Modifying Bradley–Terry and other ranking models to allow ties

Geometric Distribution ◽

Discrete Distributions ◽

Ranking Models

Abstract Models derived from distributions of order-statistics are useful for modelling ranked data. The well-known Bradley–Terry (BT) and Plackett–Luce (PL) models can be derived from the order statistics of the exponential distribution but cannot handle ties. However, ties often occur in sports, and the ability to accommodate them leads to more useful ranking models. In this paper, we use discrete distributions, principally the geometric distribution, to obtain modified BT and PL models and some others that allow tied ranks. Our methodology is introduced for some mathematically tractable and some less tractable distributions and is illustrated using test match cricket.

Characterizations of Twenty (2020-2021) Proposed Discrete Distributions

Pakistan Journal of Statistics and Operation Research ◽

10.18187/pjsor.v17i4.3902 ◽

2021 ◽

pp. 847-884

Author(s):

G.G. Hamedani ◽

Mahrokh Najaf ◽

Amin Roshani ◽

Nadeem Shafique Butt

Keyword(s):

Poisson Distribution ◽

Geometric Distribution ◽

Discrete Distribution ◽

Gumbel Distribution ◽

Rayleigh Distribution ◽

Discrete Distributions ◽

Lindley Distribution ◽

Lomax Distribution ◽

Two Parameter

In this paper, certain characterizations of twenty newly proposed discrete distributions: the discrete gen- eralized Lindley distribution of El-Morshedy et al.(2021), the discrete Gumbel distribution of Chakraborty et al.(2020), the skewed geometric distribution of Ong et al.(2020), the discrete Poisson X gamma distri- bution of Para et al.(2020), the discrete Cos-Poisson distribution of Bakouch et al.(2021), the size biased Poisson Ailamujia distribution of Dar and Para(2021), the generalized Hermite-Genocchi distribution of El-Desouky et al.(2021), the Poisson quasi-xgamma distribution of Altun et al.(2021a), the exponentiated discrete inverse Rayleigh distribution of Mashhadzadeh and MirMostafaee(2020), the Mlynar distribution of Fr¨uhwirth et al.(2021), the ﬂexible one-parameter discrete distribution of Eliwa and El-Morshedy(2021), the two-parameter discrete Perks distribution of Tyagi et al.(2020), the discrete Weibull G family distribution of Ibrahim et al.(2021), the discrete Marshall–Olkin Lomax distribution of Ibrahim and Almetwally(2021), the two-parameter exponentiated discrete Lindley distribution of El-Morshedy et al.(2019), the natural discrete one-parameter polynomial exponential distribution of Mukherjee et al.(2020), the zero-truncated discrete Akash distribution of Sium and Shanker(2020), the two-parameter quasi Poisson-Aradhana distribution of Shanker and Shukla(2020), the zero-truncated Poisson-Ishita distribution of Shukla et al.(2020) and the Poisson-Shukla distribution of Shukla and Shanker(2020) are presented to complete, in some way, the au- thors’ works.

Characterizing the geometric distribution using expectations of order statistics

10.2307/3214277 ◽

1987 ◽

Vol 24 (2) ◽

pp. 534-539 ◽

Cited By ~ 4

Author(s):

A. C. Dallas

Keyword(s):

Order Statistics ◽

Conditional Expectation ◽

Geometric Distribution ◽

Discrete Distributions

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.

EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990258 ◽

2010 ◽

Vol 24 (2) ◽

pp. 245-262 ◽

Cited By ~ 33

Author(s):

Tiantian Mao ◽

Taizhong Hu

Keyword(s):

Order Statistics ◽

Geometric Distribution ◽

Random Variables ◽

The Other ◽

Stochastic Orders ◽

Exponential Random Variables ◽

Discrete Counterpart ◽

Similarities And Differences ◽

Geometric Variables

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

On characterizations of the geometric distribution by independence of functions of order statistics

10.2307/3214133 ◽

1986 ◽

Vol 23 (1) ◽

pp. 227-232 ◽

Cited By ~ 9

Author(s):

R. C. Srivastava

Keyword(s):

Order Statistics ◽

Geometric Distribution ◽

Random Variables ◽

Weak Form ◽

Discrete Distributions ◽

Independence Properties

Let X1, · ··, Xn, n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y1 ≦ Y2 ≦ · ·· ≦ Yn denote the corresponding order statistics. Define Rn = Yn – Y1 and Zn = Σj=2n (Υj – Y1). Then it is well known that (i) Y, and Rn are independent and (ii) Y1 and Zn are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.

The characterization of distributions by order statistics and record values — a unified approach

10.2307/3213643 ◽

1984 ◽

Vol 21 (2) ◽

pp. 326-334 ◽

Cited By ~ 39

Author(s):

Paul Deheuvels

Keyword(s):

Order Statistics ◽

Simple Proof ◽

Order Statistic ◽

Continuous Distribution ◽

Record Values ◽

Discrete Distributions ◽

Unified Approach ◽

Characterization Of Distributions

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successive jth record values and associated to an i.i.d. sequence are such that and are independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such that Xk, n and Xk+1, n – Xk, n are independent for some k ≧ 2 (where Xk, n is the kth order statistic of X1, ···, Xn) are degenerate.

Characterizing the geometric distribution using expectations of order statistics

10.1017/s002190020003117x ◽

1987 ◽

Vol 24 (02) ◽

pp. 534-539 ◽

Cited By ~ 1

Author(s):

A. C. Dallas

Keyword(s):

Order Statistics ◽

Conditional Expectation ◽

Geometric Distribution ◽

Discrete Distributions

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.

The characterization of distributions by order statistics and record values — a unified approach

10.1017/s0021900200024712 ◽

1984 ◽

Vol 21 (02) ◽

pp. 326-334

Author(s):

Paul Deheuvels

Keyword(s):

Order Statistics ◽

Simple Proof ◽

Order Statistic ◽

Continuous Distribution ◽

Record Values ◽

Discrete Distributions ◽

Unified Approach ◽

Characterization Of Distributions

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successivejth record valuesandassociated to an i.i.d. sequence are such thatandare independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such thatXk, nandXk+1,n–Xk, nare independent for somek≧ 2 (whereXk, nis thekth order statistic ofX1, ···,Xn) are degenerate.

On characterizations of the geometric distribution by independence of functions of order statistics

10.1017/s0021900200106448 ◽

1986 ◽

Vol 23 (01) ◽

pp. 227-232

Author(s):

R. C. Srivastava

Keyword(s):

Order Statistics ◽

Geometric Distribution ◽

Random Variables ◽

Weak Form ◽

Discrete Distributions ◽

Independence Properties

Let X1, · ··, X n , n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y 1 ≦ Y 2 ≦ · ·· ≦ Y n denote the corresponding order statistics. Define Rn = Yn – Y 1 and Z n = Σ j=2 n (Υ j – Y1). Then it is well known that (i) Y, and Rn are independent and (ii) Y 1 and Zn are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.