By Rueda R.

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Welcome to new territory: A direction in likelihood versions and statistical inference. the concept that of likelihood isn't really new to you in fact. you might have encountered it on the grounds that adolescence in video games of chance-card video games, for instance, or video games with cube or cash. and also you learn about the "90% probability of rain" from climate studies.

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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.

### A Bayesian Alternative to Parametric Hypothesis Testing by Rueda R.

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