Fractile

A scale parameter determines the location of fractiles of the distribution relative to some specified point, often the value of the location parameter. 

Fox, C. L., 22 679 Foxing, 11 409 FPAT file, 13 230, 235 Fractal gelation model, 23 63-64 Fractal objects, formation of, 23 63 Fractile distribution, 26 1021 Fractional carbonation, gallium extraction by, 12 345... 

Statisticians have introduced the notion of fractile (quantile), which is a special argument of the density function. The quantile, xq, of order q (0 [Pg.32]

Let us have another look at the table of Appendix B and understand why we may need two -values with the same confidence level P and the same total risk a. If we compare columns 2 and 3 we note that one can use a fractile of the order q = 0.975, which means that a risk oe/2 remains that both sides of the symmetrical region around the estimated parameter do not include the true value. On the other hand the fractile of the order g = 0.95 may be useful for constructing a one-sided interval hence concentrating the risk a on one side. 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

It has been reported (Krause et al., 1987 1991) that in German established buildings (private homes,) the arithmetic mean TVOC concentration is in the order of 400 pg/m, whereas the 50 % fractile is in the order of 320 pg/m. ... 

The distinction between confidence intervals and toleranee intervals is (hat civnlidenee interv als refer to estimates of the population statistics (usually the mean) while tolerance intervals are concerned with proportions or fractiles of the population. Thus the term tolerance as used here should be distinguished from the frequently used tolerance in engineering design for dimensions and tUher factors in the construction or manufacture of some object or structure. 

The comparsion with the critical value k = Ff,oD,i.a-fractil (dependent on the significance level a) decides on the acceptance of Ho... 

If, however, more accurate information on the distribution of measured values is to be given, for example to calculate the 5% fractile, then significantly more tests are required. 

If, however, the 5% fractile can be given safely, this value can then be used to estabhsh a reduction factor for the inhomogeneity of the fabric. 

The chi square fractile values are calculated and put into a table. In order to evaluate the test results the values will be specified with respect to an agreed confidence level a = 95% and an appropriate number of degrees of freedom put in the Annex 1. 

The fractile Up of the standardised basic variable of the maximum value distribution (dependent on the parameter C only) is given as... 

The climatic actions may often be described by the Extreme value distributions. The characteristic value of a climatic action is defined in Eurocodes as the upper fractile of the probabilistic distribution for the basic time period corresponding to the 2% probabihty of atmual exceeding. The design value of a climatic action is considered as 0,996% fractile of the probabilistic distribution for structures in the reUabUity class RC2. 

For design and production purposes the specified characteristic strength f corresponds to the 5%-fractile of the theoretical strength distribution of the considered concrete class (Fig. 1). In practice, the fraction below the specified fr, will he smaller or hi er than 5%. Designating by 9 the fiaction of test results below fr, in the offered strength distribution (Fig. 1), it follows that ... 

Schweizerische Bimdeshahnen AG (Swiss Federal Railway company) Accumulated operating test time Train Recorder Unit Significance level 100(1 - a) % is the confidence level at which confidence intervals and limits are calculated The a fractile of the cumulative /2 (chi- square) distribution with v degrees of freedom... 

P(l) A material property shall wherever possible, be represented by a characteristic value X ( corresponding to a fractile or other hmiting condition in the assumed statistical distribution of the particular property of the material, specified by relevant standards and tested under specified... 

CR = cj(c + cj = 0.571. The issue now is to translate this critical ratio into a critical fractile of the demand distribution. Recall from the preceding text that D N(10,000,3,121). Thus, another table look-up (or Excel computation) gives us Q = 10,558, the value that accumulates a total probability of 0.571 under the normal distribution pdf with mean 10,000 and standard deviation 3,121. 

A less rigorous approach to finding a (Q, R) solution would be to solve for Q and R separately. Note that z = R - MoltV dlt gives a fractile of the distribution of demand over the lead time. Thus, we could set R to achieve a desired in-stock probability, along the lines of the newsvendor problem solution discussed earlier (i.e., to accumulate a given amount of probability under the DLT distribution). In this setting, the in-stock probability is typically referred to as the cycle service level (CSL), or the expected in-stock probability in each replenishment cycle. Specifically, for normally distributed lead-time demand, DLT, we set... 

The approximation developed by Shang and Song (2003) uses echelon holding costs, as defined earlier, and starts from the optimal echelon N base-stock level, which—by the authors reference to Chen and Zheng (1994)— is given by computing the critical fractile... 

Shang and Song demonstrate that approximate solutions can be found for each upstream echelon e 1,2,..., N - 1 in the serial supply chain from two easily computed newsvendor-type fractiles, Q] and 0 , which they prove to be... 

When there is a tradeoff between the holding cost and the penalty cost for failing to immediately satisfy uncerfain demand, we should employ a "critical fractile-based" solution to exploit this tradeoff. 

For a specified internal pressure and test temperature the measured times-to-failure exhibit a logarithmic normal distribution. To attribute a certain time-to-failure to the applied test condition (hoop stress and temperature) a fractile of this distribution is chosen, for example, the 5 % ffac-tile, i.e. 95 % of the pipes will have time-to-failure at the specified test... 

The standard gives a formula for the calculation of n assuming that a reliable estimate of the hue coefficient of variation is known, for example, from qnality control of the manufacturer. Let c be the coefficient of variation of a very large number N of laboratory quality control measurements of thickness on randomly taken specimens. Let 197 5 be the 97.5-fractile of the /-distribntion for A degrees of freedom. The number n can then be calculated according to n = (19 5 c/0.05). A reliable estimate is obtain for , if A n. 

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