SIMULATION - PROBLEM SET Problems 1 to refer the following random sample of 15 data points: 8.0, 5.1,., 8.6, 4.5, 5.6, 8.1, 6.4,., 7., 8.0, 4.0, 6.5, 6., 9.1 The following three bootstrap samples of the empirical distribution have been simulated. Each sample is of size 15. Sample 1:.,.,., 4.0, 4.0, 4.0, 6.5, 7., 8.1, 8.6, 8.8, 9.1, 9.1, 9.1, 9.1 Sample :.,., 5.1, 5.1, 5.1, 5.6, 6., 6.4, 6.5, 7., 7., 8.0, 8.0, 8.0, 8.1 Sample :.,.,., 4.0, 4.5, 4.5, 5.1, 6.5, 7., 8.0, 8.1, 8.1, 8.1, 9.1, 9.1 1. Find +,, where +œsample variance (unbiased form), and,œ variance of the empirical distribution A).0 B). C).4 D).6 E).8. Suppose that ) is the mean of the distribution, and it is being estimated by the sample mean. Use the three bootstrap samples to estimate the QWI of the estimator. A).01 B).0 C).0 D).04 E).05. Suppose that ) is the probability TÒ\ Ÿ &Ó, and it is being estimated by the proportion of data points in the random sample that are Ÿ&. Use the three bootstrap samples to estimate the QWI for this estimator. A).018 B).00 C).0 D).04 E).06 4. The mean of a distribution is being estimated, using the sample mean of a random sample as an estimator. A sample of size is drawn: B œ!ß B œ. QWIÐJ/ Ñ is the mean square error of the estimator when the empirical distribution is used. Find the exact value of QWIÐJ / Ñ. A).15 B).50 C).75 D).500 E).65 5. With the bootstrapping technique, the underlying distribution function is estimated by which of the following? (A) The empirical distribution function (B) A normal distribution function (C) A parametric distribution function selected by the modeler (D) Any of (A), (B) or (C) (E) None of (A), (B) or (C) 6. You are given a random sample of two values from a distribution function J : 1 You estimate Z+<\ ( ) using the estimator 1 (\ \ ), where \œ \ 1 \. œ1 Determine the bootstrap approximation to the mean square error. (A) 0.0 (B) 0.5 (C) 1.0 (D).0 (E).5
7. You are given a random sample of two values from a distribution function J : 1 You estimate Z+<\ ( ) using the estimator 1\ (, \ ) œ (\ \ ), where \œ \ \ 1 1. œ1 Determine the bootstrap approximation to the mean square error. (A) 0.0 (B) 0.5 (C) 1.0 (D).0 (E).5 8. The following random sample of size 5 is taken from the distribution of \: 1,, 4, 7, 10 Bootstrap approximation of the mean square error of estimators is to be based on the following 6 resamplings of size 5 from the empirical distribution: Resample 1 : 1, 1, 4, 7, 7 Resample :, 4, 4, 7, 10 Resample : 1, 4, 4, 10, 10 Resample 4 :,,, 4, 10 Resample 5 : 4, 4, 7, 7, 10 Resample 6 : 1, 7, 7, 10, 10 The median of \ is estimated by the third order statistic of a sample. Find the bootstrap approximation to the estimator of the median using the 6 resamplings. 9. \ is the random variable denoting the number of claims in one day. The following is a sample of the number of claims occurring on 5 randomly chosen days: 0 4 5 7 The following estimator from a sample of 8 days is used to estimate TÐ\ Ÿ Ñ, the probability days with 4 claims or less of 4 or less claims in one day: )s œ 8 The bootstrap approximation is applied to estimate the mean square error of )s using the following 8 samples simulated from the empirical distribution of the original sample: Sample 1 0 0 7 7 Sample 4 5 Sample 7 5 4 5 Sample 4 4 0 7 5 Sample 5 7 5 5 Sample 6 4 5 4 7 Sample 7 0 0 4 7 Sample 8 5 0 4 7 5 Find the bootstrap approximation to the mean square error of the estimator.
