10. Using the technique outlined in Section 3, find the maximum likelihood estimate for 0, Suppose we have a collection of n randomly sampled data points, x, x,X3,...,Xn that we know came from an exponential distribution, but we don't know which one. Recalling that exponential distributions are indexed by values of , we can determine which exponential function our data came from if we can accurately estimate 0 We do this by using a process called maximum likelihood estimation. Here's how it works. 1. Calculate the likelihood function for the parameter, L(0). This is given by f(x) x f(x) x... x f(x) =f(x,), =1 where f(x) is the exponential distribution function associated to given value of e and the IP xnotation indicates taking the product of all terms (this is analogous to summation notation.) The term P(X-x) represents the probability that the random variable X has value x. Our maximum likelihood estimate of 8,, will be the value of 0 that maximizes the likelihood function. We know how to maximize a function: take the derivative, find critical points, and choose the one that maximizes the function (in this case, L(e)). Most of the time, taking the derivative of a likelihood function is difficult (why?). So we add one step to simplify the problem. 2. Calculate the log-likelihood function LL(e) In[(e)] The log transformation is increasing, so the maximum of L(e) will also be the maximum of LL(0) In[(0)]. More importantly, this function is much easier to take the derivative of. 3. Take the derivative of the log-likelihood function, LL (0), and use that find the value of e that maximizes LL(0). This is your maximum likelihood estimate, ê
10. Using the technique outlined in Section 3, find the maximum likelihood estimate for 0, Suppose we have a collection of n randomly sampled data points, x, x,X3,...,Xn that we know came from an exponential distribution, but we don't know which one. Recalling that exponential distributions are indexed by values of , we can determine which exponential function our data came from if we can accurately estimate 0 We do this by using a process called maximum likelihood estimation. Here's how it works. 1. Calculate the likelihood function for the parameter, L(0). This is given by f(x) x f(x) x... x f(x) =f(x,), =1 where f(x) is the exponential distribution function associated to given value of e and the IP xnotation indicates taking the product of all terms (this is analogous to summation notation.) The term P(X-x) represents the probability that the random variable X has value x. Our maximum likelihood estimate of 8,, will be the value of 0 that maximizes the likelihood function. We know how to maximize a function: take the derivative, find critical points, and choose the one that maximizes the function (in this case, L(e)). Most of the time, taking the derivative of a likelihood function is difficult (why?). So we add one step to simplify the problem. 2. Calculate the log-likelihood function LL(e) In[(e)] The log transformation is increasing, so the maximum of L(e) will also be the maximum of LL(0) In[(0)]. More importantly, this function is much easier to take the derivative of. 3. Take the derivative of the log-likelihood function, LL (0), and use that find the value of e that maximizes LL(0). This is your maximum likelihood estimate, ê
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
Related questions
Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
The first picture is the actual question i have to answer using section 3 which is shown in the second photo. Clean legible handwriting would be very much appreciated as i have already recieved some answers i can not read or understand
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning