T/sub n/ be the life length of the consecutive-k-out-of-n:F system. The authors prove that for large values of n, the distribution of the n/sup 1/(ka)/T/sub n/, is satisfactorily approximated by a Weibull distribution with the same scale parameter and shape parameter k times the original shape parameter.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Combined Effect of Flaw Distribution and Diameter Variation on the Statistics of Glass Fiber Strength

This paper examines the combined effects of flaw‐severity distribution and fiber diameter distribution on the fracture statistics of glass fibers. The approach adopted is to assume specific forms (Gaussian and Weibull) for the diameter distribution, a Weibull distribution for the true strength, and then to calculate the measured strength distributions. Exact analytical results are obtained in the cases of flawless fibers, uniform size flaws, and when true strength and fiber diameter are Weibull distributed. Approximate analytical expressions have been derived for the general case. A procedure is described to obtain the parameters of the true strength distribution from the knowledge of measured strength distribution and the coefficient of variation of fiber diameters. Implications of these results for the fracture statistics of high‐strength glass fibers are discussed.

The goodness of fit of the weibull and rayleigh distributions to the distributions of observed wind speeds in a topographically diverse area

AbstractThe goodness of fit of the Weibull and Rayleigh distributions to the observed wind speed distribution was investigated at seven coastal weather stations markedly influenced by local topography and possessing relatively long periods of record. Wind speeds will he Rayleigh or Weibull distributed if four conditions specifying the circular normal distribution for wind velocity are met. These conditions were not met at the seven stations because of the influences of local topography and storm systems. In all cases the null hypothesis that there is no difference between the Weibull or Rayleigh distributions and the observed distribution had to be rejected with the large sample sizes employed. The transformation of wind velocity involved in deriving the Weibull distribution helps make the distribution more circular normal and the Weibull distribution gave better fits than the Rayleigh distribution. The fits were generally best al those stations with the most circular normal wind velocity distributions and the lowest percentage of calms.

Competing Risk and Multiple Comparison Analysis for Bearing Fatigue Tests

New developments in statistical inference for the two parameter Weibull distribution are explained in the terminology of bearing endurance testing. Specifically the developments are: 1. Procedures for conducting an exact test of whether the Weibull shape parameter and a percentile, such as the L10 life, differs significantly among κ (2≤κ≤10) groups of censored life test results. These procedures have application in the very common circumstance wherein one wishes to evaluate the effect of several levels of a design or environmental variable upon fatigue life. Given the results of a censored life test conducted at each level of the variable, it is necessary to decide whether the observed differences in life are real or are explainable by random variation. 2. Procedures for setting confidence limits on the shape parameter and the L10 life of a Weibull distributed failure mode when observations are censored through the failure of the bearing by means of another competing failure mode. These procedures have ap...