The model, the relevant algorithm, formulae involved and useful explanation of your calculations

School of Computer Science and Mathematics

 Faculty of Science, Engineering and Computing

Time Series Analysis and Further Inference with Applications Including Insurance Models (ST6300)

Retake Coursework   -   2019-20

Submission deadline of this coursework:

Instructions to Candidates:

Answer Both Questions 1 and 2

1. Q1 and Q2 each carries 100 marks
2. Use 4 decimal places in your calculations

Candidates are required to write their solutions VERY CLEARY, and showing ALL calculations- (Page numbering is also required)

 All of your answers must be incorporated into a single document (Word or Pdf)

1.  The assessment must be submitted electronically via Canvas
2. Any assignment showing evidence of copying may result in a mark of zero

General Guidelines to Candidates

• As this is a coursework assessment there is no extra time allocated for students with a SOSN.
• If you experience technical difficulties, e.g. access and upload issues, or identify a potential error in a question please email the module leader [email protected] .
• This is an open book assessment so you may consult your notes, textbooks and the Internet.
• You must not collaborate with someone else on this assessment, it should be wholly your own   work. Your work will be checked for evidence of plagiarism and/or collusion using Turnitin.
• If you include graphics in your answer, please embed these into the Word document (e.g. a photograph of a hand-drawn graphic). The source of any copied and pasted figures should be cited. If you have embedded any scanned/photographed diagrams and graphs, then your submission file cannot exceed 100MB. The model, the relevant algorithm, formulae involved and useful explanation of your calculations

Total number of pages including this page: 4

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1:    The monthly sales of personal computers by an electric outlet for the past 8 years are tabulated 

        below: 

   (The data is read column-wise)

Year	2012	2013	2014	2015	2016	2017	2018	2019
	↓	↓	↓	↓	↓	↓	↓	↓
January	128	124	135	132	138	157	161	165
February	120	114	123	131	142	146	147	161
March	116	110	114	115	121	127	134	143
April	108	103	105	112	124	123	124	135
May	85	93	99	101	109	107	112	114
June	96	98	101	96	106	111	122	112
July	95	96	100	110	122	125	124	120
August	93	94	104	114	110	112	119	115
September	118	125	126	131	130	124	120	162
October	124	123	125	129	132	145	137	160
November	124	124	132	138	147	143	152	168
December	134	137	142	141	150	158	158	169

 

Plot the time series and comment on your visual observation of the graph. Then, by applying the following two methods, determine the outlet’s sales forecasts of computers for the months of 2012-2019, where applicable.

1: The Multiplicative Decomposition Method.

2: The Single Exponential Smoothing Method (SES).

In each of the above two methods you need to include at least the following set of guidelines:

(a): The model, the relevant algorithm, formulae involved and useful explanation of your calculations.

(b): Your Excel worksheets compactly presented with suitable titles and column headings.

(c):  Calculations of the optimum values (where applicable, using the Excel Solver) for the parameter involved.

(d): The sales forecasts for periods 2012-2019, where applicable, and the corresponding forecast errors.  

(e):  The Mean Squared Error (MSE) and the Mean Absolute Percentage Error (MAPE), for the test period of November 2012to December 2019 (inclusively).                                          

In addition, the following needs to be provided:

(f):  The comparative assessment table for the above two forecasting techniques using the MSE and the

       MAPE. 

(g):  For each method, illustrate its forecast plot individually.  Furthermore, in a single diagram, also show all

forecast plots and critically compare them with the time series plot of the data.  Hence, using both assessment criteria table in (f) and the forecast plots, decide on the optimum forecasting method for the sales of personal computers. 

(h): Forecast of the monthly sales of personal computers by the electric outlet for the year 2020, using

       your optimal model. 

Retake Coursework Continued … 

2: A random sample  is drawn from a distribution with probability density function with mean and variance and, respectively.   

An estimator of given by for a constant and the sample mean, is being considered

(a) Show that and. (b) Show that the bias of this estimator in terms of   is (c) For show that is an asymptotically unbiased and a consistent estimator of (d) Given that that   ,  use this to find the Cramér-Rao lower bound for the variance of an unbiased estimator of. (e) For determine the efficiency of (that is, the ratio of the Cramér-Rao lower bound to the MSE of ), and comment on its value. You may use the shortcut formula You may also use either of the two formulae below to derive the Cramér-Rao lower bound for the variance an unbiased estimator                                                                                      

End of Retake Coursework