Technology & Engineering
Maximum Likelihood Estimation
Maximum Likelihood Estimation is a statistical method used to estimate the parameters of a model. It aims to find the values of the model's parameters that maximize the likelihood of observing the given data. This technique is widely used in fields such as engineering and technology to make inferences about unknown parameters based on observed data.
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7 Key excerpts on "Maximum Likelihood Estimation"
eBook - PDF- Ian C. Marschner(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
In our discussions about estimation in Chapter 2, we relied on being able to choose estimates that made intuitive sense based on the situation under study. For example, we used the sample mean to estimate the population mean, and the sam-ple variance to estimate the population variance. In more complicated situations, it is usually not possible to find an intuitively natural estimate such as the sample mean or variance, and the principle of Maximum Likelihood Estimation provides a generally applicable criterion on which to base our parameter estimate. While it is straightforward to define the MLE as the parameter value that achieves the highest likelihood, there are a number of potentially important details that we need to be aware of. The discussion above is framed as though there could only ever be one value of the parameter that maximises the likelihood function. While it is often the case that there is only one unique maximum for the likelihood function, it is possible in complicated situations for there to be two or more values of the parameter that achieve the highest possible value for the likelihood function. In such situations we say that the MLE is not unique, and it may not be clear which parameter value to use as our estimate. While this is a cause for concern when it arises, it turns out that in most contexts of interest the MLE is unique and this potential complication does not arise. Another important detail in relation to our definition of the MLE is our choice of the parameter space, that is, the set of all possible values that the parameter can have. When finding the MLE we must consider only those values of the parameter that are within the parameter space, even though the function L ( θ ) may be evaluable for values of θ that are not in the parameter space.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
Indeed, estimates the expected log-likelihood of a single observation in the model. The method of maximum likelihood estimates θ 0 by finding a value of θ that maximizes . This method of estimation is a maximum likelihood estimator ( MLE ) of θ 0 : A MLE estimate is the same regardless of whether we maximize the likelihood or the log-likelihood function, since log is a monotone transformation. For many models, a maximum likelihood estimator can be found as an explicit function of the observed data x 1 , …, x n . For many other models, however, no closed-form solution to the maximization problem is known or available, and a MLE has to be found numerically using optimization methods. For some problems, there may be multiple estimates that maximize the likelihood. For other problems, no maximum likelihood estimate exists (meaning that the log-likelihood function increases without attaining the supremum value). In the exposition above, it is assumed that the data are independent and identically distributed. The method can be applied however to a broader setting, as long as it is possible to write the joint density function ƒ ( x 1 ,…, x n | θ ), and its parameter θ has a finite dimension which does not depend on the sample size n . In a simpler extension, an allowance can be made for data [[Homogeneity (statistics) |heterogeneity]], so that the joint density is equal to ƒ 1 ( x 1 | θ ) · ƒ 2 ( x 2 | θ ) · … · ƒ n ( x n | θ ). In the more complicated case of time series models, the independence assumption may have to dropped as well. A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters. Properties Maximum likelihood is the extremum estimator based upon the objective function ________________________ WORLD TECHNOLOGIES ________________________ and its sample analogue, the average log-likelihood . The expectation here is taken with respect to the true density f (·| θ 0 ).