SIMULATION - PROBLEM SET SOLUTIONS 1. Sample variance œ W œ ÐB BÑ œ Ò B ÐBÑ Ó œ Ò '$&Þ* Ð'ÞÑ Ó œ Þ$( Þ Variance of the empirical distribution is œ œ Z +<Ò\ Ó œ ÐB BÑ œ Ò B ÐBÑ / Ó œ W œ $Þ*&&. Then œ œ +, œ W W œ W œ Þ)Þ Answer: E \ â \. The estimate of the distribution mean is the sample mean. In this example, <œ$ is the number of bootstrap samples that are available. Also, ) is the distribution mean of the empirical distribution, which is equal to the sample mean B of the original random sampleß B œ 'Þ!. The values of the estimator for the three samples are the sample means of the bootstrap $Þ$ $Þ$ â *Þ *Þ samples - œ 'Þ& ß 'Þ$ ß and 'Þ!). Then the estimated MSE is $ [ Ð'Þ& 'Þ!Ñ Ð'Þ$ 'Þ!Ñ Ð'Þ!) 'Þ!Ñ Ó œ Þ!$( Þ Answer: D. The value of ) is the probability in the empirical distribution that \ Ÿ &. This is œ Þ'(, since 4 of the 15 original sample points are Ÿ &. The values of the estimator for 4 œ ß ß $ are the proportions in each bootstrap sample of the points that are Ÿ&. These are 6 6 œþß œþ$$ßand œþ. Then, the estimated MSE is $ [ ÐÞ Þ'(Ñ ÐÞ$$ Þ'(Ñ ÐÞ Þ'(Ñ Ó œ Þ!() Þ Answer: A 4. The parameter being estimated is the distribution mean, and the mean of the empirical B B distribution is.5. The estimator being used is the sample mean,. Thus, for the 4 pairs ÐBßBÑ, we have estimator values!ßþ&ßþ&ß, and the estimate of the MSE is ÒÐ! Þ&Ñ ÐÞ& Þ&Ñ ÐÞ& Þ&Ñ Ð Þ&Ñ Ó œ Þ. Answer: A 5. Answer: A. 6. Suppose that a random sample from a distribution is given: \ ß \ ß ÞÞÞß \ 5, and suppose that the sample is used to estimate some parameter of the distribution. If the estimator is )s, then the bootstrap approximation to the mean square error of this estimator is I Ðs) ) Ñ. In this expression ) is the value in the empirical distribution of the parameter being estimated, and the expected value is taken within the empirical distribution. In this case, the parameter being $ estimated is the distribution variance. The mean of the empirical distribution is œ, and the variance of the empirical distribution is ÒÐ Ñ Ð$ Ñ Ñœ.
6 continued To find the expectation, we must consider all samples of size from the empirical distribution (since the empirical distribution was based on a sample of size itself). The mean square error of the estimator is I Ðs ) Ñ. From the empirical distribution on the set Öß $, there are four possible samples, which are Ð\ ß \ Ñ À Ðß Ñ ß Ðß $Ñ ß Ð$ß Ñ Ð$ß $Ñ (\ \ ) and. For each sample, we must calculate 1 œ for that sample. For instance, for the sample Ðß Ñ, we have \ œ, and (\ \ ) œ!. œ1 The bootstrap approximation to the mean square error of the estimator is the average of the 4 values of (\ \ ) that we get. This is summarized in the following table. œ1 Sample Ð\ ß \ Ñ Ðß Ñ Ðß $Ñ Ð$ß Ñ Ð$ß $Ñ Ð\ \Ñ œ!! Since each of the four possible samples is equally likely to occur, the bootstrap estimate of ÒÐ! Ñ Ð Ñ Ð Ñ Ð! Ñ Óœ. Answer: B 1 7. Since the sample consists of only sample points, there are only 4 possible, equally likely bootstrap samples: (1) ß () ß $ () $ß (4) $ß $. We are trying to estimate Z+<Ð\Ñ, so we find the variance of the empirical distribution. The empirical distribution is a -point random variable and its variance is ÒÐ Ñ Ð$ Ñ Óœ. The bootstrap estimate of the MSE of the estimator s) is s IÒÐ) Ñ Ó. To find this expectation we first find )s œ ( \ \ ) for each of the samples: (1) ß p s) œ! () ß $ p s) œ () $ß p s) œ (4) $ß $ p s) œ!. The bootstrap estimate is the average of the values of Ðs) Ñ for the samples: ÒÐ! Ñ Ð Ñ Ð Ñ Ð! Ñ Óœ. Answer: C œ1 8. The median of the empirical distribution is ) œ. Resample s) Ðs ) Ñ 1, 1, 4, 7, 7 4 Ð Ñ œ!, 4, 4, 7, 10 Ð Ñ œ! 1, 4, 4,!,! Ð Ñ œ!,,, 4, 10 $ Ð$ Ñ œ 4, 4, 7, 7, 10 ( Ð( Ñ œ * 1, 7, 7, 10, 10 ( Ð( Ñ œ *!!! * * ' The bootstrap estimate of MSE( )sñ is œ $Þ(.
$ 9. In the empirical distribution, the actual value of the parameter being estimated is ) œ & œ Þ', since three of the values in the 5-point empirical distribution are Ÿ. We calculate s) and Ðs) ) Ñ œ Ðs) Þ'Ñ for each simulated sample. s) ) œþ' Ðs) Þ'Ñ Sample 1 0 0 7 7 Þ'! Sample 4 5 Þ) Þ! Sample 7 5 4 5 Þ Þ! Sample 4 4 0 7 5 Þ'! Sample 5 7 5 5 Þ Þ! Sample 6 4 5 4 7 Þ'! Sample 7 0 0 4 7 Þ) Þ! Sample 8 5 0 4 7 5 Þ Þ! & Þ! The average value of Ðs) Þ'Ñ is ) œ Þ!& ; this is the bootstrap approximation to the mean square error of )s based on the 8 samples.