Anant Godbole
East Tennessee State University, Mathematics and Statistics, Faculty Member
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Consider a lecture class with a population of N students. Suppose we keep track of the order of students called upon to answer a question. Each student on the roster has l friends before his/her name and l friends after his/her name;... more
Consider a lecture class with a population of N students. Suppose we keep track of the order of students called upon to answer a question. Each student on the roster has l friends before his/her name and l friends after his/her name; these may be considered to be students who are lexicographically close. A klmatch occurs when two students, who are in each other’s list of 2l friends or are themselves, are called upon within the k previous questions. A large number of such occurrences might indicate that the professor is not selecting students at random. Let Xn denote the number of kl-matches within the first n questions asked by the professor, where each student has a full window of 2l C 1 friends A. Godbole ( ) Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN, USA e-mail: godbolea@etsu.edu K. Grzesik Department of Biostatistics and Computational Biology, University of Rochester, New York, NY, USA e-mail: Katherine_Grzesik@urmc.rochester.edu H. Shappell Department of Biostatistics, The Johns Hopkins University, Baltimore, MD, USA e-mail: hshap@bu.edu © Springer Science+Business Media LLC 2018 J. Glaz, M.V. Koutras (eds.), Handbook of Scan Statistics, https://doi.org/10.1007/978-1-4614-8414-1_45-1 1
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ABSTRACT
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Givenp↦ (0,1), we consider a sequence of {0,1}-valued random variables (a) that have an i.i.d. Bernoulli (p) distribution or (b) which evolve according to a stationary ergodic 2-state Markov chain with transition probabilities given by ,... more
Givenp↦ (0,1), we consider a sequence of {0,1}-valued random variables (a) that have an i.i.d. Bernoulli (p) distribution or (b) which evolve according to a stationary ergodic 2-state Markov chain with transition probabilities given by , and with stationary distribution . Lehmann (1986) proved that the conditional run test possesses certain optimality properties if used as a criterion to discriminate
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Let R = Rn denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two... more
Let R = Rn denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two negative binomial random variables with parameters p and q (=1-p). Using the representation of R as
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Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple... more
Let Nn,k denote the number of recurrent success runs of length k≥2 in a sample of size n drawn with replacement from a dichotomous population. The exact distribution of Nn,k has recently been obtained in closed algorithmically simple form; we discuss the programming of these algorithms for values of n that are large, but not so large that asymptotic results can be invoked. Using the conditional distribution of Nn,k we derive a test for randomness and compare it with standard procedures based on runs, ranks, and variances. The simulation results showed that the new test is significantly more powerful in detecting certain types of clustering. Applications in neurology and reliability are provided.
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... Author Keywords: Bernoulli trials; success runs of length k; discrete distributions of order k; occupancy models. Article Outline. References. ... Keywords: Bernoulli trials, success runs of length k, discrete distributions of order... more
... Author Keywords: Bernoulli trials; success runs of length k; discrete distributions of order k; occupancy models. Article Outline. References. ... Keywords: Bernoulli trials, success runs of length k, discrete distributions of order k, occupancy models. ...
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Research Interests: Mathematics, Combinatorics, Partitions, Partition, Permutations, and 5 moreAlphabet, Array, Permutation, ARRAYS, and Logarithm
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Consider a stationary Markov chain with state space consisting of the ξ -letter alphabet set Λ= {a1, a2, ···, aξ }. We study the variables M=M(n, k) and N=N(n, k), defined, respectively, as the number of overlapping and non-overlapping... more
Consider a stationary Markov chain with state space consisting of the ξ -letter alphabet set Λ= {a1, a2, ···, aξ }. We study the variables M=M(n, k) and N=N(n, k), defined, respectively, as the number of overlapping and non-overlapping occurrences of a fixed periodic k-letter word, and use the Stein–Chen method to obtain compound Poisson approximations for their distribution.
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... Mood, AM, FA Graybill, and DC Boes (1974), An Introduction to the Theory of Statistics, 3rd Edition, (McGraw Hill, New York). Panaretos, J. and E. Xekalaki (1986), On some distributions arising from certain generalized sampling... more
... Mood, AM, FA Graybill, and DC Boes (1974), An Introduction to the Theory of Statistics, 3rd Edition, (McGraw Hill, New York). Panaretos, J. and E. Xekalaki (1986), On some distributions arising from certain generalized sampling schemes, Commun. Statist.- Theor. ...
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... ABSTRACT Let M = Mn.k and N = Nn,k denote, respectively, the number of overlapping and non-overlapping success runs of fixed length k in n Bernoulli (p) trials. We derive an alternative formula for P(M = x) which is substantially... more
... ABSTRACT Let M = Mn.k and N = Nn,k denote, respectively, the number of overlapping and non-overlapping success runs of fixed length k in n Bernoulli (p) trials. We derive an alternative formula for P(M = x) which is substantially Copyrieht O 1992 by Marcel Dekker, Inc. ...