Stat/Math 430 Mathematical Statistics Final Exam Take Home Distributed May 6, Due May 14, 2015 by 3 pm to 306 SM

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1 Stat/Math 430 Mathematical Statistics Fial Exam Take Home Distributed May 6, 05 Due May 4, 05 by 3 pm to 306 SM Istructios:. Show all work (o your ow paper or via a compiled RMarkdow file). You may receive partial credit for partially completed problems, so you should try all parts of all problems. Tur i ONLY your FINAL solutios please. Please write solutios legibly!. The exam is ope book ad ope otes. You may also refer to your homework, homework solutios, hadouts, ad your exams, as well as ay other items posted o our course Moodle site. 3. You may ot use ay exteral refereces/texts ad o olie media (i.e. closed geeral iteret). 4. You may use a calculator, Excel, R/RMarkdow, or ay other statistical computig program to help with calculatios. Accessig related help files (for example?porm) is permitted. A template is provided for RMarkdow to assist with data etry. 5. You may ot discuss the exam with ayoe but Prof. Wagama util after the exam has bee tured i by all studets. If you ask me a questio which warrats clarificatio, I will sed my reply to all studets i the class. 6. You may take as much time as you like ad may complete the exam i multiple sittigs. 7. Suggestio: Poit values per problem are displayed below if that helps you allocate your time amog problems. 8. Office hours: Posted o Moodle. For other times, just sed me a , ad we ll arrage a time to meet for your questios. 9. Good luck! Problem Total Possible Poits

2 . Estimatio ad The Power of Simulatio () Adapted from Rice Type Cout Probability RV Starchy gree ( ) Cosider the followig data for two categorical variables related to theoretical probabilities about Starchy white ( ) geetics of plat offsprig from self-fertilized heterozygotes (this meas AB x AB as described i Sugary gree ( ) 3 class). The two variables are cotet (starch/sugar) ad color (gree/white) from which four types of Sugary white 3. 5( ) 4 plats arise. The parameter is bouded betwee 0 ad ad is related to the likage of the uderlyig gees. The data is take from Fisher 958. a. We wat to test whether or ot the data is cosistet with the stated theoretical probabilities. Derive the MLE for, deoted ˆ, ad fid its value based o the data. b. Perform a appropriate test procedure to determie if the data is cosistet with the stated theoretical probabilities at a sigificace level of Be sure to show work to validate your steps, ad report your fial real-world coclusio. Now suppose we are really iterested i due to its coectio to likage properties. I fact, we wat to form a cofidece iterval for. Assumig our large-sample theory applies toˆ, as a MLE, the we eed to derive or estimate its variace i order to costruct the cofidece iterval for. The derivatio is ivolved, so perhaps we could use simulatio to estimate its variace. c. Explai how you would use a simulatio to estimate the variace ofˆ. Be sure you address how radomizatio would be employed ad state the ecessary iputs for the simulatio to ru. (Thik carefully about this!) I strogly suggest you write pseudocode to show what steps you would use i your simulatio. As you probably suspect, workig withˆ is ivolved, so we could cosider alterate estimators of. d. If we used oly, the radom variable associated with the cout i the first cell, to come up with a atural estimate of, what would your estimator be? Deote this estimator ~. e. Evaluate the estimator ~ o this data set. How does this estimator compare toˆ, based o this data? f. Usig properties of the margial distributio of, show that ~ is ubiased for ad fid a expressio for its variace. (Simulatio is ot eeded here as a exact expressio ca be foud).. A Little Bit More Estimatio (3) Suppose,,..., is a radom sample of observatios from a Pareto distributio with pdf i the geeral case of: f ( x) where x 0, ad 0, ad 0, otherwise. Assume that is kow. x We will also assume that for our derivatios, but that it is ukow. a. Fid the method of momets (MoM) estimate of. b. Now assume is kow, but is ot (just for this part of the problem). Fid the MoME for. Is the MoME ubiased? Show your work. If it is ot ubiased, fid a ubiased versio of it, if possible. c. Fid the MLE of. d. Idetify a sufficiet statistic for, ad justify your choice. e. Is your MLE miimal sufficiet? Explai i oe setece.

