On estimating the reliability after sequentially estimating the mean: The exponential case

Sequential estimation problems for the mean parameter of an exponential distribution has received much attention over the years. Purely sequential and accelerated sequential estimators and their asymptotic second-order characteristics have been laid out in the existing literature, both for minimum risk point as well as bounded length confidence interval estimation of the mean parameter. Having obtained a data set from such sequentially designed experiments, the paper investigates estimation problems for the associatedreliability function. Second-order approximations are provided for the bias and mean squared error of the proposed estimator of the reliability function, first under a general setup. An ad hoc bias-corrected version is also introduced. Then, the proposed estimator is investigated further under some specific sequential sampling strategies, already available in the literature. In the end, simulation results are presented for comparing the proposed estimators of the reliability function for moderate sample sizes and various sequential sampling strategies.     

On the existence of moments of partially restricted reduced form coefficients

In this paper ridgelike Bayesian estimators of structural coefficients have been used to form the partially restricted reduced form estimators. These partially restricted reduced form estimators are simple in form and possess finite sampling moments and risk in contrast to other restricted reduced form estimators that possess no finite moments and have infinite risk relative to quadratic loss functions. The usual k-class implied partially restricted reduced form estimators with 0≦k≦1 do not posses finite moments unless the degree of overidentification (or the excess of sample size over the number of coefficients) of the structural equation being estimated is suitably restricted.     

Almost unbiased product-type estimator

In sample survey methods the use of product estimators was suggested byMurthy [1964] andSrivastava [1966] and were found to serve good purpose provided the two variables viz. the main variable under study and the auxiliary variable have a very high negative correlation between them. The product estimators suggested by them are biased. In the present paper the author has obtained unbiased product estimators (to the first degree of approximation) with the help of the technique developed byQuenouille [1956] and has established that this new estimator is better than the other product estimator in the mean square error sense.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

A true simulation study of three estimators at equal protection of respondents in randomized response sampling

In this study, three different estimators for estimating the proportion of a sensitive attribute in survey sampling are compared at equal protection of the respondents. The three estimators considered are due to Odumade and Singh (2009, Comm. Statist. Theory Methods) , Singh and Sedory (2011, Sociological Methods and Research) and a new estimator obtained by minimizing a chi‐squared distance. A SAS Macro is developed to compare these three estimators using a simulation study at equal protection of the respondents. A set of data from a real face‐to‐face interview was collected using two decks of cards and has been analyzed. The results are discussed.     

On the variance of the ratio estimator for midzuno-sen sampling scheme

Summary For the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. An expression for the same is derived in this note.     

Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under Type-I and Type-II censoring

In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum likelihood estimators [^(q)]₁{\hat{\theta}_{1}} and [^(q)]₂{\hat{\theta}_{2}} of the parameters θ ₁ and θ ₂ have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively. Here, we prove that these estimators are stochastically monotone with respect to θ ₁ and θ ₂, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters θ ₁ and θ ₂. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions under truncation, which is also of independent interest.     

The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.     

Minimum norm quadratic estimators of variance components

In this note we consider the classes of quadratic estimators ofLamotte [1973] for estimating the variance components and derive the forms of the minimum norm quadratic estimators in the classes of quadratics not considered byC.R. Rao [1971a, 1972].     

Ratio cum product method of estimation

Summary In this paper methods of estimation which may be considered as combination of ratio and product methods have been suggested. The mean square errors of these estimators utilizing two supplementary variables are compared with (i) simple unbiased estimator (p=0), (ii) usual ratio and product methods of estimation (p=1) and (iii) multivariate ratio and multivariate product estimators (p=2), wherep is the number of supplementary variables utilized. Conditions for their efficient use have been obtained for each case. Extension to general case ofp-variables has been briefly discussed. A new criteria for the efficient use of product estimator have been obtained.     

Relative efficiencies of some simple Bayes estimators of coefficients in a dynamic equation with serially correlated errors - II

Generalized least squares estimators, with estimated variance-covariance matrices, and maximum likelihood estimators have been proposed in the literature to deal with the problem of estimating autoregressive models with autocorrelated disturbances. In this paper we compare the small sample efficiencies of these estimators with those of some approximate Bayes estimators. The comparison is done with the help of a sampling experiment applied to a model specification. Though these Bayes estimators utilize very weak prior information, they out-perform the sampling theory estimators in every case we consider.     

