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The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete ...
Assume that we want to estimate an unobserved population parameter on the basis of observations . Let f {\displaystyle f} be the sampling distribution of x {\displaystyle x} , so that f ( x ∣ θ ) {\displaystyle f(x\mid \theta )} is the probability of x {\displaystyle x} when the underlying population parameter is θ {\displaystyle \theta } .
Another popular M-estimator is maximum-likelihood estimation. For a family of probability density functions f parameterized by θ, a maximum likelihood estimator of θ is computed for each set of data by maximizing the likelihood function over the parameter space { θ } .
Consider the estimator of θ based on binomial sample x~b(θ,n) where θ denotes the probability for success. Assuming θ is distributed according to the conjugate prior, which in this case is the Beta distribution B( a , b ), the posterior distribution is known to be B(a+x,b+n-x).
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events.
Gumbel has also shown that the estimator r ⁄ (n+1) for the probability of an event — where r is the rank number of the observed value in the data series and n is the total number of observations — is an unbiased estimator of the cumulative probability around the mode of the distribution. Therefore, this estimator is often used as a ...
In statistics and econometrics, the maximum score estimator is a nonparametric estimator for discrete choice models developed by Charles Manski in 1975. Unlike the multinomial probit and multinomial logit estimators, it makes no assumptions about the distribution of the unobservable part of utility .
The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.