Random Variables Xi Research Articles

Characterization of laws by balancing weighting against a binary operation

LetL(X) be the law of a positive random variableX, andZ be positive and independent ofX. Solution pairs (L(X), L(Z)) are sought for the in-law equation\(\hat X \cong X \circ Z\) where\(L(\hat X)\) is a weighted law constructed fromL(X), and ° is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ° is the maximum operation a complete solution in terms of a product integral is found for arbitrary weighting. Examples are given. An identity for the length biasing operator is used when ° is multiplication to establish a general solution in terms of an already solved inverse equation. Some examples are given.

Characterizations, length-biasing, and infinite divisibility

SupposeL(X) is the law of a positive random variableX, andZ is positive and independent ofX. Admissible solution pairs (L(X),L(Z)) are sought for the in-law equation\(\hat X \cong X o Z\) ∘Z, where\(L\left( {\hat X} \right)\) is a weighted law constructed fromL(X), and ∘ is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ∘ is addition and the weighting is ordinary length biasing, the class of admissibleL(X) comprises the positive infinitely divisible laws. Examples are given subsuming all known specific cases. Some extensions for general order of length-biasing are discussed.

Likelihood ratio tests for symmetry against one-sided alternatives

A random variableX is said to have a symmetric distribution (about 0) if and only ifX and −X are, identically distributed. By considering various types of partial orderings between the distributions ofX and −X, one obtains various notions of skewness or one-sided bias. In this paper we study likelihood ratio tests for testing the symmetry of a discrete distribution about zero against the alternatives, (i)X is stochastically greater than −X; and (ii) pr(X=j)≥pr(X=−j) for allj>0. In the process, we obtain maximum likelihood estimators of the distribution function under the above alternatives. The asymptotic null distributions of the test statistics have been obtained and are of the chi-bar square type. A simulation study was performed to compare the powers of these tests with other tests.

Branching random walk with a critical branching part

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timekn withkn/n→β. We attach to each vertexv of this tree a random variableX(v) and define\(S(v) = \Sigma _{w \varepsilon \pi (0,v)} X(w)\), where π(0,v) is the unique path in the family tree from its root tov. FinallyMnis the maximal displacement of the branching random walkS(·), that isMn=max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn−1/2Mn converges in distribution. In particular {n−1/2Mn}n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Let the random variableX be normal distributed with known varianceσ 2>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl p (θ, d)=|θ-d| p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.

Packing random items of three colors

Consider three colors 1,2,3, and forj≤3, considern items (X i,j)i≤n of colorj. We want to pack these items inn bins of equal capacity (the bin size is not fixed, and is to be determined once all the objects are known), subject to the condition that each bin must contain exactly one item of each color, and that the total item sizes attributed to any given bin does not exceed the bin capacity. Consider the stochastic model where the random variables (X i,jj)i≤n,j≤3 are independent uniformly distributed over [0,1]. We show that there is a polynomial-time algorithm that produces a packing which has a wasted space≤K logn with overwhelming probability.

Some Optimal Selection Procedures for Dependent Normal Populations

We consider k dependent normal populations having known equal variance. The problem is to select the population associated with the largest mean. Let π1, ..., πk be k populations. We assume that the random variable Xi associated with πi ( i = 1, ..., k). Let X1, ..., Xk be k—variate normal distribution with unknown mean μ1, ... , μk and with equal variance σ2 and with correlation matrix { ρi j} For two case : (a) ρi j = aiaj for i ≠ j and (b) ρi j = ρ, for i ≠ j, we construct some optimal subset selection procedure to select a subset of the k dependent normal population containing the largest population mean. AMS ( 1980) subject classification : Primary 62F07 ; Secondary 62030

Distributions determined by conditioning on a single order statistic

LetX 1,X 2, …,X n(n ⩾ 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), −∞ ⩽a <b ⩽ ∞, the support ofF andX 1:n ⩽X 2:n ⩽ … ⩽X n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a +) andE {g(X)}. Then for some positive integers, 1 <s ⩽n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and $$F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)$$ . This leads to characterization of several distribution functions. A general form of this result is also stated.

An infinite-dimensional law of large numbers in Cesaro's sense

Let (X k) k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>−1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1[ the sequenceV n =(1/A α )∑0⩽k⩽n A α−1 X k converges almost surely toE(X) if and only if ‖X‖1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Deniel and Derriennic (case α=1/2).

Selection from Logistic populations

Assume k(k≥ 2) independent populations π1, π2μk are given. The associated independent random variables Xi,(i= 1,2,…k) are Logistically distributed with unknown means μ1, μ2, μk and equal variances. The goal is to select that population which has the largest mean. The procedure is to select that population which yielded the maximal sample value. Let μ(1)≤μ(2)≤…≤μ(k) denote the ordered means. The probability of correct selection has been determined for the Least Favourable Configuration μ(1)=μ(2)==μ(k – 1)=μ(k)–δ where δ &gt; 0. An exact formula for the probability of correct selection is given.

Laws of large numbers for weighted sums of random variables and random elements

Consider the weighted sums\(S_n = \sum\limits_{k = 1}^\infty {a_{nk} X_k } \) of a sequence {Xn} of independent random variables or random elements inD [0,1]. For convergence ofSn in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {Xn} is uniformly bounded by a random variableX,i.e.P(¦Xn¦⩾x)⩽P(¦X¦⩾x) for allx>0. Our paper aims at trying to drop this restriction.

