Question

Pivotal quantity for a double exponential distribution: 5 points Assume y follows a double exponential distribution (as above), where is the parameter of interest and is unknown, and o is known to be 1. Show that Y -u is a pivotal quantity.

Answer 1

Pivotal quantity or pivot is the function of sample observations and parameter such that its distribution does not depend on unknown parameter.

Double exponential distribution is also known as Laplace distribution whose pdf is given as in two Cases due to 'modulus' sign.

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Case I , we can clite, vol Combining Case I & f (o) = te where w =y-u. .. Distribution of w is free from any untrown - povrameter value. Thus, W =Y-u is a protal quantly In general, the result is - if X N Double exponential ( 8 then - IX-U N exponential (6) Here, 6 & known. and o=1. Definition of Pratal quantity: Protal Quantly or Prot & and unknown parameters such distribution aves not depend function that on the of sample volues its probability unknown parameters