Maximum likelihood estimate

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VLE DATA... 

MAXIMUM LIKELIHOOD ESTIMATION OF OAnAMETEPS FoOM VLE DCONTROL PARAMETERS K ERE SET AS FOLLOWS -... 

Enderiein J, Goodwin P M, Van Orden A, Ambrose W P, Erdmann R and Keller R A 1997 A maximum likelihood estimator to distinguish single molecules by their fluorescence decays Chem. Phys. Lett. 270 464-70... 

Figure 7 shows that for the maximum likelihood estimator the variance in the slope estimate decreases as the telescope aperture size increases. For the centroid estimator the variance of the slope estimate also decreases with increasing aperture size when the telescope aperture is less than the Fried parameter, ro (Fried, 1966), but saturates when the aperture size is greater than this value. 

Figure 7. The variance in the slope estimate versus aperture size for the centroid and maximum likelihood estimators for turbulence defined by ro = 0.25.

Weighted regression of U- " U- °Th- Th isotope data on three or more coeval samples provides robust estimates of the isotopic information required for age calculation. Ludwig (2003) details the use of maximum likelihood estimation of the regression parameters in either coupled XY-XZ isochrons or a single three dimensional XYZ isochron, where A, Y and Z correspond to either (1) U/ Th, °Th/ Th and... 

The above implicit formulation of maximum likelihood estimation is valid only under the assumption that the residuals are normally distributed and the model is adequate. From our own experience we have found that implicit estimation provides the easiest and computationally the most efficient solution to many parameter estimation problems. 

This choice of Qi yields maximum likelihood estimates of the parameters if the error terms in each response variable and for each experiment (eu, i=l,...N j=l,...,w) are all identically and independently distributed (i.i.d) normally with zero mean and variance, o . Namely, (e,) = 0 and COV(s,) = a I where I is the mxm identity matrix. 

Excoffier L, Slatkin M. Maximum-likelihood estimation of molecular haplotype frequencies in a diploid population. Mol Biol Evol 1995 12 921-927. 

Finally, it is interesting to note that biases can be introduced by data fitting at low counts even with the use of ordinarily unbiased estimators like the maximum likelihood estimator [37],... 

Darouach, M., Ragot, R., Zasadzinski, M., and Krzakala, G. (1989). Maximum likelihood estimator of measurement error variances in data reconciliation. IFAC, AIPAC Symp. 2, 135-139. 

If it is assumed that the measurement errors are normally distributed, the resolution of problem (5.3) gives maximum likelihood estimates of process variables, so they are minimum variance and unbiased estimators. 

Now, if (m2 > g), the solution of Eq. (10.24), under the assumption of an independent and normal error distribution with constant variance can be obtained as the maximum likelihood estimator of d and is given by... 

One type of common robust estimator is the so-called M-estimator or generalized maximum likelihood estimator, originally proposed by Huber (1964). The basic idea of an M-estimator is to assign weights to each vector, based on its own Mahalanobis distance, so that the amount of influence of a given point decreases as it becomes less and less characteristic. 

If the errors are normally distributed, the OLS estimates are the maximum likelihood estimates of 9 and the estimates are unbiased and efficient (minimum variance estimates) in the statistical sense. However, if there are outliers in the data, the underlying distribution is not normal and the OLS will be biased. To solve this problem, a more robust estimation methods is needed. 

An alternative approach to these methods is to obtain the influence function directly from the error distribution. In this case, for the maximum likelihood estimation of the parameters, the i/ function can be chosen as follows ... 

In order to determine the maximum likelihood estimator of /i, we have... 

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. 

Maximum likelihood estimators also have another desirable property invariance. Let us denote the maximum likelihood estimator of the parameter 6 by single-valued function of 6, the maximum likelihood estimator of f(0) is /([Pg.904]

In the above analysis, y was considered to be a reaction rate. Clearly, any dependent variable can be used. Note, however, that if the dependent variable, y, is distributed with constant error variance, then the function z will also have constant error variance and the unweighted linear least-squares analysis is rigorous. If, in addition, y has error that is normal and independent, the least-squares analysis would provide a maximum likelihood estimate of A. On the other hand, if any transformation of the reaction rate is felt to fulfill more nearly these characteristics, the transformation may be made on y, ru r2 and the same analysis may be applied. One common transformation will be logarithmic. 

If this procedure is followed, then a reaction order will be obtained which is not masked by the effects of the error distribution of the dependent variables If the transformation achieves the four qualities (a-d) listed at the first of this section, an unweighted linear least-squares analysis may be used rigorously. The reaction order, a = X + 1, and the transformed forward rate constant, B, possess all of the desirable properties of maximum likelihood estimates. Finally, the equivalent of the likelihood function can be represented b the plot of the transformed sum of squares versus the reaction order. This provides not only a reliable confidence interval on the reaction order, but also the entire sum-of-squares curve as a function of the reaction order. Then, for example, one could readily determine whether any previously postulated reaction order can be reconciled with the available data. 

The optimal parameter p can be found by maximum-likelihood estimation, but even the optimal p will not guarantee that the Box—Cox transformed values are symmetric. Note that all these transformations are only defined for positive data values. In case of negative values, a constant has to be added to make them positive. Within R, the Box—Cox transformation can be performed to data of a vector Jt as follows ... 

The parameter estimation for the mixture model (Equation 5.25) is based on maximum likelihood estimation. The likelihood function L is defined as the product of the densities for the objects, i.e.,... 

Ordinary least squares regression requires constant variance across the range of data. This has typically not been satisfied with chromatographic data ( 4,9,10 ). Some have adjusted data to constant variance by a weighted least squares method ( ) The other general adjustment method has been by transformation of data. The log-log transformation is commonly used ( 9,10 ). One author compares the robustness of nonweighted, weighted linear, and maximum likelihood estimation methods ( ). Another has... 

Decision Analysis in Medicine Electron Microscope Design Emulsion Chemistry Finite Element Analysis Helicopter Blade Motion Maximum Likelihood Estimation Genetic Studies of Family Resemblance Large Scale Integrated Circuit Design Resolving Closely Spaced Optical Targets... 

This model assumes that any dosage effect has the same mechanism as that which causes the background incidence. Low-dose linearity follows directly from this additive assumption, provided that any fraction of the background effect is additive no matter how small. A best fit curve is fitted to the data obtained from a long-term rodent cancer bioassay using computer programs. The estimates of the parameters in the polynomial are called Maximum Likelihood Estimates (MLE), based upon the statistical procedure used for fitting the curve, and can be considered as best fit estimates. Provided the fit of the model is satisfactory, the estimates of these parameters are used to extrapolate to low-dose exposures. 

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). 

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