Maximum Likelihood ML Estimation

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form, 

If we now further assume that measurements from different experiments are independent, the joint probability density function for the all the measured responses is simply the product, 

The Loglikelihood function is the log of the joint probability density function and is regarded as a function of the parameters conditional on the observed responses. Hence, we have 

At this point let us assume that the covariance matrices (E,) of the measured responses (and hence of the error terms) during each experiment are known precisely. Obviously, in such a case the ML parameter estimates are obtained by minimizing the following objective function 

Therefore, on statistical grounds, if the error terms (e,) are normally distributed with zero mean and with a known covariance matrix, then Q( should be the inverse of this covariance matrix, i.e., 

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The converged parameter values represent the Least Squares (LS), Weighted LS or Generalized LS estimates depending on the choice of the weighting matrices Q,. Furthermore, if certain assumptions regarding the statistical distribution of the residuals hold, these parameter values could also be the Maximum Likelihood (ML) estimates. 

The values of the elements of the weighting matrices R, depend on the type of estimation method being used. When the residuals in the above equations can be assumed to be independent, normally distributed with zero mean and the same constant variance, Least Squares (LS) estimation should be performed. In this case, the weighting matrices in Equation 14.35 are replaced by the identity matrix I. Maximum likelihood (ML) estimation should be applied when the EoS is capable of calculating the correct phase behavior of the system within the experimental error. Its application requires the knowledge of the measurement... 

The deconvolved or restored object that we seek is the most probable number-count set nm. This is called the maximum-likelihood (ML) estimate of the object. It obeys simply... 

The most popular approach is supervised because a region of interest has to be defined in the background of the images in order to extract n samples. Afterward, the choice of the estimator depends on what kind of data is available. For complex images, the optimal maximum likelihood (ML) estimator of a2 is given by28... 

Maximum likelihood (ML) estimation can be performed if the statistics of the measurement noise Ej are known. This estimate is the value of the parameters for which the observation of the vector, yj, is the most probable. If we assume the probability density function (pdf) of to be normal, with zero mean and uniform variance, ML estimation reduces to ordinary least squares estimation. An estimate, 0, of the true yth individual parameters (pj can be obtained through optimization of some objective function, 0 (0 ). ModeP is assumed to be a natural choice if each measurement is assumed to be equally precise for all values of yj. This is usually the case in concentration-effect modeling. Considering the multiplicative log-normal error model, the observed concentration y is given by ... 

Accurate results may be obtained by maximum likelihood (ML) estimation or Bayesian estimation if one is using a formal probability model (e.g., a normal model) and the missing values are MAR when dealing with missing data. Since both ML and Bayesian approaches rely on the complete data likelihood, the function linking the observed and missing data to the model parameters, the probability model is key. 

Equation (87) is a nice demonstration of the maximum-probability principle. After registration of data the estimate of the unknown parameter is chosen, which would give rise to the observed data with greatest probability. This principle is very general and maximum-likelihood (ML) estimation as well as many other reconstruction methods (maximum entropy, etc.) follow from this principle [65]. 

Also under OLS assumptions, the regression parameter estimates have a number of optimal properties. First, 0 is an unbiased estimator for 0. Second, the standard error of the estimates are at a minimum, i.e., the standard error of the estimates will be larger than the OLS estimates given any other assumptions. Third, assuming the errors to be normally distributed, the OLS estimates are also the maximum likelihood (ML) estimates for 0 (see below). It is often stated that the OLS parameter estimates are BLUE (Best Linear Unbiased Predictors) in the sense that best means minimum variance. Fourth, OLS estimates are consistent, which in simple terms means that as the sample size increases the standard error of the estimate decreases and the bias of the parameter estimates themselves decreases. 

The formulation of PE problems is usually based on the concept of maximum-likelihood (ML) estimation. Therefore, one obtains objective functions of a rather specific form. Quite often, the observations of the model output y are assumed to be affected with errors that are normally distributed with zero mean and known covariance matrix U. Then, the minimization of the weighted least-squares term where e denotes the difference between y... 

If the covariance matrices of the response variables are unknown, the maximum likelihood parameter estimates are obtained by maximizing the Loglikeli-hood function (Equation 2.20) over k and the unknown variances. Following the distributional assumptions of Box and Draper (1965), i.e., assuming that i= 2=...=En= , it can be shown that the ML parameter estimates can be obtained by minimizing the determinant (Bard, 1974)... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

Also in Section 3.2, several estimation procedures are defined, such as method of moments (MOM), maximum likelihood (ML), and least squares (LS). Criteria are reviewed that can be used to evaluate and compare alternative estimators. 

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). 

Maximum likelihood ML A general statistical procedure to estimate one or more parameters (e.g., recombination fraction) of a distribution provided that the distribution is specified. 

Left-censored data are characteristic of many bioassays due to the inherent limitation of the presence of a lower limit of detection and quantification. An ad hoc approach to dealing with the left-censored values is to replace them with the Unfit of quantification (LOQ) or LOQ/2 values. Alternatively, one can borrow information from other variables related to the missing values and use MI to estimate the left-censored data. In addition, the left-censored mechanism can be incorporated directly into a parametric model, and a maximum likelihood (ML) approach can be used to estimate the parameters (21). 

Equation (4.2) is called a residual variance model, but it is not a very general one. In this case, the model states that random, unexplained variability is a constant. Two methods are usually used to estimate 0 least-squares (LS) and maximum likelihood (ML). In the case where e N(0, a2), the LS estimates are equivalent to the ML estimates. This chapter will deal with the case for more general variance models when a constant variance does not apply. Unfortunately, most of the statistical literature deals with estimation and model selection theory for the structural model and there is far less theory regarding choice and model selection for residual variance models. 

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 CARES code can estimate Weibull parameters by either maximum likelihood (ML) or le st squares (LS). Since insufficient volume flaw data existed at RT and 1000°C, parameters obtained at these temperatures were neither reasonable nor consistent with each other. The volume flaw... 

Galiatsatou (Fig 38.5) estimates median and 95% confidence intervals of retiu n level estimates for wave heights, when extreme value parameters are estimated with (a) maximum likelihood (ML), (b) Bayesian with noninformative prior distributions, and (c) L-moments (LM) estimation procedures. 

The Weibull parameters m and do are estimated from a sample of strength measurements di, d2,..., cr . The Maximum Likelihood (ML) and the General Linear Regression (GLR)... 

The estimation of AFS/FS-TARMA models is typically accomplished within a maximum likelihood (ML) framework. In the FS-TARMA case, and under Gaussian innovations, the log-likelihood fimction is (Spiridonakos and Fassois 2014b)... 

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