Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.     

Large sample prediction regions with estimated parameters

We obtain and asymptotic correction for the coverage probability of prediction regions when the parameter are estimated for dependent observations. Both stationary adn non-stationary type models are considered as applications. Asymptotic power of theses regions is also studied briefly.     

A note on stratification and gain in precision in estimating diversity from large samples

Several indices of entropy have been suggested in the literature as weighted diversity measures of a population with respect to a classification process. Among them, Shannon's entropy and Havrda -Charvát's non-additive entropies of order a, have been exhaustively used.

When the population is finite but too large to be censused, the diversity with respect to a given classification process must be estimated from a sample.

In this note, on the basis of an asymptotic study of the sample indices in the stratified random sampling, we are going to confirm that when we deal with large samples one can guarantee a gain in precision from stratified random over simple random sampling. This gain becomes considerable when the ‘inaccuracy" (as intended by Kerridge and Rathie and Kannapan) between the frequency vector in each stratum and that in the whole population, varies greatly from stratum to stratum.     

A nonparametric estimator of the number of classes based on a stratified random sample

Under stratified random sampling, we develop a k^th-order bootstrap bias-corrected estimator of the number of classes θ which exist in a study region. This research extends Smith and van Belle’s (1984) first-order bootstrap bias-corrected estimator under simple random sampling. Our estimator has applicability for many settings including: estimating the number of animals when there are stratified capture periods, estimating the number of species based on stratified random sampling of subunits (say, quadrats) from the region, and estimating the number of errors/defects in a product based on observations from two or more types of inspectors. When the differences between the strata are large, utilizing stratified random sampling and our estimator often results in superior performance versus the use of simple random sampling and its bootstrap or jackknife [Burnham and Overton (1978)] estimator. The superior performance is often associated with more observed classes, and we provide insights into optimal designation of the strata and optimal allocation of sample sectors to strata.     

Large sample behavior of the Bernstein copula estimator

Bernstein polynomial estimators have been used as smooth estimators for density functions and distribution functions. The idea of using them for copula estimation has been given in Sancetta and Satchell (2004). In the present paper we study the asymptotic properties of this estimator: almost sure consistency rates and asymptotic normality. We also obtain explicit expressions for the asymptotic bias and asymptotic variance and show the improvement of the asymptotic mean squared error compared to that of the classical empirical copula estimator. A small simulation study illustrates this superior behavior in small samples.     

Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution

This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. depending on the parameters and the probability of selecting a question. We showed that our proposed methods have better efficiencies than Land et al.’s randomized response model in some conditions. When the sizes of stratified populations are not given, other estimators are suggested using a stratified double sampling. For the proportional allocation, the difference between two variances in the stratified sampling and the stratified double sampling is given with the known rare unrelated attribute.     

Empirical process of long-range dependent sequences when parameters are estimated

In this paper, we study the asymptotic behavior of empirical processes when parameters are estimated, assuming that the underlying sequence of random variables is long-range dependent. We show completely different phenomena compared to i.i.d. situation, as well as compared to ordinary empirical processes of long-range dependent sequences. Applications include Kolmogorov–Smirnov and Cramer–Smirnov–von Mises goodness-of-fit statistics.     

Large sample results for tests of association ii. multinomial and stratified sampling

This paper deals with the asymptotics of a class of tests for association in 2-way contingency tables based on square forms in cell frequencies, given the total number of observations (multinomial sampling) or one set of marginal totals (stratified sampling). The case when both row and column marginal totals are fixed (hypergeometric sampling) was studied in Kulinskaya (1994), The class of tests under consideration includes a number of classical measures for association, Its two subclasses are the tests based on statistics using centralized cell frequencies (asymptotically distributed as weighted sums of central chi-squares) and those using the non-centralized cell frequencies (asymptotically normal). The parameters of asymptotic distributions depend on the sampling model and on true marginal probabilities. Maximum efficiency for asymptotically normal statistics is achieved under hypergeometric sampling, If the cell frequencies or the statistic as a whole are centralized using marginal proportions as estimates for marginal probabilities, the asymptotic distribution does not differ much between models and it is equivalent to that under hypergeometric sampling. These findings give an extra justification for the use of permutation tests for association (which are based on hypergeometric sampling). As an application, several well known measures of association are analysed.     

