What is the likelihood function for a random sample of size 3 from an exponential distribution, with observations .3, .7, and .2? Find this by evaluating the above likelihood function appropriately. Do the arithmetic to simplify it. L(? | .3, .7, .2) = 3. For practice, express the preceding likelihood function with observations .3, .7, and .2 as a product of exponential distributions evaluated at the observations.
Math 425/525, F. Example of Maximum Likelihood Estimation using Observations of an Exponential Random Variable by Prof. T. Fiore Name: Date: Likelihood Function for a Random Sample from an Exponential Distribution of Size 3, and a Graphical Determination of the Maximum Likelihood Estimate for Specific Observations 1. What is the likelihood function for a random sample of size 3 from an exponential distribution? Simplify it. L(θ | x1, x2, x3) = 2. What is the likelihood function for a random sample of size 3 from an exponential distribu- tion, with observations .3, .7, and .2? Find this by evaluating the above likelihood function appropriately. Do the arithmetic to simplify it. L(θ | .3, .7, .2) = 3. For practice, express the preceding likelihood function with observations .3, .7, and .2 as a product of exponential distributions evaluated at the observations. L(θ | .3, .7, .2) = 4. Now let’s quickly use GeoGebra to easily plot and explore the likelihood function L(θ | .3, .7, .2) you found in Exercise 2 and its logarithm. (a) Google “Geogebra Calculator Suite”. This is an easy-to-use plotter directly in your browser, no installation needed! (b) In the left corner type in the formula for the likelihood function from Exercise 2, use the letter x in place of θ (unfortunately Geogebra requires us to use x). Just type in the formula, leave out L. Exponents are made using the carat symbol. To get out of an exponent, press the right arrow key on your keyboard once or twice. After you type in the formula, press Enter, and the plot is done! (c) In the next field that opens up in the left corner, type in ln( ) of your formula from Exercise 2. Press Enter. Now you have both plots on one graph! (d) On the two curves, click on the dots that are the maximuma, and notice the x-coordinate for both (which is actually your θ-coordinate). (e) Below, draw by hand one coordinate axes and label the vertical axis L and ln(L), and label the horizontal axis θ. Include only quadrants I and IV because we are focusing on the allowable parameter range θ > 0. Next draw by hand the curves you see in Geogebra on this one graph. Indicate the input on the horizontal θ axis at which the maxima are attained. 5. Use the preceding graphs to visually determine the maximum likelihood estimate of the pa- rameter θ if you have observed .3, .7, and .2. 2 6. What is the formula for the maximum likelihood estimator for the parameter θ for an exponen- tial distribution? Use the table in the notes. Be careful to properly use capital or lowercase letters! 7. Use the formula for the maximum likelihood estimator for the parameter θ for an exponential distribution to find the maximum likelihood estimate based on the observations .3, .7, and .2. 8. Compare your answer using the formula with the answer you found using the graphs. Why is this estimate called the maximum likelihood estimate for θ? Refer to the graphs. Derivation of Maximum Likelihood Estimator for the Parameter θ of an Expo- nential Distribution 9. Use Calculus 1 to find the formula for the maximum likelihood estimator for the parameter θ of an exponential distribution. Remember: instead of optimizing L(θ | x1, . . . , xn), we optimize the logarithm ln(L(θ | x1, . . . , xn)) to avoid messy products. This problem will take you 1 or 2 pages of work. The steps are: (a) Write down L(θ | x1, . . . , xn) using its definition and simplify it. (b) Then apply natural logarithm to both sides and simplify it using the log rules. (c) Then differentiate both sides with respect to θ. In other words, differentiate ln(L(θ | x1, . . . , xn)), and avoid the mistake of saying the derivative is equal to the original function! When you take the derivative, on the left you also need to write: d dθ ln(L(θ | x1, . . . , xn)) = 3 (d) Next set the derivative equal to 0, and solve for θ in terms of x1, . . . , xn to find the critical point. (e) Apply the First Derivative Test to the critical point to confirm that you have a maximum at the critical point. (f) Finally, write θ̂ = formula in terms of capital X1, . . . , Xn where the formula is your formula for the critical point, but with lowercase xi’s replaced by capital Xi’s. (g) Check your answer: did you get the same formula as the relevant entry in the table of maximum likelihood estimators in the notes? 4