3 3. Quadratic Regressio with Costraits (6) Suppose you sample pairs of observatios i a settig where you believe a regressio model is appropriate ad you believe that the relatioship betwee Y ad is give by E Y ) x x, ( i i i i=,,. Additioally, you kow the setup of the experimet is such that 3 x 0 (symmetric i x i aroud 0). Fially, you assume the remaiig regressio assumptios are met homoscedasticity (costat variace), idepedece, ad ormality of the error terms. a. Determie the relevat least squares equatios for the model ad use them to solve for explicit forms for ˆ ad ˆ. b. Fit the model to the data set displayed i the table below ad report the estimates of ˆ ad ˆ. c. Determie a ubiased estimator for, give its formula for this settig, ad obtai its value for this data set. (You do ot eed to prove that it is ubiased. ) d. Treatig the x values as give or coditioed o (i.e. you may treat them as costats), prove that ˆ ad ˆ are ubiased estimators for their respective parameters. e. Fid the variaces of ˆ ad ˆ as fuctios of. For this data set, which estimator, ˆ or ˆ, will have the lowest variace? Regressio Data: Y Note: You may use R for basic computatios here (for example to fid a mea or SD), ad ideed you ca use R to CHECK your computatios for part b. ad c., but you must show work for derivig the estimates ad performig the various computatios for this data set. 4. Likelihood Ratio ad UMP Tests (6) Suppose,,..., is a radom sample from a Poisso( ) distributio. a. Fid the likelihood ratio test for testig H 0 : 0 vs. H A : 0. Be sure to clearly state your test statistic, the distributio of the test statistic, ad the form of the rejectio regio for a give sigificace level. You should NOT eed to use the asymptotic result to get a distributio for the test statistic (but do that if you ca t get oe aother way). b. Complete problem 0. i your textbook (page 555). c. Argue that the likelihood ratio test you foud i part a. is UMP usig your results from b. (Hit: it may help to pick a specific value uder the alterative).

4 5. Geeral Testig (3) Two city coucil members are iterested i estimatig the proportio of voters who pla to vote i favor of a secod term for them. The first city coucil member obtais a radom sample of 5 voters ad fids that 3 say they pla to vote i favor of a secod term for the coucil member. The secod city coucil members hires a surveyig firm, which obtais a radom sample of 50 voters ad fids that 6 pla to vote i favor of a secod term for the coucil member. a. For the first coucil member, is there sigificat evidece that the proportio of voters who pla to vote i favor of a secod term is greater tha /3? As part of your respose, compute a appropriate p- value, ad explai what it measures ad meas i layma s terms. Use a sigificace level of b. For the secod coucil member, obtai a 90% cofidece iterval for the proportio of voters who pla to vote i favor of a secod term for that coucil member. As part of your respose, explai what the 90% cofidece level refers to. c. Igorig the coucil members re-electio chaces, which coucil member has more reliable results based o the data collected ad aalysis above? Why? Explai. 6. Bayesia Iferece (8) adapted from Chihara ad Hesterberg Suppose,,..., is a radom sample from a Uiform(0, ) cotiuous distributio. A prior desity for is selected to be the Pareto distributio, givig the prior the followig form: g( ) where 0, ad 0, ad 0, otherwise. a. Explai why a ormal prior o would ot make sese i this cotext. b. Derive the posterior distributio for based o the radom sample of observatios. You may use a proportioality argumet ad drop costats, but eed to show some work for obtaiig the posterior distributio. c. Assume that you observe the followig four observatios from the Uiform(0, ) distributio: 6, 6, 8, 0, ad that the Pareto prior o has parameters 0. 3 ad 5. Fully specify the posterior distributio for. d. State the formula for the Bayes estimate of ad fid its value i the same settig as part c.

5 7. Assessig the Fit (3)- data take from Rice A study examied birds which were feedig ad researchers couted the umber of hops betwee flights for each bird. The frequecy of each cout of umber of hops is reported i the table show: # of Hops Frequecy We wat to fit a appropriate model to this data. a. Lookig closely at the data, explai why a Poisso model would ot be appropriate to fit to this data. b. Fid the MLE for p whe fittig a geometric distributio to this data. I other words, fit a geometric distributio to the data, ad report the best-fittig p. Note: i your textbook, the geometric is defied i terms of the trial umber of the (first) success, where p is the probability of success. c. Assess the fit from part b. usig a appropriate GRAPH. Be sure to commet o the fit ad explai your choice of graph. d. Assess the fit from part b. usig a appropriate iferece procedure (with sigificace level 0.0). Be sure to check all relevat coditios, show justificatio for your work, ad report a fial coclusio.