Consider a sample of x1, x2, ., xn, observations from a...

Consider a sample of $x_{1}, x_{2}, ., x_{n},$ observations from a Weibull distribution with parameters a and $\beta$ and density function $$f i x)=\left\{\begin{array}{ll}\int \alpha \beta x^{\beta-1} e^{-\alpha x^{A}} & x>0 \\ 0, & \text { elsewhere }\end{array}\right.$$ for $\alpha, 3>0$ (a) Write out the likelihood function. (b) Write out the equations which when solved give the maximum likelihood estimators of $a$ and ,3.

Consider a sample of $x_{1}, x_{2}, ., x_{n},$ observations from a Weibull distribution with parameters a and $\beta$ and density function
$$f i x)=\left\{\begin{array}{ll}\int \alpha \beta x^{\beta-1} e^{-\alpha x^{A}} & x>0 \\
0, & \text { elsewhere }\end{array}\right.$$ for $\alpha, 3>0$
(a) Write out the likelihood function.
(b) Write out the equations which when solved give the maximum likelihood estimators of $a$ and ,3.

Key Concepts

Likelihood Function

The likelihood function is the joint probability (or density) of the observed data, considered as a function of the parameter(s). In statistical inference, it represents how plausible particular values of the parameters are given the observed sample, and it is the foundation for methods such as maximum likelihood estimation.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used to estimate the parameters of a statistical model. It involves finding the parameter values that maximize the likelihood function, thereby making the observed data most probable. The process typically requires taking the derivative of the likelihood (or log?likelihood) with respect to the parameters and setting these derivatives equal to zero to solve for the estimators.

Log-Likelihood Function

The log-likelihood function is the natural logarithm of the likelihood function. Taking the logarithm often simplifies the mathematical manipulation because it converts products into sums, which makes differentiation easier when deriving the estimation equations. It preserves the location of the maximum, meaning that the parameter values that maximize the likelihood also maximize the log-likelihood.

Score Equations

Score equations are obtained by differentiating the log-likelihood function with respect to the parameters and setting the derivatives equal to zero. These equations, also known as the first-order conditions of optimality, provide the necessary conditions that the maximum likelihood estimators must satisfy. Solving these equations yields the parameter estimates that maximize the likelihood function.

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