A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

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Partial orders and minimization of records in a sequence of independent random variables

Journal of Applied Probability ◽

10.1239/jap/1032374747 ◽

1999 ◽

Vol 36 (4) ◽

pp. 965-973

Author(s):

Raúl Gouet ◽

Jaime San Martín

Keyword(s):

Partial Order ◽

Likelihood Ratio ◽

Hazard Rate ◽

Random Variables ◽

Partial Orders ◽

Parametric Models ◽

Independent Random Variables ◽

Expected Number ◽

Monotone Likelihood Ratio ◽

Hazard Rate Ordering

Given independent random variables X1,…,Xn, with continuous distributions F1,…,Fn, we investigate the order in which these random variables should be arranged so as to minimize the number of upper records. We show that records are stochastically minimized if the sequence F1,…,Fn decreases with respect to a partial order, closely related to the monotone likelihood ratio property. Also, the expected number of records is shown to be minimal when the distributions are comparable in terms of a one-sided hazard rate ordering. Applications to parametric models are considered.

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Partial orders and minimization of records in a sequence of independent random variables

Journal of Applied Probability ◽

10.1017/s0021900200017782 ◽

1999 ◽

Vol 36 (04) ◽

pp. 965-973

Author(s):

Raúl Gouet ◽

Jaime San Martín

Keyword(s):

Partial Order ◽

Likelihood Ratio ◽

Hazard Rate ◽

Random Variables ◽

Partial Orders ◽

Parametric Models ◽

Independent Random Variables ◽

Expected Number ◽

Monotone Likelihood Ratio ◽

Hazard Rate Ordering

Given independent random variables X 1,…,X n , with continuous distributions F 1,…,F n , we investigate the order in which these random variables should be arranged so as to minimize the number of upper records. We show that records are stochastically minimized if the sequence F 1,…,F n decreases with respect to a partial order, closely related to the monotone likelihood ratio property. Also, the expected number of records is shown to be minimal when the distributions are comparable in terms of a one-sided hazard rate ordering. Applications to parametric models are considered.

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On the Convolution of Heterogeneous Bernoulli Random Variables

Journal of Applied Probability ◽

10.1017/s0021900200008391 ◽

2011 ◽

Vol 48 (03) ◽

pp. 877-884 ◽

Cited By ~ 1

Author(s):

Maochao Xu ◽

N. Balakrishnan

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Random Variables ◽

Likelihood Ratio Order ◽

Hazard Rate Order ◽

Reversed Hazard Rate ◽

Bernoulli Random Variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

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On the monotone likelihood ratio property for the convolution of independent binomial random variables

Discrete Applied Mathematics ◽

10.1016/j.dam.2009.03.002 ◽

2009 ◽

Vol 157 (11) ◽

pp. 2562-2564

Author(s):

Andrey Rukhin ◽

Carey E. Priebe ◽

Dennis M. Healy

Keyword(s):

Likelihood Ratio ◽

Random Variables ◽

Monotone Likelihood Ratio ◽

Binomial Random Variables

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On improvements of the order of approximation in the Poisson limit theorem

Journal of Applied Probability ◽

10.1017/s0021900200103808 ◽

1996 ◽

Vol 33 (01) ◽

pp. 146-155 ◽

Cited By ~ 1

Author(s):

K. Borovkov ◽

D. Pfeifer

Keyword(s):

Limit Theorem ◽

Random Variables ◽

Poisson Approximation ◽

Order Of Approximation ◽

Operator Semigroups ◽

Bernoulli Random Variables ◽

Usual Rate ◽

Poisson Limit ◽

Poisson Measures ◽

Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

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