Optimal instruments when the disturbances are small

Single-equation instrumental variable estimators (e.g., the k-class) are frequently employed to estimate econometric equations. This paper employs Kadane's (1971) small-σ method and a squared-error matrix loss function to characterize a single-equation class of optimal instruments, $\text{A}$. $\text{A}$ is optimal (asymptotically for a small scalar multiple, σ, of the model's disturbance) in that all of its members are preferred to all non-members. From this characterization it is shown all k-class estimators and certain iterative estimators belong to $\text{A}$. However, non-iterative principal component estimators [e.g., Kloek and Mennes (1960)] are unlikely to belong to $\text{A}$. These latter instrumental variable estimators have been advocated [see Amemiya (1966) and Kloek and Mennes (1960)] for estimating ‘large’ econometric models.     

An efficient estimator using auxiliary information

The use of ratio and product methods of estimation using auxiliary information for estimating the mean of a finite population is well known.Srivastava [1967] andReddy [1973] proposed ratio-cum-product type estimators. This paper proposes a transformed estimator which is even more efficient than these estimators for a wide range of the value of the correlation coefficient between the main and auxiliary variables.     

On interpenetrating samples of unequal sizes

Summary Koop [1967] proved that interpenetrating samples of unequal sizes are more efficient than those with equal sizes for estimating a finite population total. After observing that there is a serious lacuna present in his proof, a correct proof has been suggested. The optimum choice of individual sample sizes has also been discussed for a given (i) total sample size, (ii) cost and (iii) precision, with an assumed cost structure. Finally, the resulting estimators have been compared with those based on a single sample.     

INVARIANCE OF THE UNORDERED BEST ESTIMATOR IN THE T,–CLASS FOR MIDZUNO'S AND FOR IKEDA–SEN'S SAMPLING PROCEDURES

MIDZUNO'S sampling procedure is considered where the first (n – 1) draws are carried out with simple random sampling without replacement and the nth draw with varying probabilities. It is shown that for this scheme, the best estimator in the HORVITZ–THOMPSON (1952) T_t–class of linear estimators exists and rejects the last draw. When MURTHY'S technique of unordering of an ordered estimator is employed, the rejected draw is restored and the unordered estimator is obtained. Surprisingly, this unordered estimator is the same as the unordered best estimator in the T₁–class, derived for IKEDA–SEN'S sampling procedure.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

Some results concerningHorvitz-Thompson's T 1-class of estimators

Summary The question of availability of a unique best estimator inHorvitz-Thompson's T ₁-class of estimators is dealt with in general and a simple sampling scheme yielding such an estimator is found out; the efficiency of the same is studied and a method of improving on it is also discussed.     

Optimum allocation in Bayesian stratified two-phase sampling when there arem-attributes

The problem of optimally allocating a sample amongk-strata at the second phase of a two-phase sampling procedure when the sampling is for proportion has been discussed byNewbold [1971] and then by the author [Khan, 1972].This paper deals with optimum allocation of the sample amongk-strata at the second phase of a two-phase sampling procedure, when the sampling is form-attributes. The problem is to estimate the proportion of each attribute in the population. Here (m–1) dimensional Dirichlet distribution is taken as the prior distribution.     

Unbiased product estimators

Summary The paper presents a comparative study of product estimators proposed byRobson [1957] andMurthy [1964]. It is seen that the Robson's estimator gives a better performance.     

A Note on the Theme of Too Many Instruments*

The difference and system generalized method of moments (GMM) estimators are growing in popularity. As implemented in popular software, the estimators easily generate instruments that are numerous and, in system GMM, potentially suspect. A large instrument collection overfits endogenous variables even as it weakens the Hansen test of the instruments’ joint validity. This paper reviews the evidence on the effects of instrument proliferation, and describes and simulates simple ways to control it. It illustrates the dangers by replicating Forbes [American Economic Review (2000) Vol. 90, pp. 869–887] on income inequality and Levine et al. [Journal of Monetary Economics] (2000) Vol. 46, pp. 31–77] on financial sector development. Results in both papers appear driven by previously undetected endogeneity.