Parametric distribution

Such Bayesian models could be couched in terms of parametric distributions, but the mathematics for real problems becomes intractable, so discrete distributions, estimated with the aid of computers, are used instead. The calculation of probability of outcomes from assumptions (inference) can be performed through exhaustive multiplication of conditional probabilities, or with large problems estimates can be obtained through stochastic methods (Monte Carlo techniques) that sample over possible futures. 

An alternative to choice of a parametric distribution is to rely on a nonparametric distribution. The simplest such distribution is the empirical distribution, which assigns equal probability to each datum in a specified dataset. Considerable... 

If we are to use a log-normal distribution (or any other parametric distribution), values have to be assigned to the parameters, based on data or some rational argument. For the log-normal distribution, given the characterization of /< and a as log-scale mean and standard deviation, an obvious approach is to transform values in some suitable dataset to logarithms and use the sample mean (of the logarithms) to estimate fi, and sample standard deviation to estimate o. However, as for distributions of many types, there is more than 1 reasonable approach for estimating lognormal parameters. Below, a brief account is provided of estimation procedures and criteria for evaluation of estimation procedures. 

Parametric Distributions Useful for Environmental Risk Assessment... 

Maximum likelihood (ML) is the approach most commonly used to fit a parametric distribution (Madgett 1998 Vose 2000). The idea is to choose the parameter values that maximize the probability of the data actually observed (for fitting discrete distributions) or the joint density of the data observed (for continuous distributions). Estimates or estimators based on the ML approach are termed maximum-likelihood estimates or estimators (MLEs). 

When enough data are available, the need to assume a specific parametric distribution can be avoided by using the empirical distribution. The empirical distribution based on n observations is the distribution that assigns equal probability (1/n) to each observed value. A particular focus of a workshop on distribution selection (USEPA 1998) was considerations for choosing between the use of parametric distribution functions. .. and empirical distribution functions. That report of the workshop emphasizes case-specific criteria. 

When a parametric distribution is fitted, each datum contributes to the estimate of each parameter or percentile. Whether this is good or not depends on whether the distribution to be fitted is reasonable. If it is assumed that one can identify the true distribution, the data will be used in a way that is in some sense optimal. In the real world, where the best distribution is uncertain, it may happen that estimated frequencies for one tail of a distribution are sensitive to observations on the other tail, e.g., estimates of high concentration percentiles are sensitive to observed low concentrations. 

Capabilities are available in risk assessment software for inducing rank correlations among variables with arbitrary parametric distributions (Warren-Hicks and Moore 1998 Vose 2000). Also see Vose for a discussion of the envelope method for handling dependencies. 

Statistical tests will have relatively low power. In particular, there will be low power for testing the fit of a parametric distribution. 

The probability of selecting the most appropriate parametric distribution from a set of candidate distributions will be comparatively low. 

The corrected / -values at any given locus can then be obtained using an adjusted distribution that accounts for any inflation observed. The structured association approach differs from the genomic approach in that it estimates the population structure while genomic control assumes a particular parametric distribution of the value of the test statistic (70). Compared with the structured association, the genomic control approach is computationally simple and can be applied to both scanning and validation stages. 

If a parametric distribution (e.g. normal, lognormal, loglogistic) is fit to empirical data, then additional uncertainty can be introduced in the parameters of the fitted distribution. If the selected parametric distribution model is an appropriate representation of the data, then the uncertainty in the parameters of the fitted distribution will be based mainly, if not solely, on random sampling error associated primarily with the sample size and variance of the empirical data. Each parameter of the fitted distribution will have its own sampling distribution. Furthermore, any other statistical parameter of the fitted distribution, such as a particular percentile, will also have a sampling distribution. However, if the selected model is an inappropriate choice for representing the data set, then substantial biases in estimates of some statistics of the distribution, such as upper percentiles, must be considered. 

Structural uncertainties in a distribution can be represented by empirical distributions or a mixture of a discrete and parametric distribution, as described previously. Unless there is a need to partition the distribution into zero and non-zero components, failure to account for the frequency of zero values can lead to overestimation of the mean of the overall distribution. 

