True error

If the assumption of normality is grossly violated, ML estimates of the parameters can only be obtained using the error-in-variables" method where besides the parameters, we also estimate the true (error-free) value of the measured variables. In particular, assuming that Ey i is known, the parameters are obtained by minimizing the following objective function... 

The above equation cannot be used directly for RLS estimation. Instead of the true error terms, e , we must use the estimated values from Equation 13.35. Therefore, the recursive generalized least squares (RGLS) algorithm can be implemented as a two-step estimation procedure ... 

To reject the null hypothesis erroneously although it is true (error of first kind, false-negative, risk a). 

Not to reject the null hypothesis by erroneously though the alternative hypothesis is true (error of second kind, false-positive, risk / ). 

To make a more formal assessment of the possible errors (a combination of numerical modeling errors) in the calculations of DREAM-SOFC, here we treat the other eight results as outcomes from eight repetitions of a virtual experiment, the differences being a result of experimental uncertainty. With this assumption, we can calculate the 90% confidence interval for the true error in the DREAM results using the t-distribution as ... 

The point (Xj, Yj) denotes the i-th observation. The true error or residual is Yj-(Po+PiXi), the difference between the observed Y and the true unknown value Po+PiXj. The observed residual ej is Yj-fbo+l Xj Y — Y(, which is the difference between the observed Yj and the estimated Yi = b0 + blXi. The problem is now to obtain estimates b0 and bi from the sample for the unknown parameters p0 and Pi. 

Signihcant Data (if this is to be distinguished from critical data) are required to have a predetermined acceptable accuracy (e g., has a maximum of 5% error rate). This can be established by a randomly drawn sample so long as a small risk is accepted that, even though the sample strongly indicates that the error rate is below the predetermined acceptable level, in fact the true error rate is above the predetermined acceptable level. This is an inevitable consequence of using a sample. The only alternative is a 100% check, as above. 

The objective of statistical samphng is to establish likely values for the true error rate in the population of data being considered. If the tme error rate was known, the probabihties of given numbers of errors in samples could be obtained mathematically using standard statistical distributions. Statistical inference allows the reverse process — from an observed error rate in a sample likely and possible true error rates can be inferred. Likely data population error rates are defined by the 99% single upper confidence limit, and possible data population error rates by the 99.9% single upper confidence limit on the sample error rate. 

TABLE 14A.1 Likely "True" Error Rates (%) for Observed Error Rates (%) in Samples of Given Sizes with Target Errors Rate of at Most 5% observed Error Rate in Sample ... 

Note %. True error rate is defined at 99% single upper confidence limit. 

Any errors in the simulation package would be more likely to result in test failures rather than mask a true error iu the DCS configuration. 

Values (literature cf. ref. 63) determined from reversion frequencies of mutants and hence represent true error rates (to be distinguished from population numbers of mutants, i.e., so-called mutant frequencies). 

It seems clear that until new data address the unresolved systematic errors afflicting the derivation of the primordial helium abundance, the true errors must be much larger than the statistical uncertainties. For the comparisons between the predictions of SBBN and the observational data to be made in the next section, I will adopt the Olive, Steigman Walker (2000 OSW) compromise Yp = 0.238 0.005 the inflated errors are an attempt to account for the poorly-constrained systematic uncertaintiess. 

But 2 (a2) is the sum of the squares of the true errors. The true errors are unknown. By the principle of least squares, when measurements have an equal degree of confidence, the most probable value of the observed quantities are those which render the sum of the squares of the deviations of each observation from the mean, a minimum. Let 2(va) denote the sum of the squares of the deviations of each observation from the mean. If n is large, we may put 2(a 2) = S(v2) but if n is a limited number,... 

Having determined the value of s, the true standard deviation of the average value is related to the true error for an infinite simulation by ... 

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