Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην εκτίμηση των παραμέτρων στο μοντέλο της διπαραμετρικής εκθετικής κατανομής με παράμετρο θέσης μ και παράμετρο κλίμακος σ. Θεωρούμε ένα δείγμα n τυχαίων μεταβλητών, καθεμία από τις οποίες ακολουθεί την διπαραμετρική εκθετική κατανομή. Λογοκρίνουμε κάποιες αρχικές παρατηρήσεις και έστω ότι τερματίζουμε το πείραμά μας πριν αποτύχουν όλες οι συνιστώσες. Τότε προκύπτει ένα διπλά διακεκομμένο δείγμα διατεταγμένων παρατηρήσεων. Η εκτίμηση των παραμέτρων της διπαραμετρικής εκθετικής κατανομής, γίνεται από το συγκεκριμένο δείγμα. Πρώτα μελετάμε κάποιες βασικές έννοιες της Στατιστικής και της Εκτιμητικής και βρίσκουμε εκτιμητές για τις παραμέτρους. Πιο συγκεκριμένα, βρίσκουμε αμερόληπτο εκτιμητή ελάχιστης διασποράς, εκτιμητή μέγιστης πιθανοφάνειας, εκτιμητή με την μέθοδο των ροπών και τον βέλτιστο αναλλοίωτο εκτιμητή σε συγκεκριμένη κλάση, αντίστοιχα και για τις δύο παραμέτρους. Σαν βελτίωση των προηγούμενων εκτιμητών, ακολουθούν οι εκτιμητές τύπου Stein και, ολοκληρώνοντας, ασχολούμαστε με πρόβλεψη κατά Bayes για μια μελλοντική παρατήρηση / The present master thesis deals with the estimation of the location parameter μ and the scale parameter σ of the two-parameter exponential distribution. A sample n of random variables from the two-parameter exponential distribution is assumed. Part of the initial variables is censored and the experiment is terminated before all the components fail. A doubly censored sample emerges from which the two-parameter exponential distribution's parameters are estimated. First of all, basic Statistics' concepts are studied in order to estimate the parameters. More specifically, the Minimum Variance Unbiased Estimator (MVUE), the Maximum Likelihood Estimator (MLE), the estimator based on the Method of Moments and the best affine equivariant estimator are computed for both the parameters. To improve the previous estimators, the Stein method is used and to conclude the Bayes prediction is used for future observation

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This project is associated with statistical methods to find the unknown parameters of a model. It is the statistical investigation of the algorithm with respect to accuracy (the Cramer-Rao bound and L-curve technique) and optimization of the algorithmic parameters. This project aims to estimate the true temperature (final temperature) of a certain liquid in a container by using initial measurements (readings) from a temperature probe with a known time constant. Basically, the final temperature of the liquid was estimated, before the probe reached its final reading. The probe obeys a simple first-order differential equation model. Based on the model of the probe and the measurement data the estimate was calculated of the ’true’ temperature in the container by using a maximum likelihood approach to parameter estimation.  The initial temperature was also investigated. Modelling, analysis, calculations, and simulations of this problem were explored.

MSc (Statistics) / Department of Statistics / Daily peak electricity data from ESKOM, South African power utility company for the period, January 1997 to December 2013 consisting of 6209 observations were used in this dissertation. Since 1994, the increased electricity demand has led to sustainability issues in South Africa. In addition, the electricity demand continues to rise everyday due to a variety of driving factors. Considering this, if the electricity generating capacity in South Africa does not show potential signs of meeting the country’s demands in the subsequent years, this may have a significant impact on the national grid causing it to operate in a risky and vulnerable state, leading to disturbances, such as load shedding as experienced during the past few years. In particular, it is of greater interest to have sufficient information about the extreme value of the stochastic load process in time for proper planning, designing the generation and distribution system, and the storage devices as these would ensure efficiency in the electrical energy in order to maintain discipline in the grid systems. More importantly, electricity is an important commodity used mainly as a source of energy in industrial, residential and commercial sectors. Effective monitoring of electricity demand is of great importance because demand that exceeds maximum power generated will lead to power outage and load shedding. It is in the light of this that the study seeks to assess the frequency of occurrence of extreme peak electricity demand in order to come up with a full electricity demand distribution capable of managing uncertainties in the grid system. In order to achieve stationarity in the daily peak electricity demand (DPED), we apply a penalized regression cubic smoothing spline to ensure the data is non-linearly detrended. The R package “evmix” is used to estimate the thresholds using the bounded corrected kernel density plot. The non-linear detrended datasets were divided into summer, spring, winter and autumn according to the calender dates in the Southern Hemisphere for frequency analysis. The data is declustered using Ferro and Segers automatic declustering method. The cluster maxima is extracted using the R package “evd”. We fit Poisson GPD and stationary point process to the cluster maxima and the intensity function of the point process which measures the frequency of occurrence of the daily peak electricity demand per year is calculated for each dataset. The formal goodness-of-fit test based on Cramer-Von Mises statistics and Anderson-Darling statistics supported the null hypothesis that each dataset follow Poisson GPD (σ, ξ) at 5 percent level of significance. The modelling framework, which is easily extensible to other peak load parameters, is based on the assumption that peak power follows a Poisson process. The parameters of the developed i models were estimated using the Maximum Likelihood. The usual asymptotic properties underlying the Poisson GPD were satisfied by the model. / NRF