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Estimation Maximum Likelihood Estimates Industrial Engineering

Estimation Maximum Likelihood Estimates Industrial Engineering. p. (. x. ). q. q. =. ×. ×. ×. L. (. ). p. (. x. ). p. (. x. ). p. (. x. ). q. q. q. 1. 2. n. Discrete Case.

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Estimation Maximum Likelihood Estimates Industrial Engineering

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  1. Estimation Maximum Likelihood EstimatesIndustrial Engineering

  2. p ( x ) q q = × × × L ( ) p ( x ) p ( x ) p ( x ) q q q 1 2 n Discrete Case • 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

  3. p ( x ) q q q = × × × L L ( ( ) ) p ( x ) p ( x ) p ( x ) q q q 1 2 n Discrete Case • 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 is just the joint probability mass function

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

  5. 0.053 0.458 0.112 0.602 0.178 0.805 0.255 1.151 0.347 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).

  6. l = × × × L ( ) f ( x ) f ( x ) f ( x ) l l l 1 2 n 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). Our object is to estimate the true but unknown parameter l.

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

  8. - l - l - l = l × l × × × l x x x e e e 1 2 n å - l x = l n e l = × × × i L ( ) f ( x ) f ( x ) f ( x ) - l = l l l l 9 3 . 961 1 2 n e MLE (Exponential)

  9. MLE (Exponential)

  10. l L ( ) = 0 ¶ l ( ) MLE (Exponential) • 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,

  11. q = q Ln ( ) LN L ( ) Log Likelihood • 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.

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

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

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

  15. n å - = x 0 i l n å = x i l MLE (Exponential)

  16. n å - = x 0 i l n å = x i l 1 = x l MLE (Exponential)

  17. n å - = x 0 i l n å = x i l x 1 = x l $ 1 l = MLE (Exponential)

  18. i X i 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 x Sum = 3.961 X-bar = 0.440 $ 1 l = MLE (Exponential)

  19. i X i 1 0.053 2 0.112 3 0.178 $ 1 l = = 2 . 27 4 0.255 0 . 44 5 0.347 6 0.458 7 0.602 8 0.805 9 1.151 x Sum = 3.961 X-bar = 0.440 $ 1 l = MLE (Exponential)

  20. = X 19 . 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:

  21. Failure Data

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