On the ratio of the expected maximum of a martingale and theL p-norm of its last term

For eachp>1, the supremum,S, of the absolute value of a martingale terminating at a random variableX inLp, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1).The maximum,M, of a mean-zero martingale which starts at zero and terminates atX, satisfiesES≦(Γ(q))1/q‖X‖p (q=p(p-1)-1), whereσq is the unique solution of the equationt = ‖Z −t ‖q for an exponentially distributed random variableZ with mean 1.σp has other characterizations and satisfies limp‖q− 1σq =c withc determined bycec+1 = 1. Equalities in (1) and (2) are attainable by appropriate martingales which can be realized as stopped segments of Brownian motion. A presumably new property of the exponential distribution is obtained en route to inequality (2).

The optimality of radial flight

A target makes a loud noise which can be detected at a great distance and then the target becomes quiet. A potential pursuer who may have heard the noise will be assumed to head directly toward the position of the loud noise. The target will have no knowledge of the initial bearing to the pursuer. Given a constant speed strategy for the target what path should the target take to minimize the chance that he is redetected by the pursuer after he has ceased making the noise? Let C1 be the tactic of a straight line flight from the initial source of noise. Let C2 be any other constant speed tactic taken by the target. The minimal distance between the target and the pursuer for each tactic Ci is a random variable Xi dependent on the initial heading of the target. Let FXi be the distribution function for the random variable Xi. We will prove that FX2 (x) ⩾ FX1 (x) for all x. Thus, for any distance x, there is a larger probability for the target to come within x miles of the pursuer if the target follows any non-radial flight strategy than if he follows a radial flight strategy. Note that the pursuer need not travel at a constant speed. Finally we will drop the restriction of constant speed from the target.

Some characterizations of distributions by regression models for ordinal response data

Let a random variableX be classified intok classes. By doing so, a new random variable is obtained, measured on ordinal scale. If this variable is a response variable in certain regression models for ordinal response data, the distribution ofX is characterized by the models. In this paper, characterizations of the distribution ofX by the proportional odds model and the proportional hazards model are given.

Wrapped distributions and measurement errors

For a random variableX and α>0 letUα≔nα (X)−X, wherenα(x)=n∈Z iffx∈(nα−α/2,nα+α/2]. Random variables of this type are important in the theory of measurement errors. We derive formulas for the distribution ofUα and apply them to the case X∼N(θ,σ2). General conditions for the unimodality ofUα are given. The correlation of the measurement errorsX−E (X) andUα (X) is seen to beO (αj) withj depending on the smoothness and asymptotic behavior of the density ofX. This gives a precise sense to the assertion that scale errors upwards and downwards are averagely well-balanced. In the normal case the density ofUα is shown to be constant up to\(o (e^{ - C\alpha ^{ - 2} } )\), as α→0.

Admissibility of some preliminary test estimators for the mean of normal distribution

Let a random variableX follow ap-variate normal distributionNp(θ, Ip) with an unknownp×1 vector θ andp×p identity matrixIp. The admissibility of a preliminary test estimator using AIC (Akaike's Information Criterion) procedure will be shown ifp=1 and its inadmissibility will be shown ifp≧3 under the loss function based on Kullback-Leibler information measure. Furthermore the two sample case is also considered.

Improving predictive distributions

Consider a sequence of decision problems S1, S2, ... and suppose that in problem Si the statistician must specify his predictive distribution Fi for some random variable Xi and make a decision based on that distribution. For example, Xi might be the return on some particular investment and the statistician must decide whether or not to make that investment. The random variables X1, X2, ... are assumed to be independent and completely unrelated. It is also assumed that each predictive distribution Fi assigned by the statistician is a subjective distribution based on his information and beliefs about Xi. In this context, the standard Bayesian approach provides no basis for evaluating whether the statistician's subjective predictive distribution for Xi is good or bad, and does not even recognize this question as being meaningful. In this paper we describe models in which the statistician can study his process for specifying predictive distributions, identify bad habits, and improve his predictions and decisions by gradually breaking these habits.

On a characterization of the exponential distribution by spacings

LetX be a non-negative random variable with probability distribution functionF. SupposeX i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X i+1,n−Xi,n) and (n−j)(X j+1,n−Xj,n) for somei, j andn, (1≦i<j<n).

A characterization of the exponential distribution by spacings

Let X 1, X 2, ···, Xn be a random sample of size n from a population with probability density function f(x), x &amp;gt; 0, and let X 1, n &amp;lt; X 2, n &amp;lt; ··· &amp;lt; Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − X i−1, n ) and (n − i)(X i+1, n − X i, n ) for one i and one n with 2 ≦ i ˂ n.

A characterization of the exponential distribution by spacings

Let X1, X2, ···, Xn be a random sample of size n from a population with probability density function f(x), x &gt; 0, and let X1, n &lt; X2, n &lt; ··· &lt; Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − Xi−1, n) and (n − i)(Xi+1, n − Xi, n) for one i and one n with 2 ≦ i ˂ n.