A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

BAYESIAN SMOOTHING WITH OCCASIONAL JUMPS

Consider a sequence of independent random variables X ₁, X ₂,…,X _n observed at n equally spaced time points where X _i has a probability distribution which is known apart from the values of a parameter θ _i ∈ R which may change from observation to observation. We consider the problem of estimating θ = (θ₁, θ₂,…,θ _n ) given the observed values of X ₁, X ₂,…,X _n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.     

Nonparametric estimation of survival functions for incomplete observations when the life time distribution is proportionally related ot the censoring time distribution

Let X₁, X₂…,X_n be a random sample from [ILM0001] and let Y₁, …,Y_n be a random sample from [ILM0002]. Then instead of observing a complete sample X₁,…X_n, we can only observe the pairs Z_i. = min(X_i.,Y_i) and [ILM0003] In this paper, we consider estimation of survival function [ILM0004] when [ILM0005], where β is an unknown positive real number.

     

The parametric hypothesis test of multiple uniform populations

X₁, X₂, …, X_k are k(k ? 2) uniform populations which each X_i follows U(0, θ_i). This note shows the test statistic for the null hypothesis H₀: θ₁ = θ₂ = ??? = θ_k by using the order statistics.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

Minimizing the expected sample range

Let X_i be i.i.d. random variables with finite expectations, and θ_i arbitrary constants, i=1,…,n. Y_i=X_i+θ_i. The expected range of the Y's is R_n(θ₁,…,θ_n)=E(maxY_i-minY_i. It is shown that the expected range is minimized if and only if θ₁=?=θ_n. In the case where the X_i are independently and symmetrically distributed around the same constant, but not identically distributed, it is shown that θ₁=?=θ_n are not necessarily the only (θ₁,...,θ_n) minimizing R_n. Some lemmas which are applicable to more general problems of minimizing R_n are also given.     

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Locally best invariant and locally minimax test of independence

Let X₁ X₂ … X_N be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sG_i) [(1–p) I + pE] Diag ($sG_i), $sG_i> 0, i = 1… p, 1>p>=1/p–1. Write r_ij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H₀ : p = 0 against the alternative H₁ : p>0 where $$ and $sG₁,…, $sG_p are unknown, the test which rejects H₀ for large value of $$ r_ij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

Test of equality of normal means in the absence of independent estimator of variance

Let X_l,…,X_n be normally and independently distributed with means θ_l,…,θ_nand a cornmorl variance. Thus there are n observations and n+i unknwon parameters. A test of the null hypothesis that, the θ_i's are all zero and the alternative that the vector (θ_l,…,θ_n) lies in a convex cone with its vertex a.t the origin is connsidered in this paper. It is shown that under a mild condition the likelihood ratio test is possible. The ordinary one sided t - test belongs to the class of tests considered in this paper. The hypothesis of equality of means against the simple order alternative can be tested in certain cases .     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

A One Sided Test Based on Sample Quasi Ranges

Abstract

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F _i (x) = F[(x ? μ _i )/θ _i ], i = 1,…, k (k ≥ 2). Here μ _i (?∞ < μ _i  < ∞) is the location parameter, θ _i (θ _i  > 0) is the scale parameter and F(?) is any absolutely continuous cdf, i.e., F _i (?) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H ₀ ? θ₁ = · = θ _k against the simple ordered alternative H _A  ? θ₁ ≤ · ≤ θ _k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X _i1,…, X _in be a random sample of size n from the population π _i and R _ir  = X _i:n?r  ? X _i:r+1, r = 0, 1,…, [n/2] ? 1 be the sample quasi range corresponding to this random sample, where X _i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W _r  = max_1≤i<j≤k (R _jr /R _ir ). The test is reject H ₀ for large values of W _r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ _j /θ _i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W _r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.     

Stochastic orderings related to ranking in an nth order record process

Let X₁, X₂, …, X_m be successive observations on m objects, numbered 1,2, …, m. If X₁ belongs to the n largest observations among X₁,X₂,…,X_i, object i is called a record, i = 1,2,…,m; n?1.In investigating the influence of the ranking of the objects on the expected number of records, a hierarchy of stochastic order relations between random variables arises.It is these order relations and their relationship with known stochastic orderings that are studied in this paper.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

Testing independence with additional information

Let X = (X^j : j = 1,…, n) be n row vectors of dimension p independently and identically distributed multinomial. For each j, X^j is partitioned as X^j = (X^j₁, X^j₂, X^j₃), where p_i is the dimension of X^j_i with p₁ = 1,p₁+p₂+p₃ = p. In addition, consider vectors Y^j_i, i = 1,2j = 1,…,n_i that are independent and distributed as X¹_i. We treat here the problem of testing independence between X¹₁ and X¹₃ knowing that X¹₁ and X¹₂ are uncorrected. A locally best invariant test is proposed for this problem.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.     

On the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.