METHOD OF MAXIMUM LIKELIHOODAnother procedure for extimating parameters, it views the pdf or pmf as a function in theparameter given the observed values. This function is called the likelihood, and if is maximize dwith repsect to the parameter to provide an estimate of the parameter.Example X has a binomial(100, p) distribution, and it a value of 40 is observed for X.Find the likelihood function – a function of pL(p) = P (X = 40|p) =Maximize it with respect to p.∂L(p)/∂p =Establish that the value found is indeed a maximum of the likelihood.1Often it is easier to maximize the log of the likelihood function, than to maximize the likelihoodfunction itself. consider the normal example, where Y1, . . . Ynare iid normal with a mean µ and avariance σ2. Suppose we observe values, x1, . . . , xnand we wish to estimate µ.In this example, we have a continuous pdf rather than a pmf.Find the likelihood function – a function of µP (X1∈ dx1, . . . , Xn∈ dxn|µ) = f(x1, . . . , xn|µ)dx1· · · dxn=We ignore the infinitesimal region and maximize the likelihood function:L(µ) = f(x1, . . . , xn|µ)Alternatively we could maximize the log-likelihood function (WHY?):l(µ) = log(f (x1, . . . , xn|µ))Maximize it with respect to µ.∂l(µ)/∂µ =Establish that the value found is indeed a maximum of the
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