n/{p(1 − p)}. ˙ dθ The C-R lower bound for the variance of unbiased estimator of θ(= p2 ) is ( dp )2 /I(p) = 4p3 (1 − p)/n. ∏n (b) Note L(p) = j=1 pXj (1 − p)1−Xj . This yields p = X/n. Hence θ = (p)2 = X 2 /n2 . (c) Note Ep (X 2 ) = n ∑ ∑ 2 Ep (Xi ) + i=1 2 Ep (Xi Xj ) = nEp (X1 ) + (n2 − n)Ep (X1 X2 ) = np + (n2 − n)p2 . 1≤i̸=j≤n Hence Ep (θ) = p2 + p(1 − p)/n ̸= p2 ‚ i.e θ is a biased estimator for θ with bias p(1 − p)/n. ∗ ∗ (d) We draw bootstrap sample X1 ‚
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reliability of eyewitness testimony. In general‚ these factors are separated into two categories – estimator variables and system variables. Estimator variables Estimator variables are factors that can impact a person’s ability to observe and recollect events. These variables cannot‚ however‚ be controlled by the criminal justice system‚ according to the American Bar Association. Some of the most common estimator variables include the following: • The age of the witness • The lighting in the area where
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1. German economy overview Germany is the largest national economy in Europe‚ the fourth-largest by nominal GDP in the world‚ and fifth by GDP (PPP) in 2008. Since the age of industrialisation‚ the country has been a driver‚ innovator‚ and beneficiary of an ever more globalised economy. Germany is the world’s second largest exporter with $1.474 trillion‚ €1.06 trillion exported in 2011 (Eurozone countries are included). Exports account for more than one-third of national output. Germany
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generalized version of the normal- gamma distribution is conjugate to the half-normal likelihood and give the moments of this new distribution. The bias and coverage of the Bayesian posterior mean estimator of the halfnormal location parameter are compared with those of maximum likelihood based estimators. Inference for the half-t distribution is performed using Gibbs sampling and model comparison is carried out using Bayes factors. A real data example is presented which demonstrates the fitting of
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.2.3 Time series models Time series is an ordered sequence of values of a variable at equally spaced time intervals. Time series occur frequently when looking at industrial data. The essential difference between modeling data via time series methods and the other methods is that Time series analysis accounts for the fact that data points taken over time may have an internal structure such as autocorrelation‚ trend or seasonal variation that should be accounted for. A Time-series model explains
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therefore‚ does not replay events just as they happened. Rather‚ there is subjectivity in how people remember things and people they saw. In general‚ the factors that may impact this subjectivity can be split into two categories – estimator variables or system variables. Estimator
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Decomposing Portfolio Value-at-Risk: A General Analysis Winfried G. Hallerbach *) Associate Professor‚ Department of Finance Erasmus University Rotterdam POB 1738‚ NL-3000 DR Rotterdam The Netherlands phone: +31.10.408 1290 facsimile: +31.10.408 9165 e-mail: hallerbach@few.eur.nl http://www.few.eur.nl/few/people/hallerbach/ final version: October 15‚ 2002 forthcoming in The Journal of Risk 5/2‚ Febr. 2003 *) I’d like to thank Michiel de Pooter and Haikun Ning for excellent programming
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Subject CT3 Probability and Mathematical Statistics Core Technical Syllabus for the 2014 exams 1 June 2013 Subject CT3 – Probability and Mathematical Statistics Core Technical Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in the aspects of statistics and in particular statistical modelling that are of relevance to actuarial work. Links to other subjects Subjects CT4 – Models and CT6 – Statistical Methods: use the statistical concepts
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. . . . . . . . . . Types of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Estimates and Estimators 5.1 5.2 5.3 5.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Likelihood Estimation (MLE) Algorithm . . . . . . . . . . . . . . . . . . . . . Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biases in Statistics . . . . . . . . . . . . . . . .
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variable) and the estimated Y values is as small as possible. (b) The estimators of the regression parameters obtained by the method of least squares. (c) An estimator being a random variable‚ its variance‚ like the variance of any random variable‚ measures the spread of the estimated values around the mean value of the estimator. (d) The (positive) square root value of the variance of an estimator. (e) Equal variance. (f) Unequal variance. (g) Correlation
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