(iii) Under equal cluster sampling, show that ( ) notations. (d) Attempt any four of the following:

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1 Central University of Rajasthan Department of Statistics M.Sc./M.A. Statistics (Actuarial)-IV Semester End of Semester Examination, May-2012 MSTA 401: Sampling Techniques and Econometric Methods Max. Marks: 100 Date: Duration: hours 1. (a) Define the following terms with suitable examples. (i) finite population (ii) infinite population (iii) target population (iv) study population (v) sample (vi) census (vii) sampling frame (viii) sampling units (ix) population standard deviation (x)population coefficient of variation. [10] (b) Identify Population and Sampling unit for selecting a simple random sample in each of the following cases: i. Annual fruit in a hilly district.. ii. Popularity of family planning among families having more than two children. iii. Election for a political office with adult franchise. iv. One hundred account holders of bank of India, Central University of Rajasthan branch, Kishangarh. [2] (c) i) Show that is an unbiased estimator of population total [4] (ii) Show that under SRSWR sampling, while estimating population mean (total), the minimum sample size with minimum relative standard error equal to is given by [ ] (iii) Under equal cluster sampling, show that ( ) notations. (d) Attempt any four of the following: [4] ( ) [ ( ) ] with usual [4] (i) What do you understand by multicollinearity? What are the sources of multicollinearity in a regression model? [4] (ii) Differentiate between R 2 and adjusted R 2. What is the need of adjusted R 2. [4] (iii) Distinguish between endogenous and exogenous variables. Give illustrative examples. (iv) When do you say that a model is identifiable? Distinguish between exact and over identification. [4] (v) Verify the order condition of identifiability for the equations in the following model. [4] In the above model y s are endogenous and z s are exogenous variables. What estimation method would you suggest for estimating these equations? [4] 1 P a g e

2 (vi) Distinguish between micro and macro models. [4] (vii) What are recursive models? Justify that they can be estimated, equation wise, from the data on the variables in them by OLS method. [4] 2A. (i) What is simple random sampling? Define Simple random sampling with replacement (SRSWR) and simple random sampling without replacement (SRSW). Explain the differences between them in detail. [5] (ii) Show that ( ) ( ) ( ), where ( ) and ( ) indicate the variance of the estimator of population mean in stratified sampling under proportional and Neyman allocation respectively. [10] 2B. (i) Under SRSW sampling show that the unbiased estimator of the variance ( ) is given by ( ) ( ) ( ) ; where is the sample proportion and. [8] (ii) Suppose a population consists of units. The variable takes values 1, 2, 3, 4. Select a sample of size using SRSW sampling. Show that the sample mean is an unbiased estimator of population mean. Also show that sample mean square is an unbiased estimator of population mean square error. [7] 3A. (i) What is systematic sampling? Show that the sampling variance of sample mean is given by ( ) ( ) ( ), where ( ) ( ) with ( ) and has its usual meaning. [8] (ii) In the case of SRSWR, Prove that the variance of the estimator of the population mean is ( ) where ( ) [7] 3B. (i) Describe optimum allocation method and obtain the optimum size of the sample allocation in various strata when (i) the total cost is fixed and (ii) it is necessary to take into account the within strata variances and cost of sampling from each stratum. (ii) What is ratio method of estimation? Show that the first approximation to the relative bias of the ratio estimator in SRSW is given by where [10] ( ) ( ) ( ), and are the coefficients of variation of and respectively. [5] 4A. (a) Consider a multiple regression model for which all classical assumptions hold, but in which there is no constant term. Suppose you wish to test the null hypothesis that there is no relationship between Y and X, that is: H 0: β 2 =... = β k = 0 against the 2 P a g e

3 alternative that at least one of the β's is nonzero. Develop the appropriate test statistic and state its distributions (including the numbers of degrees of freedom). [7] (b) What are the consequences of heteroscedasticity and auto-correlation in regression model. Give the remedial steps to resolve for the problems. [8] 4B. With reference to the general linear model = ; t=1,2,,n Under the assumptions show that the ordinary least squares (OLS) estimate = (X X) -1 X y is (i) Best Linear Unbiased Estimate of β. (ii) V( ) = (X X) -1 σ 2. (iii) is consistent for β. (iv) An unbiased estimate of σ 2 is given by the sum of squares of the OLS residuals divided by (n-k). [15] 5A. (a) What are instrumental variables? State the assumptions. Show, using a simple illustration, that instrumental variables method provides consistent estimates. Obtain the expression for its standard error and its estimate. [9] (b) Explain 2SLS method of estimating the coefficients of an equation in a model. How will you estimate its standard errors? [6] 5B. (a) Write an explanatory notes on simultaneity problem in econometric analysis. In this context briefly describe the test for simultaneity. [7.5] (b) Show that, for an exactly identified equation, both ILS and 2SLS estimates coincide. [7.5] 3 P a g e

