3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item On some likelihood lnference for the destructive COM-Poison cure rate model with generalised gamma lifetime(2018) Majakwara, JacobIn this thesis, based on a competing causes scenario, the destructive Conway-Maxwell Poisson (COM-Poisson) cure rate model is studied. The model assumes the occurrence of the event under study to undergo a destructive process of the initial competing causes and only the undamaged portion relating to the initial number of competing causes is recorded. This provides a real and practical way of interpreting the occurrence of the event under study in a biological system. This research assumes the distribution of competing causes to be COM-Poisson which includes some of the commonly used discrete distributions as its particular cases. Furthermore, we propose to model the lifetime by the generalised gamma distribution, which includes some of the commonly used lifetime distributions as its particular cases. The main contribution of this thesis is to develop the expectation maximisation (EM) algorithm to determine the maximum likelihood estimates (MLEs) of the model parameters and to carry out the likelihood inference assuming the data to be right censored. Model discrimination within the COM-Poisson and generalised gamma families are carried out to select a parsimonious distribution for the competing cause and the lifetime that jointly provides an adequate fit to the data. We develop the estimation procedure using both profile likelihood and complete likelihood approaches and make a comparison between the two techniques through the EM algorithm. The performance of the proposed method of inference is demonstrated by carrying out a comprehensive Monte Carlo simulation study. The flexibilities of the COM-Poisson and generalised gamma families are utilised to carry out a two-way model discrimination using the likelihood-and information-based methods. The proposed estimation technique is then applied to a real melanoma data for illustrative purpose. The results show that both the COM-Poisson and generalised gamma distributions provide additional flexibility in modelling survival data with surviving fraction. We have also shown how covariates can influence the cure rate. Most importantly, the model fits the data better when the destructive mechanism is taken into account.Item Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations(2017) Alanazi, Abdulaziz MohammedIn this thesis, various partition functions with respect to `-regular overpartitions, a special partition inequality and partition con gurations are studied. We explore new combinatorial properties of overpartitions which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition and `-regular partition functions. We provide both generating function and bijective proofs. We then establish an in nite set of Ramanujan-type congruences for the `-regular overpartitions. This signi cantly extends the recent work of Shen which focused solely on 3{regular overpartitions and 4{regular overpartitions. We also prove some of the congruences for `-regular overpartition functions combinatorially. We then provide a combinatorial proof of the inequality p(a)p(b) > p(a+b), where p(n) is the partition function and a; b are positive integers satisfying a+b > 9, a > 1 and b > 1. This problem was posed by Bessenrodt and Ono who used the inequality to study a maximal multiplicative property of an extended partition function. Finally, we consider partition con gurations introduced recently by Andrews and Deutsch in connection with the Stanley-Elder theorems. Using a variation of Stanley's original technique, we give a combinatorial proof of the equality of the number of times an integer k appears in all partitions and the number of partition con- gurations of length k. Then we establish new generalizations of the Elder and con guration theorems. We also consider a related result asserting the equality of the number of 2k's in partitions and the number of unrepeated multiples of k, providing a new proof and a generalization.Item A survey of three combinatorial problems(2016) Tissink, HenrickThis dissertation is based on three di erent combinatorial papers: 1. The rst paper is by Silvia Heubach and Tou k Mansour: Enumeration of 3-Letter Patterns in Compositions. Combinatorial Number Theory in Celebration of the 70-th Birthday of Ronald Graham. In: De Gruyter Proceedings in Mathematics. 243-264, (2007). 2. The second paper is by Daniel J. Velleman and Gregory S. Warrington: What to expect in a game of memory. American Mathematical Monthly, 120:787-805 (2013). 3. The third paper is by Mireille Bousquet-M elou and Richard Brak: Exactly Solved Models of Polyominoes and Polygons. Polygons, Polyominoes and Polycubes. In: Lecture Notes in Physics, 775:43-78, (2009).Item Jump numbers, hyperrectangles and Carlitz compositions(1999) Cheng, BoLet A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts).