Project Abstract

<p> </p><p>In recent times, lots of efforts have been made to define new probability distributions<br>that cover different aspect of human endeavors with a view to providing alternatives<br>in modeling real data. A five-parameter distribution, called Weibull-Burr XII (Wei-<br>Burr XII) distribution is studied and investigated to serve as an alternative model<br>for skewed data set in life and reliability studies. Some of its statistical properties<br>are obtained, these include moments, moment generating function, characteristics<br>function, quantile function and reliability (survival) functions. The distribution’s<br>parameters are estimated by the method of maximum likelihood. We evaluated the<br>performance of the new distribution compared with other competing distributions<br>based on application on real data and it was concluded that Weibull-Burr XII distribution<br>perfom best using BIC, AIC and CAIC. It was also concluded that the<br>distribution can be used to model highly skewed data (skewed to the right)</p><p>&nbsp;</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> INTRODUCTION<br>1.0.1 Background of the study<br>Probability distributions are recently receiving alot of attention with regards to introducing<br>new generators for univariate continuous type of probability distributions<br>by introducing additional parameter(s) to the base line distribution. This seemed<br>necessary to reflect current realities that are not captured by the conventional probability<br>distributions since it has been proven to be useful in exploring tail properties<br>of the distribution under study (Tahir, et.al; 2016).<br>This idea of adding one or more parameter(s) to the baseline distribution has been<br>in practice for a quite long time. Several distributions have been proposed in<br>the literature to model lifetime data. Some of these distributions include: a twoparameter<br>exponential-geometric distribution introduced by Adamidis and Loukas<br>in 1998 which has a decreasing failure rate. Following the same idea of the exponential<br>geometric distribution, the exponential-Poisson distribution was introduced by<br>Kus (2007) with also a decreasing failure rate and discussed some of its properties.<br>Marshall and Olkin (1997) presented a simpler technique for adding a parameter to<br>a family of distributions with application to the exponential and Weibull families.<br>Adamidis et al. (2005) suggested the extended exponential-geometric (EEG) distribution<br>which generalizes the exponential geometric distribution and discussed some<br>of its statistical properties along with its hazard rate and survival functions.<br>Some of the well-known class of generators include the following: Kumaraswamy-G<br>(Kw-G) proposed by Cordeiro and de Castro (2011), McDonald-G (Mc- G) introduced<br>by Alexander et al. (2012), gamma-G type 1 presented by Zografos and<br>Balakrishanan (2009), exponentiated generalized (exp-G) which was derived by<br>Cordeiro et al. (2013), others are weibull-power function by Tahir et. al. (2010), ex-<br>1<br>ponentiated T-X proposed by Alzaghal et al.(2013). Most recently, a NewWeibull-G<br>Family of Distributions by Tahir, (2016), The Weibull–G family of probability distributions<br>by Bourguignon et al. (2014). This research is motivated by the work<br>done by Bourguignon et al. (2014) – The Weibull–G family of probability distributions<br>who introduced a generator based on the Weibull random variable called<br>a Weibull-G family. In this research, we propose an extension of the Burr XII pdf<br>called the Weibull-Burr XII distribution based on the Weibull-G class of distributions<br>defined by Bourguignon et al (2014). i.e. we propose a new distribution with<br>five parameters, referred to as the Weibull-Burr XII (Wei-BXII) distribution, which<br>contains as special sub-models the Weibull and Burr XII distributions.<br>1.0.2 Statement of the problem<br>It has been anticipated that a generalized model is more flexible than a conventional<br>or ordinary model and its applicability is preferred by many data analysts in analyzing<br>statistical data. It is imperative to mention that through generalizations, the<br>convetional logistic distribution with only two parameters (location and scale) has<br>been propagated into type I, type II and type III generalized logistic distributions<br>which has three parameters each as indicated in Balakrishnan and Leung (1988).<br>So, there is a genuine desire to search for some generalizations or modifications of<br>the Burr XII distribution that can provide more flexibility in lifetime modeling.