4 Central University of Rajasthan Department of Statistics M.Sc./M.A. Statistics(Actuarial) IV Semester End of Semester Exam-May 2012 MSTA 402 Multivariate Analysis Time: 3 Hrs. Date: Max. 100 Marks Q. 1. a. Define a p-variate normal distribution. b. Define multiple and partial correlation coefficients with examples. c. Define Wishart distribution. What does Wishart distribution reduce to when p=1? d. If, then obtain the distribution of Q =. e. Define canonical correlation and canonical variate. f. Show that Hotelling is invariant under a nonsingular linear transformation. g. What do you mean by principal components? h. What do you understand by Factor analysis? i. Give some applications of cluster analysis technique. j. Obtain 95% confidence region for the mean of distribution when is known. (2 each) Q.2 A. (a) If, than show that. (10) (b) If, where then (i) Obtain the joint distribution of and. (ii) Compute the conditional distribution of given. (10) Q 2B (a) Derive the characteristics function of distriburtion. (10) (b) If, where and then Obtain (i) Distribution of. (ii) Distribution of and. (iii) Conditional distribution of given. (10) Q.3A. (a) How do you generate a random observation from. Obtain ML estimate of and based on a random sample of size from. (3+9) (b) Suppose are independent random variables, where is distributed as. Let be an orthogonal matrix. Show that Y C N is distributed as where v C and are independent. 1 N 1 X (8)

5 Q.3B (a) Desire the Hotelling T 2 Statistic as a function of the generalized likelihood ratio. (10) (b) Discuss in detail the use of Hotelling T 2 Statistic for sample problem. (10) Q4.A. (a) Derive the characteristic function of Wishart distribution. (10) (b) Let and A and are partitioned as below:, Q 4.B. then show that (10) (a) If are independently distributed with, then show that, where (6) Q.5A. (b) If, then obtain. (8) (c) If, then obtain the distribution of, where B is a nonsingular matrix. (6) (a) Discuss the problem of classification of an observation with a suitable example, by indicating errors and costs of misclassifications. Discuss the procedure of classification of observations into one of the two populations with given probability distributions when a priori probabilities are known. By indicating an optimal criterion, prove that the classification criterion is unique. Use this procedure to obtain the classification criteria to classify an observation into one of the two known multivariate normal populations. (20) Q 5B (a) Describe a method of obtaining principle components. Obtain first two principle components and the percentage of variation explained by them if the covariance matrix is (10) (b) If a p component vector has mean vector and covariance matrix Σ and if X and are partitioned as and. Then show that first canonical correlation between is the largest root of (10)

6 Central University of Rajasthan M.Sc./M.A. Statistics (Actuarial)-IV END OF SEMESTER EXAMINATIONS MAY, 2012 MSTA 422: LIFE AND HEALTH INSURANCE Duration: 3 hours DATE: Max. Marks Answer any four of the following subdivisions: (a) Explain graphically the importance of law of large number in prediction of future mortality rates. What are the two basic sources of mortality statistics? b) Describe reasons for paying less than asset share in payment of surrender values. (c) What procedure should a management of new life office adopt to settle assumptions for design of a new product? (d) What do you understand by true and fair value accounting? Explain briefly about the principle factors utilized to work out the accounting system. (e) Assuming yourself an executive with an annual salary of 7 lacs per annum name the necessary insurance covers you need to have, with reasons thereof. (f) Describe Contribution Method for distribution of profit in a policy. (7 Marks each) 2. Answer any three of the following subdivisions: (a) Describe briefly two main types of critical Illness products along with other products designed with variations. (b) Elaborate the terms Profit Deferral and Profit Smoothing as distribution strategies. (c) In case of Long Term Care Insurance explain briefly about 1. Stand Alone Insurance Products. 2. Long Term Care Insurance as rider and 3. Claim Triggers of Long Term Care Insurance. (d) Describe Residual Disability benefit of a Disability income Policy. Calculate Residual Indemnity of an insured whose income has gone down from to per month, assuming his monthly indemnity as (e) What do you understand by Interim Bonus and Terminal bonus? Explain the role of terminal bonus in the solvency of an insurer. (8 Marks each) 3A. What are characteristics of a Unit Linked Insurance Plan? Give formulae for calculation of number of units allocated, Offer price and bid price. Also explain about settlement of expected cost of death claim in case of a ULIP in a year t. (Marks 4+3+5) 3B. Describe model Office. Which assumptions are taken in to consideration for a model office? Explain the role of model office in pricing of a product. (Marks )

