?Consider the Weibull distribution

Chapter 7, Problem 7.4.5

(choose chapter or problem)

Consider the Weibull distribution

\(f(x)=\left\{\begin{array}{ll}\frac{\beta}{\delta}\left(\frac{x}{\delta}\right)^{\beta-1} e^{-\left(\frac{x}{\delta}\right)^{\beta},} & 0<x \\0, & \text { otherwise }\end{array}\right.\)

a. Find the likelihood function based on a random sample of size n. Find the log likelihood.

b. Show that the log likelihood is maximized by solving the following equations

\(\beta=\left[\frac{\sum_{i=1}^{n} x_{i}^{\beta} \ln \left(x_{i}\right)}{\sum_{i=1}^{n}x_{i}^{\beta}}-\frac{\sum_{i=1}^{n} \ln \left(x_{i}\right)}{n}\right]^{-1}\delta=\left[\frac{\sum_{i=1}^{n} x_{i}^{\beta}}{n}\right]^{1 / \beta}\)

c. What complications are involved in solving the two equations in part (b)?

Text Transcription:

f(x)=({^\frac{\beta}{\delta}((\frac{x}{\delta}))^{\beta-1} e^{-((\frac{x}{\delta}))^{\beta},} _0<x \\0, { otherwise }\end{array})

\beta=([\frac{\sum_{i=1}^{n} x_{i}^{\beta} \ln ((x_{i}))}{\sum_{i=1}^{n}x_{i}^{\beta}}-\frac{\sum_{i=1}^{n} \ln ((x_{i}))}{n}\right]^{-1}\delta=([\frac{\sum_{i=1}^{n} x_{i}^{\beta}}{n})]^{1 / \beta}