Maximum Probable Error

Emulsion A has a droplet size distribution that obeys the ordinary Gaussian error curve. The most probable droplet size is 5 iim. Make a plot of p/p(max), where p(max) is the maximum probability, versus size if the width at p/p(max) = j corresponds to... 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

One measure of the quality of an estimate of an average Is the confidence limits (or maximum probable error) for the estimate. For averages of Independent samples, the maximum probable error Is... 

Equation 4 also can be evaluated using the nomograph. For example, to determine the maximum probable error that will occur with 95% probability based on n 4 tests when o 20 ppm, first find the point where the diagonal and the line through n 4 and o 20 intersect then extend the line through this point and P 95% to find E 19.6 ppm. 

The electron-diffraction measurements reproduce the twist angle to 42° with an error estimate of about 5 °. This value corresponds to the angle of maximum probability. An exact location of the minimum of the potential function is not easy to derive with present knowledge, but the discrepancy between the electron-diffraction value and the one obtained by ab initio calculation is too large to be accepted. 

Weak control of the FWER, also referred to as the experimentwise error rate in some traditional statistics books, controls the maximum probability of rejecting at least one null hypothesis H0i when all Ho, = 1,..., g, are true. Weak control of the FWER is inadequate in practice because, if there exists at least one differentially expressed gene, then there is no guarantee that the probability of incorrectly inferring the nondifferentially expressed genes as differentially expressed is controlled. 

In Chapter 7 we saw that the maximum possible error allows us to estimate the worst case scenario. The maximum probable error, however, takes into account the fact that this rarely is the case, with the error usually being somewhat smaller. It is straightforward to calculate the maximum probable error using the following formulae. In each case we assume that the quantities X and Yare measured, and are used to calculate Z. The estimated absolute errors on each quantity are AX, AY and AZ respectively. 

In a particular hydrogenlike atom, spectral lines appear with wavelengths 19.440 0.007 nm and 17.358 0.005 nm. What is the maximum probable error in the difference between these wavelengths ... 

Maximum probable error 0.7 kcal. Maximum probable error 1.5 eu. In carbon tetrachloride. 

Maximum probable error in is 0.7 kcal.mole" maximum probable error in A5t is 1.5 eu. 

FIGURE 8.6 For any native phase angle p the vector triangle T ru = F/> + fn fails to close by an amount e(p). which is termed the lack of closure error. The phase angle of maximum probability, that for which e(p). is minimized when considered over all isomorphous heavy atom derivatives. 

Table 11.5 Maximum probability of committing a type I error when each hypothesis is tested at a = 0.05...

C No. of hypotheses tested at a = 0.05 Maximum probability of type 1 error... 

The resulting temperature profile is shown in Figure 2. The temperature in the primary reaction zone was estimated by linear extrapolation to 300 K at the burner. Propagation of error calculations suggests a maximum probable error of +150 K. 

Notice that (with prescribed t) tSjy represents the maximum probable error associated with the estimate of (3. This can be expressed as a percentage error (where engineers very often seek estimates that are within 5 or 10%). Using b as the base (in the absence of knowing j8), let E represent the potential percentage error associated with a t value of 2, an approximate value that is never more than 2% in error for df > 30. (For df < 30, substitution of the correct value is advised.) Then... 

The first two terms in equations (12.7) and (12.8) estimate the flow time, and the last term is a waiting time allowance based on the forecast error. Normal probability tables can be used to choose the 7 value to satisfy the service level constraint on the maximum number of tardy jobs. 

Error Maximum number of iterations exceeded in nsolv Results are probably not accurate ... 

The theoretical grounds for parameter estimation are based on the principle of maximum likelihood, which states if an event has occurred, the maximum probability should have corresponded to its realization. As a consequence, the estimators maximizing the likelihood function possess certain optimal properties (Johnson and Leone, 1977, Chap. 7 Linnik, 1961, Chap. 3). The likelihood function is composed of the distribution functions of the errors. When no information on the type of distribution is available, it is usually assumed to be normal, justified by the results of the central limit theorem (Box et al, 1978, p. 44 Linnik, 1961, pp. 71-74). For normally distributed errors, maximization of the likelihood function is reduced to minimization of the residuals, i.e., to the least-squares methods. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

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