RISK THEORY - LECTURE NOTES 1. INTRODUCTION The primary subject of Risk Theory is the development and study of mathematical and statistical models to describe and predict the behaviour of insurance portfolios‚ which are simply financial instruments composed of a (possibly quite large) number of individual policies. For the purposes of this course‚ we will define a policy as a random (or stochastic) process generating a deterministic income in the form of periodic premiums‚ and incurring financial
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the method of moments estimate of θ. (b) What is the maximum likelihood estimate of θ? (a) In general‚ let X1 ‚ X2 ‚ . . . ‚ Xn be a random sample drawn from this distribution. Since 2 1 2 1 7 µ1 = E[X] = 0 · θ + 1 · θ + 2 · (1 − θ) + 3 · (1 − θ) = − 2θ. 3 3 3 3 3 we have θ= and the MME for θ is 7 1 − µ1 ‚ 6 2 7 1 θˆ = − X‚ 6 2 1∑ Xi . In this case‚ we have X = 3/2 and n i=1 n where X = 7 1 3 5 θˆ = − · = . 6 2 2 12 (b) The likelihood function of the sample (3‚ 0‚ 2‚ 1‚ 3‚ 2‚ 1‚ 0‚ 2‚ 1) is lik(θ)
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NEKN82 EMPIRICAL FINANCE LAB 3 Report Done by: Lang‚ Qin 1988-12-05 Low Lihui Valerie 1989-09-24 Q1 Before we evaluate the actual investment performance of the five constructed portfolios for period 1992.02-2008.07‚ we firstly calculate the mean‚ variance and standard deviation of each of the portfolio using Excel. The results are generated as below: Portfolios | Z1 | Z2 | Z3 | Z4 | Z5 | Zm | Mean | 0.008490 | 0.003843 | 0.009980 | 0.000141 | 0.004840 | 0.0066 | Variance |
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model‚ using Ang and Piazzesi (2003). My contribution to the model is a credit factor that I construct using a principal component analysis of notable credit variables. After determining parameters for our model through a numerical optimization of a likelihood function‚ our model supports yield data for the past twenty years. To understand the impact of the credit factor‚ I implement impulse response functions on bond yields. I find that positive shocks to credit raise bond yields at all maturities of
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magnetometers. The proposed calibration method is written in the sensor frame‚ and compensates for the combined effect of all linear time-invariant distortions‚ namely soft iron‚ hard iron‚ sensor non-orthogonality‚ bias‚ among others. A Maximum Likelihood Estimator (MLE) is formulated to iteratively find the optimal calibration parameters that best fit to the onboard sensor readings‚ without requiring external attitude references. It is shown that the proposed calibration technique is equivalent
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Identification using Gaussian Mixture Speaker Models. IEEE Transactions on Acoustics‚ Speech‚ and Signal Processing 3(1) (1995) 72–83 4. McLachlan‚ G.‚ ed.: Mixture Models. Marcel Dekker‚ New York‚ NY (1988) 5. Dempster‚ A.‚ Laird‚ N.‚ Rubin‚ D.: Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society 39(1) (1977) 1–38 6. Reynolds‚ D.A.‚ Quatieri‚ T.F.‚ Dunn‚ R.B.: Speaker Verification Using Adapted Gaussian Mixture Models. Digital Signal Processing 10(1) (2000)
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Chapter 7 Survival Models Our final chapter concerns models for the analysis of data which have three main characteristics: (1) the dependent variable or response is the waiting time until the occurrence of a well-defined event‚ (2) observations are censored‚ in the sense that for some units the event of interest has not occurred at the time the data are analyzed‚ and (3) there are predictors or explanatory variables whose effect on the waiting time we wish to assess or control. We start with some
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SUFFICIENT DIMENSION REDUCTION BASED ON NORMAL AND WISHART INVERSE MODELS A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY LILIANA FORZANI IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY R. DENNIS COOK‚ Advisor December‚ 2007 c Liliana Forzani 2007 UNIVERSITY OF MINNESOTA This is to certify that I have examined this copy of a doctoral thesis by Liliana Forzani and have found that it is complete
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The Key Role of the Volatility for Financial Engineering Dr. Zbigniew Krysiak zbigniew.krysiak@poczta.onet.pl Associate Professor of Finance - Warsaw School of Economics Visiting Professor at Northeastern Illinois University‚ Chicago Financial Mathematics Mathematics Department at Northeastern Illinois University‚ Chicago Wednesday‚ October 3rd‚ 2012 1 Agenda • Approaches to Modeling Volatilities • Volatility Models in Capital Allocation - VaR • Application of GARCH to Modeling
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techniques‚ subspace based techniques‚ maximum likelihood techniques and the integrated techniques which combine property restoral techniques with subspace based techniques. Conventional methods are based on classical beam forming techniques and require a large number of elements to achieve high resolution. Subspace based methods are high resolution sub-optimal techinques which exploit the eigen structure of the input data matrix. Maximum likelihood techniques are optimal techniques which can perform
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