scholarly journals Stochastic monotonicity of dependent variables given their sum

Test ◽  
2021 ◽  
Author(s):  
Franco Pellerey ◽  
Jorge Navarro
Keyword(s):  
Likelihood Ratio ◽  
Random Variables ◽  
Monotonicity Property ◽  
Expected Value ◽  
Independent Random Variables ◽  
Likelihood Ratio Order ◽  
Stochastic Monotonicity ◽  
Dependent Variables ◽  
Finite Set

AbstractGiven a finite set of independent random variables, assume one can observe their sum, and denote with s its value. Efron in 1965, and Lehmann in 1966, described conditions on the involved variables such that each of them stochastically increases in the value s, i.e., such that the expected value of any non-decreasing function of the variable increases as s increases. In this paper, we investigate conditions such that this stochastic monotonicity property is satisfied when the assumption of independence is removed. Comparisons in the stronger likelihood ratio order are considered as well.

STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

2016 ◽  
Vol 31 (3) ◽  
pp. 366-380
Author(s):  
Ebrahim Amini-Seresht ◽  
Yiying Zhang
Keyword(s):  
Distribution Function ◽  
Order Statistics ◽  
Likelihood Ratio ◽  
Monotonicity Property ◽  
Independent Random Variables ◽  
Likelihood Ratio Order ◽  
Stochastic Monotonicity ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Parametric Families

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

scholarly journals On the Convolution of Heterogeneous Bernoulli Random Variables

2011 ◽  
Vol 48 (3) ◽  
pp. 877-884 ◽  
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).

A Behavioral Characterization of the Likelihood Ratio Order

2021 ◽  
Vol 3 (3) ◽  
pp. 353-366
Author(s):  
Maximilian Mihm ◽  
Lucas Siga
Keyword(s):  
Likelihood Ratio ◽  
Expected Utility ◽  
Stochastic Dominance ◽  
Likelihood Ratio Order ◽  
Finite Set ◽  
Behavioral Characterization

It is well known that stochastic dominance is equivalent to a unanimity property for monotone expected utilities. For lotteries over a finite set of prizes, we establish an analogous relationship between likelihood ratio dominance and monotone betweenness preferences, which are an important generalization of expected utility. (JEL D11, D44)

ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

2011 ◽  
Vol 25 (3) ◽  
pp. 369-391 ◽  
Author(s):  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Random Variables ◽  
Parallel Systems ◽  
Likelihood Ratio Order ◽  
Maximum Order ◽  
Hazard Rate Order

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. LetX1andX2be two independent gamma random variables withXihaving shape parameterr>0 and scale parameter λi,i=1, 2, and letX*1andX*2be another set of independent gamma random variables withX*ihaving shape parameterrand scale parameter λ*i,i=1, 2. Denote byX2:2andX*2:2the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) isp-larger than (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

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

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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