A Bayesian Alternative to Parametric Hypothesis Testing - download pdf or read online

By Rueda R.

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Let the distribution function F(x) of the random variable X be continuous and sy~aetric about the origin. , F(-x) = l-F(x) for all x ~ 0, and satisfies both assump tiQns of the theorem. Hence, only the converse needs proof. Because F(x) is symmetric about the origin, P(-x

Indeed, if T(x) is an arbitrary distribution function with T(0+) = 0 then if we write x > 0 as x = n+y where n ~ 0 is an integer and 0 ~ y < 1 , (12) l-F(x) = pn[l-T(y)] satisfies , p=l-F(1) , (ib) for all integers z ~ 0 and all x ~ 0. We can, of course, understand the mathematical reason for the need of Zl/Z 2 to be assumed to be irrational. 3 is the fact that the set {u: u = sz I + tz 2 , s,t integers} is dense on the real line if Zl/Z 2 is irrational, while this fails for a rational value of Zl/Z 2.

X-u) + = max(X-u,0). , b > 0, z ~ 0 . (34) w i t h z = 0 and Len~rna 1 . 2 . 1 Consequently, Let us r e t u r n interesting -bz m imply that G(0+) = 1 o r (kCl)/m~l , which completes the proof. to the linear case. of the exponential D K-M. Chong (1977) points out the following distribution. Let X ~ 0 be a random variable with finite expectation. Let If the distribution of X is continuous at zero, then X has a negative exponential distribution if, and only if, (38) E[(X-s)+]E[(X-t) +] = mE[(X-s-t) +] , all s,t ~> 0 , with some constant m > 0.

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A Bayesian Alternative to Parametric Hypothesis Testing by Rueda R.

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