There are often data sets used to estimate distributions of model inputs for which a portion of data are missing because attempts at measurement were below the detection limit of the measurement instrument. These data sets are said to be censored. Commonly used methods for dealing with such data sets are statistically biased. An example includes replacing non-detected values with one half of the detection limit. Such methods cause biased estimates of the mean and do not provide insight regarding the population distribution from which the measured data are a sample. Statistical methods can be used to make inferences regarding both the observed and unobserved (censored) portions of an empirical data set. For example, maximum likelihood estimation can be used to fit parametric distributions to censored data sets, including the portion of the distribution that is below one or more detection limits. Asymptotically unbiased estimates of statistics, such as the mean, can be estimated based upon the fitted distribution. Bootstrap simulation can be used to estimate uncertainty in the statistics of the fitted distribution (e.g. Zhao Frey, 2004). Imputation methods, such as... 

Discuss the methods and report the goodness-of-fit statistics for any parametric distributions for input variables that were fitted quantitatively to measured data. 

Murray, D.M. and D.E. Burmaster (1995). Residential air exchange rates in the United States empirical and estimated parametric distributions by season and climatic region. Risk Anal, 15, 459-465. 

The Weibull distribution is called a parametric distribution, i.e. it is an empirical distribution and does not concern itself with the origin of the defects. The Weibull distribution for the strength (cr) of a brittle material takes the following form... 

For the parametric bootstrap instead of resampling with replacement from the data, one constructs B samples of size n, drawing from the parametric estimate of Bpar. Here Fj,ar is the parametric distribution of F. The procedures of interest are then applied to the B samples in the same manner as for the nonparametric bootstrap. 

Finally, mention should be made of the development of information theoretic methods for characterizing product energy distributions. Surprisal analysis183 may offer a means of compacting and parametrizing distributions for a wide range of reactions, by comparing with the statistically expected distribution in each case. 

Trowbridge PR and Burmaster DE (1997) A parametric distribution for the fraction of outdoor soil in indoor dust. J Soil Contamination 6 161-168. 

BUGSAVinBUGS The distributions of structural parameters are used as inputs random interindividual effects are defined by parametric distributions as well. Markov Chain Monte Carlo (MCMC) methods are used to generate posterior probabilities. WinBUGS is the windows GUI-version of the DOS-based BUGS. 

Nonparametric analysis provides powerful results since the rehahility calculation is unconstrained to fit any particular pre-defined lifetime distribution. However, this flexibility makes nonparametric results neither easy nor convenient to use for different purposes as often encountered in engineering design (e.g., optimization). In addition, some trends and patterns are more clearly identified and recognizable with parametric analysis. Several possible methods can be used to fit a parametric distribution to the nonparametric estimated rehability functions (as provided by the Kaplan-Meier estimator), such as graphical procedures or inference procedures. See Lawless (2003) for details. We choose in this paper the maximum likelihood estimation (MLE) technique, assuming that the sateUite subsystems failure data are arising from a WeibuU piobabihly distribution, as expressed in Equations 1,2. 

From a statistical point of view, the validity of the test results based on a low test data volume and the distribution heterogeneities is considerable. For example parametrical distribution models are unverifiable therefore parametric significance tests are not applicable. 

Often we have data from several populations that we believe follow the same parametric distribution (such as the normal distribution), but may have different values of the parameter (such as the mean). The classical frequentist approach would be to analyze each population separately. The maximum likelihood estimate of the parameter for each population would be estimated from the sample from that population. Simultaneous confidence intervals such as Bonferroni, Tiikey, or Scheff6 intervals would be used for the difference between different population parameter values. These wider intervals would control the overall confidence level, and the overall significance level for testing the hypothesis that the differences between all the population parameters are zero. However, these intervals don t do anything about the parameter estimates themselves. 

Since there is significant uncertainty (aleatory and epistemic) in estimating building losses at a given intensity of GM, VFs provide the loss distribution parameters as a function of intensities. It is generally assumed that the loss distribution can be represented by parametric distribution... 

In a real-world structural identification application, where no information is available regarding the true pdfs of the input random vector, someone could use maximum likelihood estimation fitting of the environmental condition data values to a parametric distribution. The results of such a fitting of the data onto pdfs are shown in Fig. 5. Based on this fitting and after transforming the pdf of the mass load into a normal distribution by using the natural logarithm, the Hermite polynomials may be selected for the construction of the multivariate PC basis functions. 

In work [3] it is shown, that probability of size realization of any engineering property of the CRL connected with presence of pores, cracks, stratifications, etc of infringements of a polymer matrix in it is described by three-parametrical distribution of a kind ... 

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