On the negative binomial distribution and its generalizations

Vellaisamy, P.; Upadhye, N. S.

doi:10.1016/j.spl.2006.06.008

Cited by 12 publications

(8 citation statements)

References 6 publications

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“…The binary AR geometric distribution obtained agrees with the literature (Viveros et al ., ; Vellaisamy and Upadhye, ; Minkova and Omey, ). Viveros et al .…”

Section: Methodsmentioning

confidence: 99%

“…Also, for p =1, Minkova and Omey () obtained the geometric distribution related to a Bernoulli sequence exhibiting the correlation coefficient

ρ (X_{t}, X_{t - 1}) = ρ \neq 0

P false(Y = k false) = π false(1 - italicρ false) {(}{1 - π (1 - ρ))}^{k - 2} false(1 - italicπ false) false(1 - italicρ false),

in which case

\begin{matrix} P false(X_{t} = 1 false| X_{t - 1} = 0 false) = π false(1 - italicρ false), \\ P false(X_{t} = 1 false| X_{t - 1} = 1 false) = 1 - false(1 - italicπ false) false(1 - italicρ false) . \end{matrix}

In this formulation, μ = π (1− ρ ) and

{italicϕ}_{1}^{*} = ρ

, which is clearly in accordance with the Yule–Walker equations for p =1. For higher order Markovian Bernoulli trials, Vellaisamy and Upadhye () obtained the geometric related probabilities

P false(Y = k false) = \{\begin{matrix} a_{1} & k = 1, \\ a_{k} \prod_{i = 1}^{k - 1} false(1 - a_{i} false) & k ⩾ 2, \end{matrix}

where the transition probability of the process is defined by

P false(X_{k} = 1 false| X_{k...}

…”

Section: Methodsmentioning

confidence: 99%

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Binary Auto-Regressive Geometric Modelling in a DNA Context

Gouveia

Scotto

Weiß

et al. 2016

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Symbolic or categorical sequences occur in many contexts and can be characterized, for example, by integer‐valued intersymbol distances or binary‐valued indicator sequences. The analysis of these numerical sequences often sheds light on the properties of the original symbolic sequences. This work introduces new statistical tools for exploring auto‐correlation structure in the indicator sequences, for the specific case of deoxyribonucleic acid (DNA) sequences. It is known that the probability distribution of internucleotide distances of DNA sequences deviates significantly from the distribution obtained by assuming independent random placement (i.e. the geometric distribution) and that the deviations can be used either to discriminate between species or to build phylogenetic trees. To investigate the extent to which auto‐correlation structure explains these deviations, the 0–1 indicator sequence of each nucleotide (A, C, G and T) is endowed with a binary auto‐regressive (AR) model of optimum order. The corresponding binary AR geometric distribution is derived analytically and compared with the observed internucleotide distance distribution by appropriate goodness‐of‐fit testing. Results in 34 mitochondrial DNA sequences show that the hypothesis of equal observed/expected frequencies is seldom rejected when a binary AR model is considered instead of independence (76/136 versus 125/136 rejections at the 1% level), in spite of χ2‐testing tending to reject for large samples, regardless of how close observed/expected values are. Furthermore, binary AR structure also leads to a median discrepancy reduction of 90% for G, 80% for C, 60% for T and 30% for nucleotide A. Therefore, these models are useful to describe the dependences within a given nucleotide and encourage the development of a model‐based framework to compact internucleotide distance information and to understand DNA differences among species further.

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“…The binary AR geometric distribution obtained agrees with the literature (Viveros et al ., ; Vellaisamy and Upadhye, ; Minkova and Omey, ). Viveros et al .…”

Section: Methodsmentioning

confidence: 99%

“…Also, for p =1, Minkova and Omey () obtained the geometric distribution related to a Bernoulli sequence exhibiting the correlation coefficient

ρ (X_{t}, X_{t - 1}) = ρ \neq 0

P false(Y = k false) = π false(1 - italicρ false) {(}{1 - π (1 - ρ))}^{k - 2} false(1 - italicπ false) false(1 - italicρ false),

in which case

\begin{matrix} P false(X_{t} = 1 false| X_{t - 1} = 0 false) = π false(1 - italicρ false), \\ P false(X_{t} = 1 false| X_{t - 1} = 1 false) = 1 - false(1 - italicπ false) false(1 - italicρ false) . \end{matrix}

In this formulation, μ = π (1− ρ ) and

{italicϕ}_{1}^{*} = ρ

, which is clearly in accordance with the Yule–Walker equations for p =1. For higher order Markovian Bernoulli trials, Vellaisamy and Upadhye () obtained the geometric related probabilities

P false(Y = k false) = \{\begin{matrix} a_{1} & k = 1, \\ a_{k} \prod_{i = 1}^{k - 1} false(1 - a_{i} false) & k ⩾ 2, \end{matrix}

where the transition probability of the process is defined by

P false(X_{k} = 1 false| X_{k...}

…”

Section: Methodsmentioning

confidence: 99%

Binary Auto-Regressive Geometric Modelling in a DNA Context

Gouveia

Scotto

Weiß

et al. 2016

Journal of the Royal Statistical Society Series C: Applied Statistics

View full text Add to dashboard Cite

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“…It is well known that if X 1 and X 2 are iid Z + -valued rvs, and X 1 + X 2 ∼ NB distribution, then X 1 and X 2 must be geometric rvs. However, if X 1 and X 2 are independent (or dependent), but not identically distributed, then it is possible for either or both rvs to be nongeometric and still X 1 + X 2 is NB (see, Shishebor and Towhidi (2004), Vellaisamy and Upadhye (2007)). …”

Section: Theorem 4 Let Y 2 Be a Nonnegative Continuous Rv And {Nmentioning

confidence: 99%

“…Note that for the B(n, p) model the converse is also true (Vellaisamy (1996)), while it is not true for the NB(r, p) model (Vellaisamy and Upadhye (2007)). In Section 3, we explore the connections between a discrete distribution and a Poisson process.…”

Section: Introductionmentioning

confidence: 97%

“…In particular, they showed that a B(n, p) distribution arises more often as the distribution of S n of dependent and non identical Bernoulli rvs. Recently, Vellaisamy and Upadhye (2007) have established that the negative binomial NB(r, p) distribution also possesses similar properties. That is, a NB(r, p) distribution may arise as a distribution of the sum of dependent and non-geometric variables.…”

Section: Introductionmentioning

confidence: 99%

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Some Intrinsic Properties of the Gamma Distribution

Vellaisamy

Sreehari²

2010

JJSS

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Let {Yn} be a sequence of nonnegative random variables (rvs), and Sn = n j=1 Yj, n ≥ 1. It is first shown that independence of S k−1 and Y k , for all 2 ≤ k ≤ n, does not imply the independence of Y1, Y2, . . . , Yn. When Yj's are identically distributed exponential Exp(α) variables, we show that the independence of S k−1 andIt is shown by a counterexample that the converse is not true. We show that if X is a non-negative integer valued rv, then there exists, under certain conditions, a rvwhere {N (t)} is a standard (homogeneous) Poisson process, and obtain the Laplace-Stieltjes transform of Y . This leads to a new characterization for the gamma distribution. It is also shown that a G(α, k) distribution may arise as the distribution of S k , where the components are not necessarily exponential. Several typical examples are discussed.

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Statistics for Scientists and Engineers

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On the negative binomial distribution and its generalizations

Cited by 12 publications

References 6 publications

Binary Auto-Regressive Geometric Modelling in a DNA Context

Binary Auto-Regressive Geometric Modelling in a DNA Context

Some Intrinsic Properties of the Gamma Distribution

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