Estimation Maximum Likelihood Estimates Industrial Engineering

Discrete Case p x ( ) L q = × p x ( ) Suppose we have hypothesized a discrete distribution from which our data which has some unknown parameter . Let denote the probability mass function for this distribution. The likelihood function is q p x q ( ) p x n ( ) L q = × 1 2

Discrete Case L for all possible ( $ ) q ³ L ( ) q $ q Since is just the joint probability, we want to choose some which maximizes this joint probability mass function. $ q L for all possible ( $ ) q ³

Continuous Case Suppose we have a set of nine observations x1, x2, . . . X9 which have underlying distribution exponential (in this case scale parameter l = 2.0). 0.053 0.458 0.112 0.602 0.178 0.805 0.255 1.151 0.347

Continuous Case L f x ( ) l = × Suppose we have a set of nine observations x1, x2, . . . X9 which have underlying distribution exponential (in this case scale parameter l = 2.0). Our object is to estimate the true but unknown parameter l. L f x n ( ) l = × 1 2

MLE (Exponential) L f x ( ) l = × = × l e = å l e - x - x n n 1 2 n 1

MLE (Exponential)

MLE (Exponential) ¶ l L ( ) = We can use the plot to graphically solve for the best estimate of l. Alternatively, we can find the maximum analytically by using calculus. Specifically, ¶ l L ( ) =

Log Likelihood Ln LN L ( ) q = The natural log is a monotonically increasing function. Consequently, maximizing the log of the likelihood function is the same as maximizing the likelihood function itself. Ln LN L ( ) q =

å MLE (Exponential) Ln nln x ( ) l = - L f x ( ) l = × = å l e i - x n 1 2 = å - l e n x i Ln nln x i ( ) l = - å

å å å MLE (Exponential) Ln nln x ( ) l = - ¶ l ¶l Ln n x ( ) ln( = - = = 0 = - å n x i l

MLE (Exponential) n x i l - = å n x i l = å

MLE (Exponential) x l = 1 $ i X 1 0.053 2 0.112 3 0.178 4 0.255 5 0.347 6 0.458 7 0.602 8 0.805 9 1.151 Sum = 3.961 X-bar = 0.440 x l = 1 $

Experimental Data Suppose we wish to make some estimates on time to fail for a new power supply. 40 units are randomly selected and tested to failure. Failure times are recorded follow: X 1 . 19 =

Failure Data