7 4 A. Using three factor contribution method, calculate the fifteenth year surplus dividend of a hypothetical policy per 1000 face amount. The policy was issued to a male 15 years ago at his 25 years age. Gross and valuation annual premiums being and respectively. Policy reserves at fourteenth and fifteenth year are and Where q 30 = , q 30 = , i 15 =0.08 and i 15 =0.03. Fifteenth year expense charge is 5.00 ( Marks 18) 4B. (a) How many types of mortality tables are there in use? Write brief description of each of them. What adjustments are made to the derived mortality data in the process of rate making? (Marks 4+4) (b) Draw a mortality table in the standard format for a group of males living at age of 65 with mortality rate of 12% till 70 years and thereafter of 18% till 80. (Marks 10) 5A. (a) Out of 9,210,289 males living at age 45, still 8,966,618 males are living at age 50. Using these mortality data calculate net single premium for a 5 year pure endowment plan of face value 1000 assuming expected return of 5 percent. (Marks 8) (b) You are given the following extract from a 2 year select and ultimate mortality table: [x] l[x] l[x]+1 lx+2 x (i) Complete the table [x] q[x] q[x]+1 qx+2 x (ii) Find 2p[47], 2p[46]+1 and 2p47. (Marks 5+5) 5B. Calculate Best Estimate Premium and Total Premium for a non profit endowment policy of term 3 years with face value Expected mortality rate at age X, (q x ) is 5% each year for all X. The company has assumed 10% per annum return on assets. Whereas expected expenses are 50%,5% and 6% of best estimate premium respectively for year 1,2 and 3. For profit margin apply a loading of 7.5% of best estimate premium. (Marks 18)

8 Central University of Rajasthan M.Sc./M.A. Statistics (Actuarial)-IV END OF SEMESTER EXAMINATIONS MAY, 2012 MSTA 427: VALUATIONS AND LOSS RESERVING Max. Marks: 60 Tuesday 15 May 2012 Duration: hours Q. 1. Attempt all of the following: a) Describe various stages involved in the processing of a single claim. (2) b) In usual notations define and distinguish and (i, k). (2) e) Define I(i, j), D(i, j) and Q(i, j) and establish their interrelationship. (2) d) Describe in brief a real life problem that you have been studying under the minor project. Elaborate statistical tools, concepts of insurance/finance that you have been used. Indicate your major findings. (9) 2A. a) Explain the concept of Super imposed inflation and describe its role in loss reserving. b) What are indexed and face values? Comment with justification indexed values of two claims may be equal but not the face values. c) What do you mean by cash flow? Describe various cash flow adjustments. (5 each) 2B. a) Describe the nature of data that is useful to an insurance to assess the loss liabilities. b) What are the IBNR claims? Develop two appropriate statistical models for the count of such claims. c) Define relative of claim occurrence per period and its weak dependency on occurrence period. (5 each) 3A. a) Define age to age factor? How is this used in estimation of IBNR claims? b) When the cell counts are assumed to be Poisson, in usual notations obtain the MLE of. (7+8)

9 3B. a) Develop the chain ladder method of estimation of and justify for the name of it. b) Define the terms i) Q, the outstanding losses ii) R, the relative error of an estimate. Obtain the impact of b, the bias and, the coefficient of variation on RE. 4A. a) For an inflation adjusted model with inflation rate f, in usual notations prove that [ ] and, where K is an arbitrary positive constant, representing the degree of redundancy. b) Describe the payment per claim incurred model indicate the advantages of the (7+8) same. Develop estimators of and. (7+8) 4B. a) By using any suitable method to be described complete the rectangle Period of Origin Development periods b) Under the payment per claim finalization (PPCF) sub-model, obtain an estimate of probability of finalization for the development period j. (7+8)