Maximum likelihood estimators, table

Comment. Logistic tumor prevalence method is unbiased. Requires maximum likelihood estimation. Allows for covariates and stratifying variables. It may be time-consuming and have convergence problem with sparse tables (low tumor incidences) and clustering of tumors. 

A comparison of the various fitting techniques is given in Table 5. Most of these techniques depend either explicitly or implicitly on a least-squares minimization. This is appropriate, provided the noise present is normally distributed. In this case, least-squares estimation is equivalent to maximum-likelihood estimation.147 If the noise is not normally distributed, a least-squares estimation is inappropriate. Table 5 includes an indication of how each technique scales with N, the number of data points, for the case in which N is large. A detailed discussion on how different techniques scale with N and also with the number of parameters, is given in the PhD thesis of Vanhamme.148... 

Maximum likelihood estimates for the parameters of the normal distributions of differences between log(median CDio) estimates in different species shown in Figures 27.4-27.6 are shown in Table 27.1. The same comparisons are shown graphically in Figure 27.8, plotted against the logarithm of body weight ratio for the... 

Representative results are shown in Table VIII for T = 50°C pH = 5.0. The manganese concentration was varied from 0.1 to 1.0 ppm, values which are representative of those found in scrubber liquors (12). The concentration of manganese is the total value, l.e., the sum of that contributed by the calcium sulfite and that added to the solution. These results were also fit to an expression of the form of equation (2) with the maximum likelihood estimates being ... 

Table 1. Simulated Maximum-likelihood estimates of logit models...

Table 1. Maximum likelihood estimations and standard deviations for reliability and maintenance parameters in valve and actuator.

The direction data mainly come from the scan lines, which are divided into three groups, (SLOl-17, SL17-30 and SL31- 0) for the three discontinuity sets, respectively. The mean and variance of the discontinuity dip directions and dip angles are evaluated by the maximum likelihood estimation method (Table 1). Then, the frequency histograms of the orientation data are drawn out and combined with the x test the probability distribution functions of the... 

In Table 1 the value of -<5/b obtained from Maximum Likelihood Estimation (MLE) is oonpared with the value obtained from the experimental slope on the log(strencrth) vs. log(diameter) plots. Since the agreement is very good, the suggested failure distribution (equation 1) resolves the well-known problem of the discrepancy between the values of b from different sources in sinpler Weibull models. 

Thus, for any laboratory model of a structure undergoing wide-band gaussian excitations on a shake table, the statistically equivalent linear model for a given vector of outputs can be obtained directly from data analysis. The question that still remains concerns the quality of the linearized model as compared to the true non- linear model of the structure. We would like to be able to say that because we have shown the statistical linearization coefficients to be the asymptotic maximum likelihood estimates of the coefficients of a linear model, then the linear model is in some sense a projection of the true non-linear model onto linear model space. 

Similarly, the estimators of the mean and standard deviation using maximum likelihood estimation method for curtailed time data is fi= 19.4409 and(T = 1.3110. And the critical value r , is 0.5714 when significance level takes 0.1. The calculation process is also shown in Table 3. 

These results are maximum likelihood estimates for the truncated regression by country. Significance at the 1% level is shown by, at the 5% level by, at the 10% level by. The figures for France are not reliable and are shown only for completeness, see note in Table 11. 

A leading utility in tlie nortlieast had requested out-of-compliance information to better schedule outages (plant shutdowns). The time to failure, T, of a bus section was assumed to have a Weibuill distribution, tlie parameters of which were estimated by the metliod of maximum likelihood on tlie basis of observed bus section failures shown in Table 21.6.1 for tlie utility s 5x8... 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variances, we refer %2 = 401n.58333 - 201n.847071 - 201n.320506 to the chi-squared table. (Under the null hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, %2 = 4.5164, which is roughly equal to the LM statistic and leads once again to rejection of the null hypothesis. 

Another way of estimating mean (and median) WT P is to use some parametric method. This involves an assumption that the distribution of yes answers follows a specific probability model. The most commonly employed model in CVM studies is the logit model. The results of the estimation of a simple logit model are found in Table 6.7. Individual data were used for the estimation, and the dummy variable BI DYES takes the value of unity in the case of acceptance of a bid, and zero otherwise. The explanatory variable BID LIRE is simply the bids in thousands of ITL. The estimation was done by the LOGIT command of Limdep 6.0, which implied the use ofthe maximum likelihood (ML) method (see Greene, 1991, p. 484). It is evident from the table that the coefficient of BIDLIRE is... 

All the objective functions shown in Table 15.1 are derived from a least-squares regression approach as previously described, whereas the estimation method more commonly used in population pharmacokinetics and nonlinear mixed effect modeling in general is based on a maximum likelihood (ML) approach. ML is an alternative to the least-squares objective function it seeks to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function). In general terms, the likelihood function is defined as... 

The third (default) method uses a table of empirically observed transitions between amino acids (the Dayhoff PAM 001 matrix). The character-based analysis of sequence data can be initiated via the appropriate executable file (e.g. DnaPars, DnaML or ProtPars). PHYLIP comprises DnaPars and DnaML to estimate phylogenetic relationships by the parsimony method and the maximum likelihood methods from nucleotide sequences respectively. ProtPars is the parsimony program for protein sequences. 

Table 1 presents the parameters for the costs and availability calculations for each component of the system, for which initial virtual ages (Fq) have also been considered. These values were estimated from data obtained from a typical nuclear power plant using a maximum likelihood approach, as in Vanes et al (2002). Other approaches for estimating the parameters can be found in Moura et al (2008), and in Jacopino (2005). 

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