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Item A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers(MDPI, 2022-03-08) Andalib, Vahid; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceAssuming exponential lifetime and repair time distributions, we study the limiting availability 𝐴∞ as well as the per unit time-limiting profit 𝜔 of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either by an in-house repairer within an exponential patience time T or by an external expert who works faster but charges more. When there are two repair facilities, we allow the regular repairer to begin repair or to continue repair beyond T if the expert is busy. Two models arise accordingly as the expert repairs one or all failed units during each visit. We show that (1) adding a second spare to a one-unit system already backed by a spare raises 𝐴∞ as well as 𝜔; (2) thereafter, adding a second repair facility improves both criteria further. Finally, we determine whether the expert must repair one or all failed units to maximize these criteria and fulfill the maintenance management objectives better than previously studied models.Item Advanced Modeling of Longitudinal Spectroscopy Data(2014) Kundu, Madan Gopal; Harezlak, Jaroslaw; Randolph, Timothy W.; Sarkar, Jyotirmoy; Steele, Gregory K.; Yiannoutsos, Constantin T.Magnetic resonance (MR) spectroscopy is a neuroimaging technique. It is widely used to quantify the concentration of important metabolites in a brain tissue. Imbalance in concentration of brain metabolites has been found to be associated with development of neurological impairment. There has been increasing trend of using MR spectroscopy as a diagnosis tool for neurological disorders. We established statistical methodology to analyze data obtained from the MR spectroscopy in the context of the HIV associated neurological disorder. First, we have developed novel methodology to study the association of marker of neurological disorder with MR spectrum from brain and how this association evolves with time. The entire problem fits into the framework of scalar-on-function regression model with individual spectrum being the functional predictor. We have extended one of the existing cross-sectional scalar-on-function regression techniques to longitudinal set-up. Advantage of proposed method includes: 1) ability to model flexible time-varying association between response and functional predictor and (2) ability to incorporate prior information. Second part of research attempts to study the influence of the clinical and demographic factors on the progression of brain metabolites over time. In order to understand the influence of these factors in fully non-parametric way, we proposed LongCART algorithm to construct regression tree with longitudinal data. Such a regression tree helps to identify smaller subpopulations (characterized by baseline factors) with differential longitudinal profile and hence helps us to identify influence of baseline factors. Advantage of LongCART algorithm includes: (1) it maintains of type-I error in determining best split, (2) substantially reduces computation time and (2) applicable even observations are taken at subject-specific time-points. Finally, we carried out an in-depth analysis of longitudinal changes in the brain metabolite concentrations in three brain regions, namely, white matter, gray matter and basal ganglia in chronically infected HIV patients enrolled in HIV Neuroimaging Consortium study. We studied the influence of important baseline factors (clinical and demographic) on these longitudinal profiles of brain metabolites using LongCART algorithm in order to identify subgroup of patients at higher risk of neurological impairment.Item Combinatorial Patterns of D-Optimal Weighing Designs Using a Spring Balance(SSCA, 2021) Pena Pardo, Monica; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceGiven a spring balance that reports the true total weight of items plus a white noise of an unknown variance, which n subsets of n items will you weigh in order to estimate the true weights of each item with the highest possible precision? For n ≤ 6, we classify all D-optimal weighing designs according to the combinatorial patterns they exhibit (modulo permutation), we count the D-optimal designs exhibiting each pattern, and we explain how a D-optimal design for n items may arise out of a D-optimal design for (n − 1) items. For n = 7, 11 we exhibit D-optimal designs obtained from balanced incomplete block designs (BIBDs). We discuss some strategies to construct D-optimal designs of larger sizes, and pose some unsolved problems.Item Computing Limiting Average Availability of a Repairable System through Discretization(Elsevier, 2020-01) Chatterjee, Debolina; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceFormulas for limiting average availability of a repairable system exist only for some special cases: (1) either the lifetime or the repair time is exponential; or (2) there is one spare unit and one repair facility. We consider a more general setting involving several spare units and several repair facilities; and we allow arbitrary life- and repair time distributions. Under periodic monitoring, which essentially discretizes the time variable, we compute the limiting average availability. The discretization approach closely approximates the existing results in the special cases; and increases the limiting average availability as we include additional spare unit or additional repair facility.Item Depicting Bivariate Relationship with a Gaussian Ellipse(2021) Sarkar, Jyotirmoy; Rashid, Mamunur; Mathematical Sciences, School of ScienceFor data on two continuous variables, how should one depict the summary statistics (means, SDs, correlation coefficient, coefficient of determination, regression lines) so that their values can be read off easily from the depiction and potential outliers can be flagged also? We propose the Gaussian covariance ellipse as an answer that will benefit all users of statistics.Item Drilling Holes Through Balls and Cubes(Springer, 2021-04) Kaur, Jaskirat; Lally, Jasmeen; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceSuppose that we are allowed to drill a cylindrical hole of any radius through the center of a unit ball or a unit cube in a given direction. Which radius will maximize the total exposed surface area (ESA) of the finished object? If we are allowed to drill two or three mutually orthogonal cylindrical holes of the same radius, which common radius will maximize the total ESA?Item Erlang Loss Formulas: An Elementary Derivation(Springer, 2021-08) Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceThe celebrated Erlang loss formulas, which express the probability that exactly j of c available channels/servers are busy serving customers, were discovered about 100 years ago. Today we ask: “What is the simplest proof of these formulas?” As an alternative to more advanced methods, we derive the Erlang loss formulas using (1) an intuitive limit theorem of an alternating renewal process and (2) recursive relations that are solved using mathematical induction. Thus, we make the Erlang loss formulas comprehensible to beginning college mathematics students. We illustrate decision making in some practical problems using these formulas and other quantities derived from them.Item Estimating the Parameters of a Simple Linear Regression Model Without Using Differential Calculus(Susan Rivers' Cultural Institute, 2022-10-03) Sarkar, Jyotirmoy; Rashid, Mamunur; Mathematical Sciences, School of ScienceTo estimate the parameters of a simple linear regression model, students who already know calculus can minimize the total squared deviations by setting its first-order partial derivatives to zero and solving simultaneously. For students who do not know calculus, most teachers/textbooks simply state the formulas without justifying them. Students accept the formulas on faith; and for given data, they evaluate the estimates using a calculator or a statistical software. In this paper, we justify the formulas without invoking calculus. We hope the users of statistics will benefit from our proposed justifications.Item Filling Jars to Measure Time(Sciendo, 10-2021) Le, Tiffany-Chau; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceIf water is flowing at the same constant rate through each of H>3 hoses, so that any one hose will fill any one of J>2 available jars in exactly one hour, then what are the fillable fractions of a jar, and what are the measurable fractions of an hour? Learning to systematically answer such questions will not only equip readers to fluently use fractions, but also introduce or reintroduce them gently to the Queen of Mathematics – Number Theory.Item Guess the Mean: Which Method is Better?(Depauw, 2020-08) Rashid, Mamunur; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceThe mean of a set of numbers may be guessed in one of two ways: (1) as a fulcrum placed under the dot plot; or (2) as a vertical line that equalizes areas of two regions bounded by the step plot (also known as the empirical cumulative distribution function). Which of these two methods is better? We design, conduct and analyze a statistical experiment to address this question. While our findings support better performance by the latter method at the aggregate level, each individual user may respond differently to the question. We hope all users will learn both methods and determine for themselves which method they are better at. We also hope educators will empower their students by including both methods in their syllabi.