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Suppose 01 and Oz are two estimators of the parameter 0. We know that E(O1) = and E(O2) 0/2, Var(O1) 10 and Var(02) Which estimator is better? Explain your choice....

Question

Suppose 01 and Oz are two estimators of the parameter 0. We know that E(O1) = and E(O2) 0/2, Var(O1) 10 and Var(02) Which estimator is better? Explain your choice.

Suppose 01 and Oz are two estimators of the parameter 0. We know that E(O1) = and E(O2) 0/2, Var(O1) 10 and Var(02) Which estimator is better? Explain your choice.



Answers

Suppose that $\hat{\Theta}_{1}$ and $\hat{\Theta}_{2}$ are unbiased estimators of the parameter $\theta$. We know that $V\left(\hat{\Theta}_{1}\right)=10$ and $V\left(\hat{\Theta}_{2}\right)=4$ Which estimator is best and in what sense is it best? Calculate the relative efficiency of the two estimators.

Welcome to this lesson in this lesson. We have to Estimated estimated one an estimated two. We have the Expectation of estimate to one which is called to the theatre and or the perimeter Unless petition of estimated to go to the Parameter on two. The variance of the estimate one that's called to 10. And the variance of Estimated to that is you call to four. We are looking at the best estimator we are looking for which of the two estimated is the best for the Prime Minister. So we use the relative efficiency of the two estimated to determine that if a relative uh efficiency of the two parts of the two estimated It's less than one means that the perimeter one uh the estimator one is the best s later. And also if it's greater than mhm Is greater than one then The estimated two becomes the best the better of the two. Yeah. Okay so let's go on. We have the relative relative efficiency which is he called the mean square error of the first. I meet the first estimator all over the mean square error of the second estimator. So this is the call to the variance all the firsts estimated last the Bayous quit. Mhm Yeah. Yeah. The Bath of the fast estimated squared all over the variance of the second estimate last the bias of the second estimated square. Okay. Mhm. So here we have the variance which is 10 for defense plus Tobias Tobias is expectation of the fire of the estimator minus the yeah perimeter. Okay. Yeah. So we also have the variance of the second which is four alas the bayous which is recalled to the expectation of the second. Yeah minus the parameters card. Okay so the whole thing here is the bias is the bayous. Mhm This is the bayous. Okay. Mhm. Mhm. At this point we have 10 plus now the expectation For the 1st 1. Yes the perimeter. So yeah Then we have four plus the I've been telling The permit on two minus the perimeter. Okay So this is a call to 10 and that is for mhm. Mhm Yeah four minus half of the private. Okay or we can say 20 four minus the hire me 10. Okay. So with this one we see that the first the first one has no bias. The second one has some kind of my ass. So prefer if we are comparing in terms of by us then we can say the first one is the better but sometimes a little by us can make the estimator a bad test meter. Uber one that has even know by us. Okay. All right. Mhm. So this one as soon as the parameter is none negative. Uh It means that the denominator is less than the numerator and as such this is better than one. So we say that the estimated the second estimator is the better estimated. Mhm. Okay. Thanks for time. This is the end of the lesson

This lesson in this list. We have a simple size. The population A population size of two. N. So we would look at the point is debater as by one and that's about to see which of them is better estimated for the population mean? So here we will take the variance of the two point estimate us in the one way the lowest variance becomes the better estimator for the population mean. Okay. Mhm. So let's go for the first one. The bar this just record too. Mhm. Mhm. Mhm. So those too sick or too? Yeah, one of four N squared mm mm Yeah, nobody X I Yeah. Okay, so one of our four and it's quiet. Yeah. Yeah. Yeah. Yeah. Mhm. In the whole of this becomes the babies. So this becomes to n Variance all over four and squared. Yeah. And now the two councils are the four and councils one of that. So we have the variance all over two in all right, that is a bar of the first point estimator. Let's look at developed the second point estimated which is a call to uh one over end. Mhm. Yes, like the meat. So this is equal to one over and squad done decimation. O E. Oh, okay. Does the the bar. Yeah. So this is equal to one over N squared. Yes. The submission from 1 to 10 and the whole of this becomes the expectation the becomes the various it becomes the variance. Okay. Yeah. So this is equal to and mhm that's the whole of the top under down this end squared and this becomes the variance all over. And Okay, so this is the uh The 2nd point estimate. So we have the first one. Let me use red to do that. We have the first one. Okay, us the variance to N. And you're having the second one experience on N. So looking at the since N is greater. Mhm. Yeah. Down zero. Okay. Mhm. We have yeah. All right. The variants of the first one. Mhm. Problem. But on the other. Okay. Yeah, lesser than yeah, the variance of The 2nd 1. Okay. Yes. So it means that the first one should be the one that would choose to be the better estimate the the better point estimated. Okay, so we choose that because the variance it's smaller than the first one. Okay. Thanks for your time. This is the end of the lesson. Okay.

In this video, we have three diagrams on each diagrams. There are two a statistics and we are supposed to say in Easter ground which statistics is the better estimate. Oh, record of Senatore east on boys has no variability in part A. You can see that water statistics are on boy us because both are centered eggs. It'll, however, there variability off. Two is less than the variability off one. Therefore to is the better stomach. Oh, in part B was estimate or the bosses statistics have low very baby. However, only to ease on bias on one use. Negative bias. Therefore, to is the bigger cynical. In part, C statistic Born is negative bias and is not gonna bias onda statistic to has high variability. Therefore, the solution to Carsey Knees *** Well, you're going to want to index video and thanks for what

Come to this lesson and this lets we have two point estimated a rear checking whether or not they're on my ass estimate is for the population parameters that will find the variance of each parameter. Each estimated the first but we are looking at the expectation of this point is meeting what should be able to give us the population me. Yeah it virtually Azam by asked. Okay so let's see what let's see if that is she caught in that Huey chef the expectation of X one last X two along too. So we can split it into we can bring the half outside so that we have expectation of this one. The last expectation of X two then half half out Expectation of X one. The last expectation of X two. Okay. Okay. Mhm. So this would give us yeah for me Yeah. Mhm. Okay. And this will also give us this will also give us the population mean? So we have half Than two times the population and this becomes me. So the expectation of the point estimate has given us me which means it is on my ass. So it's just until I asked. Yeah. Yes. Okay let's check the second one. If The one is also I'm biased. Mhm. Yeah. Yeah. Yeah. Mhm. You're looking at the second one we have the expectation of the second estimated Which is excellent for last three times x She all over four. So expectation of this in the same we can bring the ones fourth out. Okay. And now this becomes from them before yeah Expectation of X one last three expectation of X two, This becomes one on 4. This is the population mean, than three times the population means, so this becomes one on four Times four times the population meat. And this becomes the population mean Yeah, it means the second point of tomato is also and I'm by asked estimate of the population mean this was so unbiased. The second path, we are looking at the variance of the two, so by professes to meet her, we have X one for all aspects to all of it too. So the to come out as a square, let's play both. So we have X one on 2 plug back. That's true, only truth. Yeah. Mhm. Yeah. Uh huh. So this is one on 4, the consent comes up as a square, Then the bar of X one becomes the experience Plus one on 4. The very mess. Okay, so this becomes two Time, is the variance all over four and this becomes squired. Uh two two goes into someone's, it goes into four two times. So we have the bar of the parameter as the variants all over two. Okay, that's the first one, Just go to the 2nd 1, you have the second Yeah stigmata which is where X one plus three. X. See all of the four. So with this one we can splits, you have bar that's a bar three or 4. That so this becomes one on 16, the variance for us, nine on 16 times very innocent. So this is equal to 10 times the variance all of US system, and that eventually becomes five with eight times the various. Yeah, so this is the bar of the 2nd Parameter, uh, estimate. Okay, some short time. This is the end of the lesson.


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