A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let X_i≤?≤X_m and Y_i≤?≤Y_n be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let R_i denote the rank of X_i in the combined order sample. Steck (1980) has found an expression for P(b_i≤R_i≤a_i, all i) when F = h(G), h being the incomplete beta function with parameters (α,β?α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.     

Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

The probability generating function of empty cell variable in a randomized occupancy problem

Let M_o denote the number of empty cells when n distinguishable balls are distributed independently and at random in ra cells such that each ball stays with probability p in its cell, and falls through with probability 1-p. We find the probability generating function of M_o by solving a partial differential equation satisfied by a suitable generating function. The corresponding function for the classical case p = 1 is well-known, but obtained by different methods.     

Constants on a Uniform Berry-Esseen Bound on Some Borel Sets in ℝ k via Stein's Method

For each n, k ∈ ?, let Y _i  = (Y _i1, Y _i2,…, Y _ik ), 1 ≤ i ≤ n be independent random vectors in ? ^k with finite third moments and Y _ij are independent for all j = 1, 2,…, k. In this article, we use the Stein's technique to find constants in uniform bounds for multidimensional Berry-Esseen inequality on a closed sphere, a half plane and a rectangular set.     

Poisson approximation io the distribution of empiy cells in randomized occupancy

Let M_o denote the number of empty cells when n balls are dropped independently and at random in m cells such that each ball stays in its cell with probability p and falls through with probability 1-p. A Poisson limit is known for M_o when (n/m)→∞. We find a corresponding approximation to the distribution of M_o when m is large but finite, Ihe method is elementary, and yields the rate of convergence to the limit law. Ihe results are new for the classical case (p = 1) as well.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

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 $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

Precise large deviations for the difference of two sums of END random variables with heavy tails

Assume that there are two types of insurance contracts in an insurance company, and the ith related claims are denoted by {X_ij, j ? 1}, i = 1, 2. In this article, the asymptotic behaviors of precise large deviations for non random difference ∑^n₁(t)_{j = 1}X_1j ? ∑^n₂(t)_{j = 1}X_2j and random difference ∑^N₁(t)_{j = 1}X_1j ? ∑^N₂(t)_{j = 1}X_2j are investigated, and under several assumptions, some corresponding asymptotic formulas are obtained.     

Maximum likelihood estimation in convex hull models

In this paper we consider models involving the convex hull operation of the parameter and the noise i.e. Y_i = CH(A, X_X). Then we generalize the basic models to ANOVA models; i.e. Y_ij=CH(A∪B_j,X_ij). In some cases the consistent estimators for the J U new parameters are derived. Assuming the existence of density forrandom convex sets, we derive the likelihood for the convex hull model. We then find the maximum Likelihood Estimators for the parameters. Examples for some random convex sets with finite dimensional distributions are derived to show how good these estimators are.     

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.     

Minimax subset selection for the multinomial distribution

Let (X₁,…,X_k) be a multinomial vector with unknown cell probabilities (p₁,?,p_k). A subset of the cells is to be selected in a way so that the cell associated with the smallest cell probability is included in the selected subset with a preassigned probability, P¹. Suppose the loss is measured by the size of the selected subset, S. Using linear programming techniques, selection rules can be constructed which are minimax with respect to S in the class of rules which satisfy the P¹-condition. In some situations, the rule constructed by this method is the rule proposed by Nagel (1970). Similar techniques also work for selection in terms of the largest cell probability.     

A RENEWAL THEOREM IN MULTIDIMENSIONAL TIME

Let Y_l, Y₂,… be i.i.d., positive, integer-valued random variables with means, μ. Let the sequences {Y_ij, j= 1,2,…}, i= 1,…, r be independent copies of {Y₁, Y₂,…}. For n={n₁,…, n_r.}, n₁≥1, let S_n=S?ⁿ¹_k1=1= 1 …S?^nr_kr=1 Y_ik1… Y_rkr. We show that S?^N_k=1S?^∞_k1=1…S?^∞_nr=1 P[[S_n= k] ? [μ^-r N log^r-1 (N)/(r-1)!] as N →∞.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000     

On an application of concomitants of order statistics

ABSTRACT

Let (X_i, Y_i), i = 1, …, n be a pair where the first coordinate X_i represents the lifetime of a component, and the second coordinate Y_i denotes the utility of the component during its lifetime. Then the random variable Y_{[r: n]} which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth smallest lifetime. In this paper, we present a dynamic analysis for an n component system under the above-mentioned concomitant setup.     

Coupon collector's problem and generalized Pareto distributions

Let N_n={1,2,…,n}. We sample with replacement from the set N_n assuming that each element has probability 1/n of being drawn. Let M_n be the waiting time determined by certain stoping rules in the coupon collector's problem. We investigate models for the asymptotic behavior of the excesses of M_n over the high thresholds.     

A Subset Selection Procedure for Selecting the Exponential Population Having the Longest Mean Lifetime When the Guarantee Times are the Same

ABSTRACT

Consider k(≥ 2) independent exponential populations Π₁, Π₂, …, Π _k , having the common unknown location parameter μ ∈ (?∞, ∞) (also called the guarantee time) and unknown scale parameters σ₁, σ₂, …σ _k , respectively (also called the remaining mean lifetimes after the completion of guarantee times), σ _i  > 0, i = 1, 2, …, k. Assume that the correct ordering between σ₁, σ₂, …, σ _k is not known apriori and let σ_[i], i = 1, 2, …, k, denote the ith smallest of σ _j s, so that σ_[1] ≤ σ_[2] ··· ≤ σ_[k]. Then Θ _i  = μ + σ _i is the mean lifetime of Π _i , i = 1, 2, …, k. Let Θ_[1] ≤ Θ_[2] ··· ≤ Θ_[k] denote the ranked values of the Θ _j s, so that Θ_[i] = μ + σ_[i], i = 1, 2, …, k, and let Π_(i) denote the unknown population associated with the ith smallest mean lifetime Θ_[i] = μ + σ_[i], i = 1, 2, …, k. Based on independent random samples from the k populations, we propose a selection procedure for the goal of selecting the population having the longest mean lifetime Θ_[k] (called the “best” population), under the subset selection formulation. Tables for the implementation of the proposed selection procedure are provided. It is established that the proposed subset selection procedure is monotone for a general k (≥ 2). For k = 2, we consider the loss measured by the size of the selected subset and establish that the proposed subset selection procedure is minimax among selection procedures that satisfy a certain probability requirement (called the P*-condition) for the inclusion of the best population in the selected subset.     

Estimating the mean of the selected uniform population

Let π_i(i=1,2,…K) be independent U(0,?_i) populations. Let Y_i denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ?_i, we consider the natural selection rule, according to which the population corresponding to the largest Y_i is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cY_max. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L₁ and the scale-invariant loss function L₂. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n and ρ=?₁/(?₁+?₂).