Estimating the parameters of a probability distribution involves choosing an appropriate distribution, gathering data, and then using a technique to find the best-fitting parameters. There are several methods to estimate the parameters, and I will discuss two popular ones: the Method of Moments and the Maximum Likelihood Estimation. Before applying these methods, you need to ensure that the chosen distribution is appropriate for your data. You can use goodness-of-fit tests like the Kolmogorov-Smirnov test or the Anderson-Darling test to assess the suitability of the distribution.
1. Method of Moments (MoM):
The Method of Moments estimates the parameters of a distribution by equating the sample moments with the population moments. The sample moments are calculated from the data, while the population moments are expressed in terms of the distribution parameters.
Let X1, X2, …, Xn be a random sample of size n from the distribution. The k-th sample moment is given by:
$$m_k = \frac{1}{n}\sum_{i=1}^n X_i^k$$
The k-th population moment is the expected value of Xk, and it can be calculated from the distribution:
μk = E[Xk]
To estimate the parameters, we set μk = mk for k = 1, 2, …, p, where p is the number of parameters in the distribution.
For example, let’s assume we have a sample from a normal distribution, which has parameters μ and σ2. The first and second population moments are:
μ1 = μ
μ2 = E[X2] = μ2 + σ2
Equating these with the sample moments, we get the MoM estimators for μ and σ2:
μ̂ = m1
$$\hat{\sigma^2} = m_2 - m_1^2$$
2. Maximum Likelihood Estimation (MLE):
The Maximum Likelihood Estimation estimates the parameters by maximizing the likelihood function. The likelihood function represents the probability of observing the data given the chosen distribution and its parameters. The MLE is the parameter value that maximizes the probability of observing the data.
For a sample X1, X2, …, Xn from a distribution with probability density function (pdf) f(x; θ), where θ is the parameter vector, the likelihood function is given by the product of the pdf evaluated at each data point:
$$\mathcal{L}(\theta) = \prod_{i=1}^n f(X_i;\theta)$$
To simplify calculations, we usually work with the log-likelihood function:
$$\ell(\theta) = \log \mathcal{L}(\theta) = \sum_{i=1}^n \log f(X_i;\theta)$$
The MLE estimators, θ̂, are the parameter values that maximize ℓ(θ). We can find them by solving the system of equations:
$$\frac{\partial \ell(\theta)}{\partial \theta_j} = 0, \quad j=1,2,\ldots,p$$
Continuing with the normal distribution example, its pdf is:
$$f(x;\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
The log-likelihood function is:
$$\ell(\mu,\sigma^2) = -\frac{n}{2} \log (2\pi) - \frac{n}{2} \log (\sigma^2) - \frac{1}{2\sigma^2} \sum_{i=1}^n (X_i-\mu)^2$$
Taking partial derivatives with respect to μ and σ2, and setting them to zero, we get the MLE estimators:
$$\hat{\mu} = \frac{1}{n} \sum_{i=1}^n X_i = m_1$$
$$\hat{\sigma^2} = \frac{1}{n} \sum_{i=1}^n (X_i - \hat{\mu})^2 = \frac{1}{n} \sum_{i=1}^n X_i^2 - m_1^2$$
In summary, the Method of Moments and the Maximum Likelihood Estimation are two common methods to estimate the parameters of a probability distribution from data. They may yield different estimates, and the choice between them depends on the context and the desired properties of the estimators.