1 Three loooong minutes on interval estimation with a bit on hypothesis testing erm intervalestimation new tex and pdf Dec 2 2010 In a previous section we discussed point estimation coming up with a speci c estimate for a parameter or parameters Here we discuss estimating an interval that is likely to include the true value of the parameter interval estimation Our main concern will be estimating intervals for an unknown parameter or parameters As you might suspect the estimated intervals can be used to test hypotheses so hypothesis testing and interval estimation are highly related topics In fact when we identi ed the likelihood ratio test in the max lik section we identi ed a con dence interval Rather that starting by investigating the estimation an interval for an unknown parameter I will for simplicity introduce intervals on a random variable where the distribution of that rv is known completely including the parameter values a case where there is no need to estimate an interval on any parameter but one might still want to test a hypothesis 1 1 The interval on the rv X or a statistic on X when the distribution of X is known completely Assume that the rv X has some known distribution fX x x 2x and that we know x and 2x Or more generally we know fX x and 1 1 1 A con dence interval on X Given the above assumption that we know fX x one can determine P x w X Z x w fX u x x w 1 x x 2 x du 2 x w and know x and 2 x which can without loss of generality be written P x m x X Z x m x fX u x m x Given fX u x 2 x x m x 2 x du x this is a function of only m it is not statistic or rv Now choose m such that Z x m x m x fX u x and call the resulting m m 95 2 x 2 x du 95 In which case P x m 95 x X Z x m 95 x fX u x x m 95 m 95 x 2 x du x 95 x X is a rv but the interval x m 95 x to x m 95 x is not random no data or estimation was involved in determining this interval it is not a statistic and it is not an estimated interval How would one interpret the interval x m 95 x to x m 95 x If X is randomly drawn from fX x x 2x before it is drawn it has a 95 chance of being in this xed interval once drawn it either is or is not in the interval The interval x m 95 x X x m 95 x is called a con dence interval it gives one con dence about where x will lie but di ers from many of the con dence intervals that you will encounter because this interval is not a random it does not vary Note that by construction I constrained the interval to be centered on We choose the critical level of m say m 95 or m 99 x Note a few things before we proceed One could rearrange the above interval to get 95 P Pr m 95 where Q X x x x m 95 x Q m 95 X x m 95 x is a statistic rv with some density fQ q To gure out the appropriate value for m 95 one can calculate m 95 using one s knowledge of fX x x 2x x and 2x like we did above Or one can determine it given knowledge of fQ q it mightR be di cult to m determine fQ q In terms of fQ q m 95 is that m for which m fQ q dq 95 1 X 1 Both x ways can be useful Note that if one knows fX x x 2x x 2x and Q x one can in theory determine fQ q it is a derived distribution And if one knows fQ q X 2 x X x Q x one can in theory determine fX x x 2x x x and Q x 3 Note that by construction I constrained the interval to be centered on x x plus and minus the same term I also constrained the interval to be gapless For our X density known there are many 95 con dence intervals only one of which is centered on x We will mostly want to center our con dence intervals on the expected value of the variable or parameter of interest but not always We will typically want to work with the shortest con dence interval because it is most informative interval more on this in a bit The shortest con dence interval will often by centered on the expected value of the variable or parameter For example if fX u x 2x is symmetric and unimodal with the mode equal to x like the Normal then the shortest X interval that spans 95 of the density is the one centered on x In contrast consider an asymmetric distribution or a U shaped distribution See the review question about Weird Shirley 2 2 Note that the shortest interval might consist of segments that are not contiguous 4 What I have said to here is quite general in that I did not assume a particular density for X or fQ q where Q X x x There is nothing said above that requires that X is normally distributed For fun calculate these 95 and 99 con dence intervals for X for a few di erent speci c speci cations of fX x x 2x For example do it assuming X has a normal distribution assuming X has a Poisson distribution with a mean of 3 and a mean of 6 and assuming X has some t distribution Since the Poisson is a discrete distribution the range will consist of only a nite number of points The fact that the Poisson is not a symmetric distribution brings to light the possibilty that we might sometimes prefer an interval that is not symmetric around The interval for other statistics Above we have speci ced an interval for X but we could have speci ed an interval for lots of other things as well For example we could have assumed a random sample X1 X2 Xn and considered n X Xi the interval for the mean this an interval for g X1 X2 Xn X n1 i 1 being only one of many examples Can you identify a con dence interval for X for random samples of size n continuing to assume that the density of X is known completely Doing this is a special case of some of the things we do below a hypothesis test If we randomly draw an x from fX x is a member of X x 2 x we know it However if we observe a value y where y is maybe a value of …