Tag Archives: Maximum Likelihood Estimation

A Study on The Mixture of Exponentiated-Weibull Distribution PART I (The Method of Maximum Likelihood Estimation) (Published)

Mixtures of measures or distributions occur frequently in the theory and applications of probability and statistics. In the simplest case it may, for example, be reasonable to assume that one is dealing with the mixture in given proportions of a finite number of normal populations with different means or variances. The mixture parameter may also be denumerable infinite, as in the theory of sums of a random number of random variables, or continuous, as in the compound Poisson distribution. The use of finite mixture distributions, to control for unobserved heterogeneity, has become increasingly popular among those estimating dynamic discrete choice models. One of the barriers to using mixture models is that parameters that could previously be estimated in stages must now be estimated jointly: using mixture distributions destroys any additive reparability of the log likelihood function. In this thesis, the maximum likelihood estimators have been obtained for the parameters of the mixture of exponentiated Weibull distribution when sample is available from censoring scheme.The maximum likelihood estimators of the parameters and the asymptotic variance covariance matrix have been obtained. A numerical illustration for these new results is given.

Keywords: Exponentiated Weibull Distributiom (EW), Maximum Likelihood Estimation, Mixture Distribution, Mixture of two Exponentiated Weibull Distribution(MTEW), Moment Estimation, Monte-Carlo Simulation

Maximum Likelihood Method: An Alternative To Ordinary Least Square Method In Estimating The Parameters Of Simple Weibull Distribution Using Large Samples Of Type-I Censored Observation. (Published)

This study is concerned with two-parameter Weibull distribution which is very important in life testing and reliability analysis. Two methods viz: maximum likelihood estimation (MLE) and Ordinary Least Squares (OLS) are good alternatives in estimating the parameters of a simple Weibull distribution as the sample sizes increases. These estimators are derived for Random Type-I censored samples. These methods were compared by looking at their standard errors through simulation study with sample sizes of 100, 300, 500 and 1000. It was observed that MLE stands out when estimating the parameters of the Weibull distribution as the sample size increases compared to the OLSM. We also noted that both OLSM and MLE provides asymptotically normally distributed estimator.

Keywords: Maximum Likelihood Estimation, Ordinary least square estimation, Random Type-I censoring, simulation study and Weibull distribution.

Maximum Likelihood Method: An Alternative To Ordinary Least Square Method In Estimating The Parameters Of Simple Weibull Distribution Using Large Samples Of Type-I Censored Observation (Published)

This study is concerned with two-parameter Weibull distribution which is very important in life testing and reliability analysis. Two methods viz: maximum likelihood estimation (MLE) and Ordinary Least Squares (OLS) are good alternatives in estimating the parameters of a simple Weibull distribution as the sample sizes increases. These estimators are derived for Random Type-I censored samples. These methods were compared by looking at their standard errors through simulation study with sample sizes of 100, 300, 500 and 1000. It was observed that MLE stands out when estimating the parameters of the Weibull distribution as the sample size increases compared to the OLSM. We also noted that both OLSM and MLE provides asymptotically normally distributed estimator.

Keywords: Maximum Likelihood Estimation, Ordinary least square estimation, Random Type-I censoring, simulation study and Weibull distribution.

ESTIMATION OF EXTREME VALUE AT RISK IN RWANDA EXCHANGE RATE (Published)

Estimating the probability of rare and extreme events is a crucial issue in the risk estimation of exchange rate returns. Extreme Value Theory (EVT) is a well-developed theory in the field of probability that studies the distribution of extreme realizations of a given distribution function, or of a stochastic process, satisfying certain assumptions. This work has fitted the Generalized Pareto Distribution (GPD) to the excess returns assuming the residuals are independent and identically distributed. The results are used to estimate extreme Value at Risk (VaR) in Rwanda exchange rate process.

Keywords: Confidence intervals, EVT approach, Exchange Rate, Generalized Pareto Distribution, Maximum Likelihood Estimation, Value at Risk

APPLICATION OF EXTREME VALUE THEORY FOR EXTREME QUANTILES ESTIMATION IN RWANDA EXCHANGE RATE (Review Completed - Accepted)

Estimating the probability of rare and extreme events is a crucial issue in the risk estimation of exchange rate returns. Extreme Value Theory (EVT) is a well developed theory in the field of probability that studies the distribution of extreme realizations of a given distribution function, or of a stochastic process, satisfying certain assumptions.  This work has fitted the Generalized Pareto Distribution (GPD) proposed by EVT to the excesses returns over the threshold to estimate quantiles in the tails of independent and identically distributed residuals and asymptotic properties of the estimators were given. The results were applied to estimate extreme quantiles in the Rwanda exchange rate process.

Keywords: Confidence intervals, EVT approach, Exchange Rate, Generalized Pareto Distribution, Maximum Likelihood Estimation, Quantiles estimation