On a generalization of Poisson sampling

In real-time sampling, the units of a population pass a sampler one by one. Alternatively the sampler may successively visit the units of the population. Each unit passes only once and at that time it is decided whether or not it should be included in the sample. The goal is to take a sample and efficiently estimate a population parameter. The list sequential sampling method presented here is called correlated Poisson sampling. The method is an alternative to Poisson sampling, where the units are sampled independently with given inclusion probabilities. Correlated Poisson sampling uses weights to create correlations between the inclusion indicators. In that way it is possible to reduce the variation of the sample size and to make the samples more evenly spread over the population. Simulation shows that correlated Poisson sampling improves the efficiency in many cases.     

On a further generalization of the savage test

A statistic based on the frequencies within the k+1 intervals specified by k arbitrary quantiles is proposed for a LMP test against Lehmann alternatives generalizing the Savage test for the two-sample problem. The maximum efficiency relative to the Savage test for optimally chosen k quantiles is also provided for k=l(2)l5. The asymptotic normality of the statistic follows from the asymptotic multinomial distribution of the frequencies in the classes determined by the k quantiles.     

On a recent generalization of gamma distribution

In this paper, we investigate a generalized gamma distribution recentIy developed by Agarwal and Kalla (1996). Also, we show that such generalized distribution, like the ordinary gamma distribution, has a unique mode and, unlike the ordinary gamma distribution, may have a hazard rate (mean residual life) function which is upside-down bathtub (bathtub) shaped.     

On the moments of a distribution defined by the Gaussian polynomials

An alternative method is presented to compute the moments of the probability distribution defined by the Gaussian polynomials. It computes the cumulants first.     

On a generalization of the test of endogeneity in a two stage least squares estimation

In situations that the predictors are correlated with the error term, we propose a bridge estimator in the two-stage least squares estimation. We apply this estimator to overcome the multicollinearity and sparsity of the explanatory variables, when the endogeneity problem is present.The proposed estimator was applied to modify the Durbin-Wu-Hausman (DWH) test of endogeneity in the presence of multicollinearity. To compare our modified test with the existing DWH for detection of an endogenous problem in multi-collinear data, some numerical assessments are carried out. The numerical results showed that the proposed estimators and the suggested test perform better for the multi-collinear data. Finally, a genetical data set is applied for illustration the our results by estimating the coefficients parameters in the presence of endogeneity and multicollinearity.     

On Charlier polynomials in testing Poissonity

This article discusses applications of Charlier polynomials when checking how well classic Poisson model fits the data. We show that smooth components pertaining to the polynomials can be effectively used in preliminary analysis of data at hand and we propose and discuss some tests based on them. Our extensive simulations shed new light on power behavior of the best tests in the area.     

On a problem in two-stage sampling

Beattie 1962 proposed an alternate procedure for monitoring continuous processes which, unlike existing continuous samplingplans under Mil-Std-1235B, does not require periods of 100% inspection. The procedure, which is a composite of two Cumulative (Cusum) procedures, will be analyzed in terms of Cusum average run length (ARL) performance. An algorithm is proposed for generating sample plans based upon specifications on ARL performance in each of the Cusum sampling zones and the overall sampling risks of the procedure. The algorithm is fully described, and implemented using exact expressions for the Cusum ARL. An illustrated example is provided, describing how a continuous sampling maybe selected which matches performances about the AQL for a related plan in ISO 3951.     

On the generalization of the hazard rate twisting-based simulation approach

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of the proposed method with some existing techniques.     

On a search problem in multidimensional grids

A search problem of G. O. H. Katona was solved earlier by the first author (Ruszinko, J. Statist. Plann. Inference 37 (1993), 371–383) where an unknown point x in a 2-dimensional grid has to be located using queries of type “is x = (x₁,x₂) ⩽ a = (a₁,a₂)?”. Here a is an arbitrary lattice point and x ⩽ a means that x_i ⩽ a_i (i = 1,2). In the present paper we consider the generalization of this problem for arbitrary dimension d.     

On a bayesian approach for the shiftpoint problem

Using a direct resampling process, a Bayesian approach is developed for the analysis of the shiftpoint problem. In many problems it is straight forward to isolate the marginal posterior distribution of the shift-point parameter and the conditional distribution of some of the parameters given the shift point and the other remaining parameters. When this is possible, a direct sampling approach is easily implemented whereby standard random number generators can be used to generate samples from the joint posterior distribution of aii the parameters in the model. This technique is illustrated with examples involving one shift for Poisson processes and regression models.     

On independence of Markov kernels and a generalization of two theorems of Basu

Markov kernels play an important role in probability theory and mathematical statistics, conditional distributions being the main example.     

On the effect of misspecifying the error density in a deconvolution problem

The author studies the effect of a misspecification of the error density on the mean integrated squared error (MISE) of the deconvolution estimator. He shows that the MISE converges to a certain functional which he defines. He also illustrates the fact that the limit can sometimes be infinite. Finally, he derives some guidelines for selecting the error density in order to ensure robustness properties of the procedure.     

On the minimaxiy of the maximum likelihood estimator in a multivariate problem

This study looks at the minimaxity of the maximum likelihood estimator (m.1.e), of the mean of a p-normal population, that has been given by Dahel, Giri and Lepage (1985). This estimator is computed on the basis of three independent samples: the first one is drawn from the whole vector of dimension p and the two others are based on the first p₁ and the last p₂ components respectively, such as p₁ +p₂=p.     

On a multiple endpoints testing problem

In a clinical trial comparing drug with placebo, where there are multiple primary endpoints, we consider testing problems where an efficacious drug effect can be claimed only if statistical significance is demonstrated at the nominal level for all endpoints. Under the assumption that the data are multivariate normal, the multiple endpoint-testing problem is formulated. The usual testing procedure involves testing each endpoint separately at the same significance level using two-sample t-tests, and claiming drug efficacy only if each t-statistic is significant. In this paper we investigate properties of this procedure. We show that it is identical to both an intersection union test and the likelihood ratio test. A simple expression for the p-value is given. The level and power function are studied; it is shown that the test may be conservative and that it is biased. Computable bounds for the power function are established.     

On a problem of counting by weighing

Suppose it is desired to obtain a large number N_s of items for which individual counting is impractical, but one can demand a batch to weigh at least w units so that the number of items N in the batch may be close to the desired number N_s. If the items have mean weight ωTH, it is reasonable to have w equal to ωTHN_s when ωTH is known. When ωTH is unknown, one can take a sample of size n, not bigger than N_s, estimate ωTH by a good estimator ω_n, and set w equal to ω_nN_s. Let R_n = Kp²N²_s/n + K_sn be a measure of loss, where K_e and K_s are the coefficients representing the cost of the error in estimation and the cost of the sampling respectively, and p is the coefficient of variation for the weight of the items. If one determines the sample size to be the integer closest to pCN_s when p is known, where C is (K_e/K_s)^1/2, then R_n will be minimized. If p is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given ω and ω has a prior inverted gamma distribution, the optimal sample size can be found to be the nonnegative integer closest to pCN_s + p²A(pC – 1), where A is a known constant given in the prior distribution.     

On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution

A modification of the beta-correlated binomial (BCB) distribution of Paul (1985) is proposed. This modification over- comes some theoretical difficulties encountered in the original BCB distribution.