hw7 - n be a random sample of size n from a distribution...

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APPM 4/5520 Problem Set Seven (Due Wednesday, November 9th) 1. Consider the “shifted” rate 1 exponential distribution with pdf f ( x ) = e - ( x - η ) · I ( η, ) ( x ) . This distribution is frequently encountered in reliability and product lifetime analyses. The new parameter η , which may be negative, is a location parameter which slides the familiar exponential pdf left or right depending on its sign. Find the MME (method of moments estimator) of η . 2. Let X 1 , X 2 , . . . , X n be a random sample from the distribution with pdf f ( x ; θ ) = Γ(2 θ ) [Γ( θ )] 2 x θ - 1 (1 - x ) θ - 1 I (0 , 1) ( x ) . Find the MME of θ . 3. Let X 1 , X 2 , . . . X n be a random sample from each of the distributions having the following pdfs (all are zero where not explicitly de±ned): a) f ( x ; θ ) = θ x e - θ /x !, x = 1 , 2 , . . . , 0 θ < f (0; 0) = 1 b) f ( x ; θ ) = (1 ) e - x/θ , 0 < x < , 0 < θ < c) f ( x ; θ ) = 1 2 e -| x - θ | , -∞ < x < , -∞ < θ < d) f ( x ; θ ) = e - ( x - θ ) , θ x < , -∞ < θ < In each case, ±nd the MLE ˆ θ of θ . 4. Let X 1 , X 2 , . . . , X
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Unformatted text preview: n be a random sample of size n from a distribution having pdf f ( x ; θ ) = 2 x θ 2 I (0 ,θ ] ( x ) . Find the MLE for the median of the distribution. (The median for X is the value ξ such that P ( X ≤ ξ ) = 1 / 2.) 5. Let X 1 , X 2 , . . . , X n be a random sample from the Γ( α, β ) distribution (as iven y the table handed out in class. Suppose that α is ±xed and known. (a) Find the MME of β . (b) Find the MLE of β . (c) Which estimator (MME or MLE) has smaller variance. (d) [Required for 5520 only] Show that your MLE is a consistent estimator of β . 6. [Required for 5520 only] Suppose that ˆ θ is the MLE for a parameter θ . Let τ ( θ ) be an invertible function of θ . Show that τ ( ˆ θ ) is the MLE of τ ( θ ). ie: Show that ˆ τ ( θ ) = τ ( ˆ θ )....
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This note was uploaded on 02/07/2012 for the course APPM 4520 taught by Professor Manuel during the Fall '11 term at University of Colorado Denver.

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