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explain the maximum likelihood estimator
Dec 27, 2023
explain the maximum likelihood estimator
Solution by Steps
step 1
Define the likelihood function for a set of independent and identically distributed (i.i.d.) data points x1,x2,...,xn x_1, x_2, ..., x_n given a statistical model with parameter θ \theta : L(θ)=f(x1,x2,...,xn∣θ)=∏i=1nf(xi∣θ) L(\theta) = f(x_1, x_2, ..., x_n | \theta) = \prod_{i=1}^{n} f(x_i | \theta) where f(xi∣θ) f(x_i | \theta) is the probability density function (pdf) or probability mass function (pmf) for the i i -th data point
step 2
Formulate the log-likelihood function, which is the natural logarithm of the likelihood function: ℓ(θ)=log⁡L(θ)=∑i=1nlog⁡f(xi∣θ) \ell(\theta) = \log L(\theta) = \sum_{i=1}^{n} \log f(x_i | \theta) This transformation is useful because it turns the product into a sum, making it easier to work with, especially for differentiation
step 3
Find the estimator θ^ \hat{\theta} that maximizes the log-likelihood function. This involves taking the derivative of ℓ(θ) \ell(\theta) with respect to θ \theta , setting it equal to zero, and solving for θ \theta : ddθℓ(θ)=0 \frac{d}{d\theta} \ell(\theta) = 0
step 4
Solve the resulting equation or set of equations to find the maximum likelihood estimator θ^ \hat{\theta} . In some cases, this may require numerical methods if an analytical solution is not possible
Answer
The maximum likelihood estimator θ^ \hat{\theta} is the value of θ \theta that maximizes the likelihood function L(θ) L(\theta) or equivalently the log-likelihood function ℓ(θ) \ell(\theta) .
Key Concept
Maximum Likelihood Estimation (MLE)
Explanation
MLE is a method used in statistics to estimate the parameters of a statistical model by maximizing the likelihood function, which measures how well the model explains the observed data.
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