A Resampling Approach for Under-estimating a Finite Population Total from a Censored Sample

Abstract

Suppose a finite population of N objects each of which has an unknown value μ _i  ≥ 0, i = 1, … , N of a nonnegative characteristic of interest. A random sample has been drawn, but only for a selected subset of the sample the μ-values have been observed. The subset selection procedure has been somewhat obscure, and thus the subsample is censorized rather than random. Despite that, a reliable lower bound for the population total (the sum of all μ _i ) is required which uses the statistical information contained in the data. We propose a resampling procedure to construct an under-estimate of the population total. We also consider the case when the objects of the population have unequal sampling probabilities, in particular when the population is divided into a few number of strata with constant probabilities within each stratum. A real data example illustrates the method.     

Tolerance Limits Based on the Multivariate MaxMin Chart

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises y_n such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences x_n which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.     

A note on the conditional residual lifetime of a coherent system under double monitoring

In this note, we derive some mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at time t₁ the jth failures has occurred and at time t₂ the kth failures (j < k) have not occurred yet. Based on the mixture representations, we then discuss the stochastic comparisons of the conditional residual lifetimes of two coherent systems with i.i.d. components.     

Conditional residual lifetimes of coherent systems under double monitoring

In this paper, we obtain a mixture representation for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under double monitoring. We suppose that at time t₁, j components have failed while at time t₂ the system is still alive. Based on these mixture representation, we then study stochastic comparisons of the conditional residual lifetimes of two coherent systems with independent and identical components.     

Comment on J. Berkson's paper “in dispraise of the exact test”

This paper rejects the preference expressed by Berkson for the heuristic test statistic T_N with standard normal distribution testing equality of two binomial probabilities in favour of Fisher's conditional exact test statistic T_E. Conditioning upon k₁ + k₂ = k shows that T_N admits too large first kind error probabilities. But also unconditionally T_N is systematically too large compared to T_E, giving too small critical levels at given experimental outcomes and power which is misleadingly too large. This is mainly due to the fact that T_N is not suitably corrected for continuity (at small sample sizes).     

On an improved Erdő-Rényi-type law for increments of partial sums

Let X₁, X₂,… be an independently and identically distributed sequence with ξX₁ = 0, ξ exp (tX₁ < ∞ (t ≧ 0) and partial sums S_n = X₁ + … + X_n. Consider the maximum increment D₁ (N, K) = max_{0≤n ≤ N - K} (S_{n + K} - S_n)of the sequence (S_n) in (0, N) over a time K = K_N, 1 ≦ K ≦ N. Under appropriate conditions on (K_N) it is shown that in the case K_N/log N → 0, but K_N/(log N)^1/2 → ∞, there exists a sequence (α_N) such that K_-1/2 D₁ (N, K) - α_N converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erd?s-Rényi-type laws stated by Csörg? and Steinebach (1981).     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Measuring robustness for weighted distributions: Bayesian perspective

There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ _a ={w(x):w(x)=(1−ε)w ₀(x)+εq(x),q∈Q} and Γ _g ={w(x):w(x)=w ₀ ^1−ε (x)q ^ε(x),q∈Q} wherew ₀(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ _a and Γ _g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.     

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.     

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|.     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

CRITICAL VALUES FOR INFERENCE ABOUT NORMAL DISPERSION1

Let F_N(.) be the density function of X²_N. Values of C₁/N, i= 1, 2, satisfying the twin conditions Pr (C₁≤X²_N≤C₂)=1-α and the conditional expectation of X²_N given C₁≤X²_N≤C₂ is N are tabulated for α=.2, .1, .05, .01, .005, .001, N=1(1)20(2)50(5)150(10)350. The second condition may be replaced by the condition f_N+2(C₁)=f_N+2V(C₂). The author has with him a bigger table giving C₁ and C₂ for α=.2, .1, .05, .01, .005, .001, N=1(1)350 to three decimals (to three significant digits, if some decimals are not significant). Several applications are mentioned. A practical application that is perhaps not obvious is to test whether two or more counts are distributed as independent Poisson variables. The new simple formulae used in the construction of the table are given and should prove useful in obtaining accurate values for omitted entries and in increasing the accuracy of entries.     

Simulation-assisted saddlepoint approximation

A general saddlepoint/Monte Carlo method to approximate (conditional) multivariate probabilities is presented. This method requires a tractable joint moment generating function (m.g.f.), but does not require a tractable distribution or density. The method is easy to program and has a third-order accuracy with respect to increasing sample size in contrast to standard asymptotic approximations which are typically only accurate to the first order.

The method is most easily described in the context of a continuous regular exponential family. Here, inferences can be formulated as probabilities with respect to the joint density of the sufficient statistics or the conditional density of some sufficient statistics given the others. Analytical expressions for these densities are not generally available, and it is often not possible to simulate exactly from the conditional distributions to obtain a direct Monte Carlo approximation of the required integral. A solution to the first of these problems is to replace the intractable density by a highly accurate saddlepoint approximation. The second problem can be addressed via importance sampling, that is, an indirect Monte Carlo approximation involving simulation from a crude approximation to the true density. Asymptotic normality of the sufficient statistics suggests an obvious candidate for an importance distribution.

The more general problem considers the computation of a joint probability for a subvector of random T, given its complementary subvector, when its distribution is intractable, but its joint m.g.f. is computable. For such settings, the distribution may be tilted, maintaining T as the sufficient statistic. Within this tilted family, the computation of such multivariate probabilities proceeds as described for the exponential family setting.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.     

Non-parametric predictive inference for future order statistics

This article presents non-parametric predictive inference for future order statistics. Given the data consisting of n real-valued observations, m future observations are considered and predictive probabilities are presented for the rth-ordered future observation. In addition, joint and conditional probabilities for events involving multiple future order statistics are presented. The article further presents the use of such predictive probabilities for order statistics in statistical inference, in particular considering pairwise and multiple comparisons based on two or more independent groups of data.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.