eBook - ePubStatistical Theory
A Concise Introduction
- Felix Abramovich, Ya'acov Ritov(Authors)
- 2022(Publication Date)
- Chapman and Hall/CRC(Publisher)
One would evidently be interested in “good” estimators. In this chapter, we firstly present several methods of estimation and then define and discuss their goodness. 2.2 Maximum Likelihood Estimation This is probably the most used method of estimation of parameters in parametric models. Its underlying idea is simple and intuitively clear. Recall that given a random sample Y 1,..., Y n ~ f θ (y), θ ∈ Θ, the likelihood function L (θ ; y) defined in Section 1.2 is the joint probability (for a discrete random variable) or density (for a continuous random variable) of the observed data as a function of an unknown parameter(s) θ : L (θ ; y) = ∏ i = 1 n f θ (y i). As we have discussed, L (θ ; y) is the measure of likeliness of a parameter’s values θ for the observed data y. It is only natural then to seek “the most likely” value of θ, and this is the idea behind the Maximum Likelihood Estimation. Definition 2.2 (maximum likelihood estimator) The maximum likelihood estimator (MLE) θ ^ of θ is θ ^ = arg max θ ∈ Θ L (θ ; y) —the value of θ, which maximizes the likelihood. Finding MLEs is, therefore, essentially an exercise in calculus. In practice, it is usually easier to find a maximum of a log -likelihood function l (θ ; y) = ln L (θ ; y) = ∑ i = 1 n ln f θ (y i). Question 2.1 Explain why a maximizer of the log-likelihood function is an MLE of θ. Question 2.2 Show that an MLE is necessarily a function of a sufficient statistic for θ. A series of examples below demonstrate various possible scenarios and describe the main properties of MLEs. Example 2.1 (binomial distribution) Consider again the birth data example, where Y i ~ B (1, p), i = 1,..., n and p is the probability of a. newborn girl. As we have seen in Example 1.3, the likelihood function L (p ; y) is L (p ; y) = p ∑ i = 1 n y i (1 − p) n − ∑ i = 1 n y i and the
- R. J. de Ayala(Author)
- 2022(Publication Date)
- The Guilford Press(Publisher)
86 4 Marginal Maximum Likelihood Parameter Estimation In this chapter, we present an alternative to JMLE that separates the estimation of item parameters from that of person parameters. This alternative, marginal maximum like- lihood estimation (MMLE), estimates only item parameters. 1 As such, after obtaining item parameter estimates and achieving satisfactory model–data fit, one typically pro- ceeds to estimate the person parameters using either MLE (see Appendix A) or a Bayes- ian approach. We begin this chapter with a conceptual presentation of MMLE, and then we discuss the Bayesian expected a posteriori method for estimating person locations. As we did in Chapter 3, we apply these methods to the data introduced in Chapter 2. Our fit analysis involves a graphical comparison of the empirical and predicted IRFs, with the items’ empirical IRFs providing us with evidence supporting the tenability of the model’s functional form assumption. We end by discussing the conversion of estimates from one metric to a different metric, and we introduce the concept of an instrument’s characteristic function. Marginal Maximum Likelihood Estimation In JMLE one is simultaneously determining the item and person parameter estimates that maximize the joint likelihood of the observed data. This approach has a number of practical implications. First, there is the statistical issue of trying to simultaneously estimate the item parameters and the person parameters. Andersen (1970) referred to this issue as estimating structural parameters (the item parameters) concomitant with incidental parameters (the person parameters). The gist of this issue is that one way of obtaining better (e.g., consistent, efficient) item parameter estimates is to increase the sample size. However, in JMLE increasing the sample size leads to an increase in the number of person parameters to be estimated.
eBook - PDF- Peter Westfall, Kevin S. S. Henning(Authors)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
Another use of the likelihood function is to provide a specific estimate q ˆ of q . The value q ˆ that maximizes the likelihood function L ( q | y 1 , y 2 , …, y n ) is called the MLE of q . L 0.182 0.122 0.061 0.000 32 28 24 uni03BC 20 2.00 4.67 7.33 10.00 σ FIGURE 12.8 The joint likelihood (×10 20 ) of ( m , s ) assuming the age data came from a normal distribution. 319 Likelihood Function and Maximum Likelihood Estimates Definition of the MLE If L ( q ˆ | y 1 , y 2 , …, y n ) > L ( q | y 1 , y 2 , …, y n ) for all permissible q , then q ˆ is the MLE of q . You can find the MLE in various ways. One is by inspection : Just look at the graph! That is essentially what we did in Section 12.2—we looked at the graphs and located their peaks. You can make this a little more precise by calculating the likelihoods for a list of values q , then sorting the likelihoods from largest to smallest, and then picking the q that gives you the largest. For example, using the waiting time likeli-hood where L ( l | y 1 , y 2 , …, y 10 ) = l 10 e −10 λ (2.0) , consider the Excel screenshots shown in Figures 12.9 and 12.10. After sorting, Figure 12.9 becomes Figure 12.10, which agrees with Figure 12.3, where l ˆ = 0.5 was identified as the value that maximized the likelihood function. While inspection is simple and intuitively appealing, it does not necessarily find the precise value. What if the real maximum were between 0.50 and 0.51? You can choose finer increments such as 0.001 and get an answer that is closer, but the question would remain, what if the real maximum is between 0.500 and 0.501? Further, inspection becomes less useful when there are multiple parameters, as the list of possible combinations of values increases multiplicatively with each additional parameter. A better solution for finding the MLE is to use calculus.
