Statistical estimation is the cornerstone of statistical inference, enabling us to derive insights about population characteristics using sample data. The population is described by a probability density function (pdf) $f(x;θ)$, which depends on unknown parameters θ. These parameters are often not directly observable but play a vital role in defining the population's behavior and structure. To obtain a good such estimators following method can be used: Method of Maximum Likelihood Estimator Method of Minimum Variance Method of Moments Method of Least Squares Method of Minimum Chi-square Method of Inverse Probability Method of Maximum Likelihood Estimator For each sample point $x$ let $\hat\theta(x)$ be a estimator value at which $L(\theta|x)$ attains it maximum value as a function of $\theta$ with $x$ held fixed. A maximum likelihood estimator based on sample points $x_{i}$ is $\hat\theta_{MLE}$. In practical scenarios, θ is estimated using a random s...
MLE for Pareto distribution The pdf of Pareto distribution is given by: `f(x; \alpha, \lambda) = \frac{\alpha \lambda^\alpha}{x^{\alpha+1}} \quad \text{for } x \geq \lambda.` `f(x_i, \alpha, \lambda) = \frac{\alpha \lambda^\alpha}{x_i^{\alpha+1}} \cdot I(x_i \geq \lambda), \quad x_i \in \mathbb{R} ` Writing the likelihood function `L(\alpha, \lambda) = \prod_{i=1}^n f(x_i , \alpha, \lambda)` `\Rightarrow L(\alpha, \lambda) = \prod_{i=1}^n \left( \frac{\alpha \lambda^\alpha}{x_i^{\alpha+1}} \cdot I(x_(i) \geq \lambda) \right)` `= L(\alpha, \lambda) = \alpha^n \lambda^{n\alpha} \prod_{i=1}^n \frac{1}{x_i^{\alpha+1}} \cdot \prod_{i=1}^n I(x_(i)\geq \lambda)` Finding the calculus for MLE in Pareto Distribution will be easy if we optimize it by taking log both sides because it is also a increasing function. `= \log L(\alpha, \lambda) = n \log \alpha + n \alpha \log \lambda - (\alpha+1) \sum_{i=1}^n \log x_i + \sum_{i=1}^n \log I(x_i \geq \lambda)` `= \log L(\alpha, \l...
