Optimal Estimation of a Finite Population Total Under Randomized Response Surveys

Problems of estimation of a finite population total of a variable of sensitive in nature are studied under randomized response (RR) surveys. Some optimal sampling strategies are presented under different superpopulation models.     

Variance Estimation Based on Invariance Principles

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.     

A General Approach To Nonparametric Histogram Estimation

We note that some classical functional estimation problems may be reduced to a general unique framework and study an estimator within this general framework that reduces to the classical histogram type estimators in various examples presented. The convergence in probability and the almost complete convergence of this general estimator are studied obtaining convergence conditions which reduce to the classical conditions in each case. Finally, this general framework provides conditions for the convergence of the finite dimensional distributions of the associated empirical process.     

Selecting the Best Exponential Population When the Scale Parameters are Bounded

Consider k (≥ 2) independent exponential populations with different location and scale parameters. Call a population associated with largest of unknown location parameters as the best population. For the goal of selecting the best population, it is established that if the scale parameters are completely unknown, then the indifference-zone probability requirement can not be guaranteed by any single sample decision rule which is just and translation invariant. Under the assumption that the scale parameters are bounded above by a known constant, a single sample selection procedure is proposed for which the indifference-zone probability requirement can be guaranteed. Under the same assumption, 100P*% simultaneous upper confidence intervals for all distances from the largest location parameter are also obtained.     

A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

A Note on Dirichlet One-Armed Bandits

One of the two independent stochastic processes (or ‘arms’) is selected and observed sequentially at each of n(≤ ∝) stages. Arm 1 yields observations identically distributed with unknown probability measure P with a Dirichlet process prior whereas observations from arm 2 have known probability measure Q. Future observations are discounted and at stage m, the payoff is a _m(≥0) times the observation Z _m at that stage. The objective is to maximize the total expected payoff. Clayton and Berry (1985) consider this problem when a _m equals 1 for m ≤ n and 0 for m > n(< ∝) In this paper, the Clayton and Berry (1985) results are extended to the case of regular discount sequences of horizon n, which may also be infinite. The results are illustrated with numerical examples. In case of geometric discounting, the results apply to a bandit with many independent unknown Dirichlet arms.     

A Note on the Kuks-Olman Estimator

In the linear regression model the unknown parameter vector θ is supposed to vary in a known ellipsoid. Under this parameter constraint Kuks and Olman derived an estimator by demanding a minimax property. Since sometimes the Kuks-Olman estimator takes values outside of the ellipsoid a modification is proposed in the paper. It is shown that this modified variant is a least squares estimator in the restricted model.     

Consistency of the least-squares-estimators for linear stochastic difference equations

Assumptions are given for the strong consistency in the stable case and weak consistency in the instable case of the Least-Square-Estimator of the unknown system-parameters of a inhomogeneous linear stochastic difference equation system with constant coefficients.     

Changepoint Problems Under Random Censorship

Let X ₁, …, X _[nθ], X _{[nθ] + 1}, …, X _n be a sequence of independent random variables (the “lifetimes”) such that X _j ? F ₁ for 1 ≤ j ≤ [nθ] and X _j ? F ₂ for [nθ] + 1 ≤ j ≤ n, with F ₁ ≠ F ₂ unknown. In this paper we investigate an estimator θ _n for the changepoint θ if the X's are subject to censoring. The rate of almost sure convergence of θ _n to θ is established and a test for the hypothesis θ = 0, i.e. “no change”, is proposed.     

Characterization of minimax linear estimators in linear regression

Two characterization theorems of the minimax linear estimator (Mile) are proven for the case, where the regression parameter varies only in an arbitrary ellipsoid. Furthermore, the existence, uniqueness and admissibility of Mile are shown. The explicit determination of Mile is carried out for a special case.     

A Test to Detect Inadequacy of Satterthwaite's Approximation in Balanced Mixed Models

Satterthwaite's approximation of the distribution of a nonnegative linear combination of independent mean squares is addressed in this article. A necessary and sufficient condition for the approximation to be exact is presented for the case of a general balanced mixed model. A test is subsequently developed for detecting any significant departure from this condition using the data under consideration. An example is given to illustrate the proposed methodology.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

The Harris Extended Exponential Distribution

A new lifetime distribution is proposed and studied. The Harris extended exponential is obtained from a mixture of the exponential and Harris distributions, which arises from a branching process. Several structural properties of the new distribution are discussed, including moments, generating function and order statistics. The new distribution can model data with increasing or decreasing failure rate. The shape of the hazard rate function is controlled by one of the added parameters in an uncomplicated manner. An application to a real dataset illustrates the usefulness of the new distribution.     

The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 ? p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 ? p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H ₀:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H ₀. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.     

Second-Order Approximations to Two Classes of Sequential Estimation Procedures

For the model considered by Chaturvedi, Pandey and Gupta (1991), two classes of sequential procedures are developed to construct confidence regions (which may be interval, ellipsoidal or spherical) of ‘pre-assigned width and coverage probability’ for the parameters of interest and for the minimum risk point estimation (taking loss to be quadratic plus linear cost of sampling) of the nuisance parameter. Second-Order approximations are derived for the expected sample size, coverage probability and ‘regret’ associated with the two classes of sequential procedures. A simple and direct method of obtaining the asymptotic distribution of the stopping time is provided. By means of examples, it is illustrated that several estimation problems can be tackled with the help of proposed classes of sequential procedures.     

Distance Mean and Variance Functions and Sphere Fitting

Motivated by circle fitting to spatial distributions, this paper looks at the relationship between the expected distance function and the radial density. Inversion of this relationship is given simply in dimensions 1 and 3 and generally by using Mellin transforms. Examples of variance function behaviour are given and calculations of the coefficient of sphericity shown.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.