eBook - PDFEconometric Modelling with Time Series
Specification, Estimation and Testing
- Vance Martin, Stan Hurn, David Harris(Authors)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
PART ONE Maximum Likelihood 1 The Maximum Likelihood Principle 1.1 Introduction Maximum Likelihood Estimation is a general method for estimating the para-meters of econometric models from observed data. The principle of maximum likelihood plays a central role in the exposition of this book, since a num-ber of estimators used in econometrics can be derived within this framework. Examples include ordinary least squares, generalised least squares and full-information maximum likelihood. In deriving the maximum likelihood estim-ator, a key concept is the joint probability density function (pdf) of the observed random variables, y t . Maximum Likelihood Estimation requires that the follow-ing conditions are satisfied. (1) The form of the joint pdf of y t is known. (2) The specifications of the moments of the joint pdf are known. (3) The joint pdf can be evaluated for all values of the parameters, θ . Parts ONE and TWO of this book deal with models in which all these condi-tions are satisfied. Part THREE investigates models in which these conditions are not satisfied and considers four important cases. First, if the distribution of y t is misspecified, resulting in both conditions 1 and 2 being violated, estima-tion is by quasi-maximum likelihood (Chapter 9 ). Second, if condition 1 is not satisfied, a generalised method of moments estimator (Chapter 10 ) is required. Third, if condition 2 is not satisfied, estimation relies on nonparametric meth-ods (Chapter 11 ). Fourth, if condition 3 is violated, simulation-based estimation methods are used (Chapter 12 ). 1.2 Motivating Examples This section emphasises the link between observed sample data and the prob-ability distribution from which they are drawn. This relationship is illustrated with a number of simulation examples in which samples of size T = 5 are 3
- George G. Roussas(Author)
- 2003(Publication Date)
- Academic Press(Publisher)
.In a general setting, let X 1 ,…,Xnbe i.i.d. r.v.’s with p.d.f. f (·; θ) with θ Ω, and let x 1 ,…,xnbe the respective observed values and x = (x 1 ,…,xn). The likelihood function, L (θ | x ), is given by and a value of θ which maximizes L (θ | x ) is called a Maximum Likelihood Estimate (MLE) of θ. Clearly, the MLE depends on x, and we usually write . Thus,(1)The justification for choosing an estimate as the value of the parameter which maximizes the likelihood function is the same as that given in Example 1 , when the r.v.’s are discrete. The same interpretation holds true for r.v.’s of the continuous type, by considering small intervals around the observed values.Once we decide to adopt the Maximum Likelihood Principle (i.e., the principle of choosing an estimate of the parameter through the process of maximizing the likelihood function), the actual identification of an MLE is a purely mathematical problem; namely, that of maximizing a function. This maximization, if possible at all, often (but not always) is done through differentiation. Examples to be discussed below will illustrate various points.Before embarking on specific examples, it must be stressed that, whenever a maximum is sought by differentiation, the second-order derivative(s) must also be examined in search of a maximum. Also, it should be mentioned that maximization of the likelihood function, which is the product of n factors, is equivalent to maximization of its logarithm (always with base e ), which is the sum of n
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