Let (Y_1, x_1),. . . ., (Y_n, x_n) be the data where Y_i are independently and e

Question

Let (Y_1, x_1),. . . ., (Y_n, x_n) be the data where Y_i are independently and exponentially distributed random variables in the following way: Y_i Ex(lambda x_i), i = 1, 2,. . . ,n f_Y_i = lambda x_i e^-lambda x, y 1y > 0. The known constants (x_i) are strictly positive. I have derived the maximum likelihood estimalot for lambda. I think it is Exponential distributions arc often used to model survival times. A researcher wants to find the relationship between the price of a light bulb and its life span. She samples 40 different lieht bulbs and hypothesizes the above model, whereby for i = 1,. . . .,40 x_i = 1/price of lightbulb i Y_i = life span of light bulb i (in years) The plot of the relationship between price and lifespan is given. Figure 1: 40 measurements of light bulb life spans versus their price. Given that Sigma i = 1 40 x_iy_i = 80, calculate the numeric 95% confidence interval for lambda based on asymptotic distribution of the likelihood ratio statistic. Draw this confidence interval in a plot of the likelihood ratio statistic versus lambda. You are allowed to "read off" the numeric CI from this plot, but provide the inequality that you would like to solve.

Explanation / Answer

Let (Y_1, x_1),. . . ., (Y_n, x_n) be the data where Y_i are independently and e

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