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10.10: Composite Hypotheses and Uniformly Most Powerful Tests iid Example: Let Yi, Yn N(p,02) where 02 is known but /' is unknown. We wish to test Ho pl = /lo ...

Question

10.10: Composite Hypotheses and Uniformly Most Powerful Tests iid Example: Let Yi, Yn N(p,02) where 02 is known but /' is unknown. We wish to test Ho pl = /lo versus the composite Ha p > |lo where /l0 is a fixed constant: Can we find a UMP test?

10.10: Composite Hypotheses and Uniformly Most Powerful Tests iid Example: Let Yi, Yn N(p,02) where 02 is known but /' is unknown. We wish to test Ho pl = /lo versus the composite Ha p > |lo where /l0 is a fixed constant: Can we find a UMP test?



Answers

When the sample size $n$ is very large, lower and upper confidence bounds for $\mu$ can be obtained by replacing $t^{*}$ with $z^{*}$ in the expressions from the previous two exercises. For example, a large- sample lower confidence bound for $\mu$ is given by $L=\overline{x}-z^{*} s / \sqrt{n},$ where $z^{*}$ satisfies the relation
$P\left(Z<z^{*}\right)=c$ when $Z \sim N(0,1)$ and 100$c \%$ is the prescribed confidence level $($ e.g. $, c=.95)$
(a) Show that the $z$ critical value for a one-sided (i.e., upper or lower) confidence bound for $\mu$
with confidence level $100 c \%,$ is given by $z^{*}=\Phi^{-1}(c) .$
(b) Find the one-sided $z$ critical values for $90,95,$ and 99$\%$ confidence.
(c) A certain random variable is simulated $10,000$ times. The sample mean and standard
deviation of the resulting $10,000$ values are 41.63 and $8.05,$ respectively. Calculate and
interpret a 95$\%$ lower confidence bound for the expected value of this rv.

Hi. Yeah it has been given that X one X two. So on X. In a random sample from a distribution with pdf. If X did in Egypt I'd But if X three days one by theatre it will give our miners eggs by teacher zero less than X less than infinity. And the target at 10 00. Otherwise we have to find the similar to the estimated Emily of probability Eggs less than equals two two. Okay. And also we have to show that that the Emily of this is a consistent estimator of this probability. Okay now first we find probability X less than equals two two. This is integration 0- two. F. X. Dx. Our integration 0-2 one by theater. Into the minus eggs by theater dx. So it will be too they were minus exploit theater. My nurse 02 two To be one into the about minus to write a letter. No is he then parametric function of peter we have to first we have to find the Emily of this. We consider it as Gt first we find the man similarly raised the matter of africa. Okay for that. Well right the likelihood function the likelihood function is noted by L. T. To colony X. Girl equals to the product of islands from one to N. F. X. A tender. So this is one by theater to the border and into the minus Submission inside items from 1 to N. Holding a theater. Now maximising the likelihood function equivalent to maximizing log likelihood function. So we write it log likelihood function of el sida colon. Ex girl equals two script. L theater colon X. Skirt which is minus emission X. A. All divided by tika minus in blog theater. Now to find the max to find the value of peter to get a million of theater says that it maximizes the log likelihood function. So for that we find the first order derivative of this log likelihood function which is script. L theater ex girl annual equal to zero. Okay so first you know david Davies minus submission. Excite delivery by theater square. It will be plus because one way theatre des reputable -1 by theatre square so minus and by Theatre Equals to zero. So I think I will get submission X. I deliberately and so for the situation we have theater had Emily equals two submission X. I islands from on this is an estimated will get red capitalists were debated by an equals two expert. So eggs berg is the maximum distributed or Emily of data. Okay now we have got parametric function this which is G theater equals to one minus to the minus two by theater. Okay now we consider invariance property of Emily invariance property invariance property of Emily which states that if easy Real 1-1 function 1 to 1 function then if from a sample an estimated dx is a is a maximum likelihood estimator of theater then then that implies G. O the X. Is amelia jean to death. Okay bye. Invariance property. Now, clearly here G peter with one minus to the minus two by theater. This is a proper real and 1 to 11 to one function. So Emily of jupiter will be GT to head. Emily equals two G cleared ahead. Emily that is G expression So that means 1 -1 to deliver it expert. So this one is estimated one minus Italy were minus to buy expert disease. Emily of many of 1- to the minus one minus E to level minus two by theater which is genetic. So you got the familiar probability X. Less than equal to two. Okay, now we have to find the consistent estimator of this. Gt to Okay see we have got Emily of Canada which is expert. Okay now we have to take disease. Now our question is is it it is expert is a consistent estimated of theater. So to find consistent, estimated if expert converges in probability to theater then we say that expert is a consistent estimated of theater which implies limit, intends to infinity probability. Lord expert minus theta not close less than absalon equals two. One. What limit intends to infinity probability what eggs burr minus theta Not closed. Better than equals two. Excellent equals to zero. Okay see eggs berries and consistent estimator of twitter to check it. We use T J b ships inequality. Okay first we need to consider that he had each X. I follows exponential distribution with mean theater. That means expectation of X. I equals to theater and variants of X A Equals two m square Okay, wow expectation of eggs where we have also to and variance of expert equals two 3 to square were liberated by in now by iptv ship seniority jimmy ships inequality community we have probability maude expert minus theta not close get it. Then he calls to absalon Less than equals two expectation of this more exposure minus theta whole square hole diverted by excellent square where excellent is a very small quantity everywhere I have written Excellent, excellent is a very small quantity which tends to zero. Okay and this is equal to experience of expert. All liberated by excellent square variance of expert. We have got to square brian Now equals to zero as intends to infinity. So see we have got limit, intends to infinity probability mud exposure minus theater not close get a chemicals to excellent equals to zero. So the first here I have written by the definition of consistency, We're getting limited interest in 3D poverty more expire -3. To close that doesn't explain, it goes to zero. So definitely we can write expert converges in probability. So to to expert converges improbability to data, that means expert is a consistent estimator of twitter. Okay, now see another by another theory we have we have taken a function G X. Because to 1 -1 -2 x. Now this function is continuous and real continuous at X equals to theater. And also it is a real function. Okay, now we can ride that G of X. Bar converges improbability to geo theater, Not geo of expert. What is that? 1 -1 who? By expert Converges improbability to 1 -2 -2 by theatre. So definitely we're coming to the conclusion that this is probability X less than equals two. Now we we we have found that what we already explained them to one manager could even managed to wait it out. 1 -2 -2 by expert is a consistent estimated of this. This is the answer. Thank you.


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