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4. Let T be a test statistic of 9 and let C t be a subset of t, the sample space of T. Suppose C, has the following property: Property 3. Given any fixed t and 0, the probability, Pr(TEC,) is a nondecreasing function of (i) 9-9 0 when 9~90' and (ii) 9 0 -9 when 9<9 0; more or less, this means that there exists a function of T which is stochastically increasing in I 9 - 9 0 I. 34 2. Notions in significance testing of hypotheses Sometimes, the following milder property is considered adequate for defining an extreme region; it should be noted, however, that a test having Property 3 is equally or more powerful than a test with the following property.

Let XI' ... , Xn be a random sample from the uniform distribution with density function Ix (x) I = 9" for -9::; x::; 0 . = Show that T min(X), the minimum of all the observations, is a test statistic for 9. 05 for testing the above hypotheses. 2. Let XI' ... 2). In terms of the cdf of the gamma distribution find its p-value for testing each of the following pairs of hypotheses: (a) H o:S:S;9 0 ' H1:S>So, (b) H 0: 9 ~ 9 0 , HI: S < So. 3. Let XI' ... , Xn be a random sample from the exponential distribution with density function Ix (x) = <1 e- ax for x > 0 .

A) Show that the noninformative prior of ~ is 1t(~) = 1. (b) Show that the posterior distribution of ~ is a normal distribution with mean and variance lIn. (c) Find the 95% left-sided Bayesian confidence interval for~. (d) Find the 95% left-sided classical confidence interval for~. (e) Find the 95% HPD interval for ~. 1. Introduction In this book often we perform hypothesis testing on the basis of what is defined later as p-value, as opposed to testing at a fixed nominal level. The p-value is sometimes called the 'observed level of significance' or the 'significance value'.

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