Sequential estimation of the sum of sinusoidal model parameters

Estimating the parameters of the sum of a sinusoidal model in presence of additive noise is a classical problem. It is well known to be a difficult problem when the two adjacent frequencies are not well separated or when the number of components is very large. In this paper we propose a simple sequential procedure to estimate the unknown frequencies and amplitudes of the sinusoidal signals. It is observed that if there are p components in the signal then at the k  th (k?p)$(k?p)$ stage our procedure produces strongly consistent estimators of the k   dominant sinusoids. For k>p$k>p$, the amplitude estimators converge to zero almost surely. Asymptotic distribution of the proposed estimators is also established and it is observed that it coincides with the asymptotic distribution of the least squares estimators. Numerical simulations are performed to observe the performance of the proposed estimators for different sample sizes and for different models. One ECG data and one synthesized data are analyzed for illustrative purpose.     

An application of ranked set sampling when observations from several distributions are to be included in the sample

ABSTRACT

In this article, we propose a method to estimate the common location and common scale parameters of several distributions using suitably defined ranked set sampling. Efficiency comparison of the obtained estimators with some of the standard estimators is made. Illustration of the results to real life data sets is also described.     

Maximum likelihood estimators of the location and scale parameters of the exponential distribution from a censored sample

In this note explicit expressions are given for the maximum likelihood estimators of the parameters of the two-parameter exponential distribution, when a doubly censored sample is available.     

Estimation of parameters under a random censorship model

The problem of estimation of parameters of a lifetime distribution is considered under the proportional hazards model of random censorship. Asymptotic variances of several estimators of survival function are compared in the eponential case.     

Testing the equality of correlation matrices when sample correlation matrices are dependent

We consider the situation in which a 2m×2m$2m\times 2m$ correlation matrix is naturally partitioned into a 2×2$2\times 2$ form, where each submatrix is m×m$m\times m$ and we wish to test for the equality of the two correlation matrices appearing on the diagonal of the partitioned form. Standard tests for the equality of correlation matrices do not apply since the corresponding sample correlation matrices are correlated. We obtain Wald statistics under the assumption of multivariate normality as well as extensions that apply to elliptical distributions.     

Asymptotic properties of the EPMC for modified linear discriminant analysis when sample size and dimension are both large

We deal with the problem of classifying a new observation vector into one of two known multivariate normal distributions when the dimension p and training sample size N   are both large with p<N$p<N$. Modified linear discriminant analysis (MLDA) was suggested by Xu et al. [10]. Error rate of MLDA is smaller than the one of LDA. However, if p and N   are moderately large, error rate of MLDA is close to the one of LDA. These results are conditional ones, so we should investigate whether they hold unconditionally. In this paper, we give two types of asymptotic approximations of expected probability of misclassification (EPMC) for MLDA as n→∞$n\to \infty$ with p=O(n^δ)$p=O(n^{\delta })$, 0<δ<1$0<\delta <1$. The one of two is the same as the asymptotic approximation of LDA, and the other is corrected version of the approximation. Simulation reveals that the modified version of approximation has good accuracy for the case in which p and N are moderately large.     

An efficient and fast algorithm for estimating the parameters of two-dimensional sinusoidal signals

In this paper we propose a computationally efficient algorithm to estimate the parameters of a 2-D sinusoidal model in the presence of stationary noise. The estimators obtained by the proposed algorithm are consistent and asymptotically equivalent to the least squares estimators. Monte Carlo simulations are performed for different sample sizes and it is observed that the performances of the proposed method are quite satisfactory and they are equivalent to the least squares estimators. The main advantage of the proposed method is that the estimators can be obtained using only finite number of iterations. In fact it is shown that starting from the average of periodogram estimators, the proposed algorithm converges in three steps only. One synthesized texture data and one original texture data have been analyzed using the proposed algorithm for illustrative purpose.     

