Estimator sampling distribution

Torrie G M and Valleau J P 1977 Nonphysical sampling distributions In Monte Carlo free energy estimation umbrella sampling J. Comput. Phys. 23 187-99... 

Sample quantity to estimate moisture for specific material is influenced to various levels of significance by properties such as particle-size range as well as relative amounts or moisture distributed among denoted forms of retention. Practical sample size estimates require background knowledge of parameters derived from experience for specific materials. More detailed examination of moisture-sampling aspects is provided in reference texts (Pitard). 

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. 

The molten salt standard program was initiated at Rensselaer Polytechnic Institute (RPI) in 1973 because of difficulties being encountered with accuracy estimates in the NBS-NSRDS molten salt series. The density, surface tension, electrical conductivity, and viscosity of KNO3 and NaCl were measured by seven laboratories over the world using samples distributed by RPI. The data from these round-robin measurements of raw properties were submitted to RPI for evaluation. Their recommendations are summarized in Table 2. 

As a guide for the reader, Table 1 lists some features of the experimental techniques discussed in this chapter. For each method it is specified whether it allows the determination of an average molar mass and also of the distribution. Estimates of the molar mass range, typical sample amounts and the operating... 

Torrie, G. M. Valleau, J. R, Nonphysical sampling distributions in Monte Carlo free energy estimation Umbrella sampling, J. Comput. Phys. 1977, 23, 187-199... 

Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

The only way to test it is by making a repeat measurement after the estimated time necessary for the sample distribution has expired. There are two possible explanations for differences in the result ... 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

Frequentist Criteria for Evaluating Estimators, the Sampling Distribution... 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

In general, bias refers to a tendency for parameter estimates to deviate systematically from the true parameter value, based on some measure of the central tendency of the sampling distribution. In other words, bias is imperfect accuracy. In statistics, what is most often meant is mean-unbiasedness. In this sense, an estimator is unbiased (UB) if the average value of estimates (averaging over the sampling distribution) is equal to the true value of the parameter. For example, the mean value of the sample mean (over the sampling distribution of the sample mean) equals the mean for the population. This chapter adheres to the statistical convention of using the term bias (without qualification) to mean mean-unbiasedness. 

Parameter estimates will have high statistical error. In principal, this can be accounted for by use of 2D methods, which make use of a parameter uncertainty distribution. However, approximations of the sampling distribution may be relatively poor at small sample sizes. 

Bootstrap sample A sample (e.g., 5000) obtained from an original data set by randomly drawing, with replacement, 5000 values from the original sample or a distribution estimated for that sample. 

To test the applicability of statistical techniques for determination of the species contributions to the scattering coefficient, a one-year study was conducted in 1979 at China Lake, California. Filter samples of aerosol particles smaller than 2 ym aerodynamic diameter were analyzed for total fine mass, major chemical species, and the time average particle absorption coefficient, bg. At the same time and location, bgp was measured with a sensitive nephelometer. A total of 61 samples were analyzed. Multiple regression analysis was applied to the average particle scattering coefficient and mass concentrations for each filter sample to estimate aj and each species contribution to light scattering, bgn-j. Supplementary measurements of the chemical-size distribution were used for theoretical estimates of each b pj as a test of the effectiveness of the statistical approach. 

Elemental mass distribution - The aerosol sampled by the LPI for elemental analysis was impacted on coated mylar films affixed to 25 mm glass discs. The mylar had been coated with Apiezon L vacuum grease to prevent particle bound. The LPI samples were sent to Crocker Nuclear Laboratory for elemental analysis by PIXE using a focused alpha particle beam of 3 to 4 mm diameter. Nanogram sensitivities for most elements were achieved with the focused beam. A detailed description of the PIXE focused beam technique applied to LPI samples can be found in Ouimette (13). Based upon repeated measurements of field samples, the estimated measurement error was about 15-20% or twice the minimum detection limit, whichever was larger. 

Samples distributed for analysis should have known quantity values with estimated measurement uncertainties, although, of course, these are not released to the participants until after the tests. The value of the measurand ( assigned) can be assigned by... 

Figure 5. Results from a multicanonical simulation of the 3D Lennard-Jones fluid at a point on the coexistence curve. The figure shows both the multicanonical sampling distribution PAP) (symbol o) and the corresponding estimate of the equilibrium distribution Ro(p) with p = N/V the number density (symbol ). The inset shows the value of the equilibrium distribution in the...

The extent to which any chosen sampling distribution (weight function) meets our requirements is reflected most directly in the M-distribution it implies. One can estimate that distribution from a histogram H(M) of the macrostates visited in the course of a set of MC observations. One can then use this information to refine the sampling distribution to be used in the next set of MC observations. The simplest update algorithm is of the form [122]... 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

Normally the true parameters (so-called parent population parameters) of distributions are not known. For empirical distributions they have to be estimated (symbol A) on the basis of a limited number, n, of observations (so-called sample parameters). Estimates of the most important parameters are ... 

If the sample is representative, then the second consideration is how artefacts of randomly choosing the samples lead to random variation in the estimate of statistics of interest, such as the mean or standard deviation of sample distribution of interindividual variability. While it is often assumed that a small sample of data is not or could not be representative, this is a common misconception. The issue of representativeness is one of study design and sampling strategy. If one has a representative sample, even if very small, then conventional statistical methods can be used to make inferences regarding sampling distributions of statistics, such as... 

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. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

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