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F79MA/F71MA Assessed Project
创建日期：2024-04-17 17:35:05
标签： F79MA
Assessed Project

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F79MA/F71MA  Assessed Project 

 In this project we will look at maximum likelihood estimation for a gamma model and show that computer simulation can be a very useful tool to characterise estimator performance. Project description You are a statistical trainee actuary working for an insurance company. You are part of a statistical modelling team that is developing models based on independent non identically distributed random variables. You consider the following model Xi ∼ Gamma(αωi, β) , i = {1, . . . , N} where α, β, and ω1, . . . , ωN are positive scalar parameters (the gamma distribution has several common parametrisations, here we define the p.d.f. of a random variable X ∼ Gamma(α, β) as f(x) = β α Γ(α) xα−1 exp{−βx} for x > 0, where Γ(·) denotes the gamma function, see https://mathworld.wolfram.com/GammaFunction.html). You have been asked to explore the statistical properties of the maximum likelihood estimator of β assuming that α and ω1, . . . , ωN are known. Assume that α > 2 and ∑N i=1 ωi ≥ 1. Let X = X1, . . . , XN . You decide to perform the following analyses: 1. State the likelihood function L(β; X) = f(X; β). [2 marks] 2. Derive the maximum likelihood estimator for β, denoted by βˆ(X). [2 marks] 3. Show that the bias of βˆ(X) is given by B(β) = β/(α ∑N i=1 ωi − 1). [3 marks] 4. Derive the expression for the Cramer Rao Lower Bound for biased estimator of β with bias B(β) = β/(α ∑N i=1 ωi − 1). [3 marks] 5. Let α = 4 and ωi = i for i = {1, . . . , N}. Use simulation in R to calculate the variance of βˆ(X) for β = {1, 2, 4, 8, 16} and for different sample sizes N ∈ [1, 20]. Briefly discuss your results and present appropriate graphical summaries. How close are the variances of βˆ(X) to the theoretical lower bound found in Part 4? [6 marks] 1 Your findings should be presented in the form of a report, which should: • have a clear and logical structure; • include an introduction and clearly stated conclusions that can be understood by any numerate scientist; • include detail of your mathematical calculations so that your results could be reproduced by another statistician; • include clearly labelled and correctly referenced tables and diagrams, as appropriate; • include the R code you used in an appendix (you do not need to explain individual R commands but some comments should be included to indicate the purpose of each section of code); • include citation and referencing for any material (books, papers, websites etc) used. • maximum page limit of two (2) pages (11-point font, A4 size). The R code in the appendix does not count towards this page limit. A total of 4 Marks is available for these aspects of your report. This will be marked according to the rubric given in the Appendix. [Total: 20 Marks] Notes • This assignment counts for 20% of the course assessment. • You may have face-to-face discussions with me or your colleagues, but your report must be your own work. Plagiarism is a serious academic offence and carries a range of penalties, some very serious. Copying a friend’s report or code, or copying text into your report from another source (such as a book or website) without citing and referencing that source, is plagiarism. Collusion is also a serious academic offence. You must not share a copy of your report (as a hard copy or in electronic form) or your computer code with anyone else. Penalties for plagiarism or collusion can include voiding of your mark for the course. See https://www.hw.ac.uk/students/studies/ examinations/plagiarism.htm for more details. • Your report should be submitted through Canvas on Friday 15 October 2021. Stu- dents based in the Edinburgh campus should submit by 18h00 UK time. Students based in the Malaysia campus should submit by 18h00 Malaysia time. A link to the submission page is available through the ‘Assignment’ tab of the course Can- vas page. For late submissions, 30% will be deducted for work submitted at most 5 days late. Submissions that are more than 5 days late will receive 0 marks. You will receive feedback on your submission within 15 teaching days. 2 Appendix: Rubric for marking of the report The five marks available for the exposition of your report will be awarded according to the scale below: 0 Marks • Lack of clear and logical structure will be • Conclusions missing or not suitable for a non-statistician awarded • Statistical calculations and methodology not clearly set out for the reader for • Tables and figures unclear, badly labelled or not correctly referred to • R code not included, or no comments included in it • Sources used not clearly referenced 2 Marks • Clear and logical structure will be • Conclusions generally suitable for a non-statistician awarded • Statistical calculations and methodology generally set out clearly for the reader for • Tables and figures often clear and correctly referred to • R code included with some comments • Sources used clearly referenced 4 Marks • Clear and logical structure will be • Conclusions suitable for a non-statistician awarded • Statistical calculations and methodology set out clearly for the reader for • Tables and figures clear, correctly referred to and easy to interpret • R code included with comments • Sources used clearly and correctly referenced 3 













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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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