An asymptotic expansion of Wishart distribution when the population eigenvalues are infinitely dispersed

Takemura and Sheena [A. Takemura, Y. Sheena, Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix, J. Multivariate Anal. 94 (2005) 271–299] derived the asymptotic joint distribution of the eigenvalues and the eigenvectors of a Wishart matrix when the population eigenvalues become infinitely dispersed. They also showed necessary conditions for an estimator of the population covariance matrix to be tail minimax for typical loss functions by calculating the asymptotic risk of the estimator. In this paper, we further examine those distributions and risks by means of an asymptotic expansion. We obtain the asymptotic expansion of the distribution function of relevant elements of the sample eigenvalues and eigenvectors. We also derive the asymptotic expansion of the risk function of a scale and orthogonally equivariant estimator with respect to Stein’s loss. As an application, we prove non-minimaxity of Stein’s and Haff’s estimators, which has been an open problem for a long time.     

Estimation of a rare sensitive attribute in a stratified two-stage unrelated randomized response model by using Poisson probability distribution

The present article deals with the estimation of mean number of respondents who possess a rare sensitive character in presence of known and unknown proportion of a rare unrelated non-sensitive attribute by using the Poisson probability distribution in stratified random sampling as well as in stratified random double sampling. The variance of rare sensitive character is also derived under proportional and optimal allocation methods in stratified random sampling when stratum sizes are known and unknown. The properties of the suggested estimation procedures have been deeply examined. The proposed model is found to be dominant over Lee et al. [Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution. Statistics. 2013;47:575–589] model. Numerical illustrations are presented to support the theoretical results. Results are analysed and suitable recommendations are put forward to the survey practitioners.     

A Test for the Equality of Means of Two Groups with Different Variances When the Sample Size and the Dimension are Large

We consider a class of test statistics including the Dempster trace criterion in the case of two groups without assuming equal covariance matrices. The test statistics in the class are valid when the dimension is larger than the sample size. We obtain asymptotic distributions of the test statistics in the class and use these distributions to derive the limiting power in each case. We obtain the most powerful test in the class with respect to this limiting power.     

A one-sample location test based on weighted averaging of two test statistics when the dimension and the sample size are large

We discuss a one-sample location test that can be used when the dimension and the sample size are large. It is well-known that the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non exact approaches were proposed, e.g., Dempster (1958 Dempster, A.P. (1958). A high dimensional two sample significance test. Ann. Math. Stat. 29:995–1010.[Crossref] , [Google Scholar], 1960 Dempster, A.P. (1960). A significance test for the separation of two highly multivariate small samples. Biometrics 16:41–50.[Crossref], [Web of Science ®] , [Google Scholar]), Bai and Saranadasa (1996 Bai, Z.D., Saranadasa, H. (1996). Effect of high dimension: by an example of a two sample problem. Stat. Sin. 6:311–329.[Web of Science ®] , [Google Scholar]), and Srivastava and Du (2008 Srivastava, M.S., Du, M. (2008). A test for the mean vector with fewer observations than the dimension. J. Multivariate Anal. 99:386–402.[Crossref], [Web of Science ®] , [Google Scholar]). In this article, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T²-statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. Numerical results show that our test is more stable than Hotelling’s T²-statistic and Dempster’s statistic in most parameter settings.     

Estimation of the parameters of Downton's bivariate exponential distribution using ranked set sampling scheme

The parameters of Downton's bivariate exponential distribution are estimated based on a ranked set sample. Parametric and nonparametric methods are considered. The suggested estimators are compared to the corresponding ones based on simple random sampling. It turns out that some of the suggested estimators are significantly more efficient than the ones based on simple random sampling.