<br>1.0.3 Purpose of the Study<br>Existing literature focus on generalizations or modifications of the Weibull distribution<br>that can provide more flexibility in modeling lifetime data such as; Weibull-<br>Log logistic distribution by Broderick (2016), Weibull-Lomax distribution by Tahir,<br>(2015), etc. Less attention is given to generalization of Weibull and Burr XII distributions.<br>Where the later distribution was discovered by Burr in1942 as a two<br>2<br>parameter family. An additional scale parameter was introduced by Tadikamalla in<br>1980. It is a very popular distribution for modelling lifetime data.<br>The purpose of this research focuses mainly on generalization of a Burr XII distribution<br>to a five-parameter distribution, called the Weibull-Burr XII (Wei-BurrXII)<br>distribution for modelling skewed data set (skewed to the right).<br>1.0.4 Aim and Objectives of the Study<br>The aim of this research is to study Weibull-Burr XII probability distribution and<br>investigate its properties and applications. This is expected to be achieved through<br>the following objectives by:<br>1. establishing the Weibull-Burr XII distribution;<br>2. establishing some statistical properties of Weibull-Burr XII distribution such<br>as; moments, moment generating function, quantile function, characteristics<br>function, survival function and hazard rate function;<br>3. estimating the parameters of the proposed model by the method of maximum<br>likelihood estimation;<br>4. evaluating how well theWeibull-Burr XII distribution perform when compared<br>with other Weibull–G family of distributions based on application on real life<br>data.<br>1.0.5 Significance of the study<br>Many models were introduced in the literature by extending some distributions with<br>Burr XII distribution. e.g. the Beta- Burr XII (BBXII) distribution discussed by<br>Paranaíba et al. (2011) where it was concluded that application of the Beta-BXII<br>3<br>distribution indicated that it had provided a better fit than other statistical models<br>used in lifetime data analysis, the Kumaraswamy -Burr XII distribution introduced<br>by Paranaíba et. al. (2013). Therefore, the significance of this study is mainly to<br>propose a new model (Wei-Burr XII distribution) that is much more flexible than<br>the Burr XII distribution.<br>1.0.6 Limitations of the study<br>The limitation of this research is that, it did not consider estimating parameters of<br>the Weibull-Burr XII distribution using other methods like Bayesian method. Some<br>other properties of probability distribution are also not considered in this research<br>work. e.g Rényi entropy, incomplete moments, e.t.c.<br>1.0.7 Definition of terms<br>Reliability is generally regarded as the likelihood that a product or service is<br>functional during a certain period of time under a specified operation.<br>Survival function is the probability that a patient, device, or other object of<br>interest will survive beyond a specified time. It is also known as the survivor function<br>or reliability function.<br>S(x) = Pr(an object will survive beyond time x).<br>Hazard function (also known as the failure rate, hazard rate, or force of mortality)<br>is the ratio of the probability density function to the survival function. Failure rate<br>is the frequency with which an engineered system or component fails, expressed in<br>4<br>failures per unit of time (Evans,et.al. 2000)<br>H(x)= Pr(an object will fail at time x+t given that it survive up to time x)<br>Akaike Information Criterion (AIC) is a measure of the relative quality of<br>statistical models for a given set of data. Given a collection of models for the data,<br>AIC estimates the quality of each model, relative to each of the other models. Hence,<br>AIC provides a means for model selection. Given a set of candidate models for the<br>data, the preferred model is the one with the minimum AIC value. Mathematically,<br>AIC =2k-2ll<br>Where ll is the log-likelihood function for the model and k is the number of estimated<br>parameters in the model.<br>Bayesian Information Criterion (BIC) or Schwarz criterion is also a criterion<br>for model selection among a finite set of models. The model with the lowest BIC is<br>preferred. Computed by;<br>BIC = ln(n)k-2ll where n is the sample size and k is the number of estimated<br>parameters in the model.<br>Consistent Akaike Information Criterion (CAIC) is mathematically defined<br>by<br>CAIC = -2ll+ 2kn/(n-k-1) where ll = log likelihood.<br>5 <br></p>