Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

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.     

Teacher shortages, teacher contracts and their effect on education in Africa

Summary.  To enhance primary enrolment rates, many African countries have launched large teacher recruitment programmes in recent years. Given tight budgetary constraints, teachers are no longer employed in civil service positions, but on the basis of (fixed term) contracts typically implying considerably lower salaries and a sharply reduced amount of professional training. We analyse the effect of this change on educational quality in Niger, Togo and Mali, on the basis of very informative data, which are comparable across these countries. We use a variety of estimation techniques, including a non-parametric estimation of quantile treatment effects. Our results demonstrate that contract teachers tend to reduce inequalities in student outcomes. Overall, the effects are positive in Mali, somewhat mixed in Togo and negative in Niger. This ordering is consistent with theoretical expectations related to the manner in which contract teacher programmes were implemented differently in each of the three countries under study.     

Nonparametric regression estimation in a null recurrent time series

We derive an asymptotic theory of nonparametric estimation for a time series regression model Z_t=f(X_t)+W_t, where {X_t} and {Z_t} are observed nonstationary processes, and {W_t} is an unobserved stationary process. The class of nonstationary processes allowed for {X_t} is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process {W_t} is assumed to be linear and stationary.     

Notes on odds ratio estimation for a randomized clinical trial with noncompliance and missing outcomes

The odds ratio (OR) has been recommended elsewhere to measure the relative treatment efficacy in a randomized clinical trial (RCT), because it possesses a few desirable statistical properties. In practice, it is not uncommon to come across an RCT in which there are patients who do not comply with their assigned treatments and patients whose outcomes are missing. Under the compound exclusion restriction, latent ignorable and monotonicity assumptions, we derive the maximum likelihood estimator (MLE) of the OR and apply Monte Carlo simulation to compare its performance with those of the other two commonly used estimators for missing completely at random (MCAR) and for the intention-to-treat (ITT) analysis based on patients with known outcomes, respectively. We note that both estimators for MCAR and the ITT analysis may produce a misleading inference of the OR even when the relative treatment effect is equal. We further derive three asymptotic interval estimators for the OR, including the interval estimator using Wald’s statistic, the interval estimator using the logarithmic transformation, and the interval estimator using an ad hoc procedure of combining the above two interval estimators. On the basis of a Monte Carlo simulation, we evaluate the finite-sample performance of these interval estimators in a variety of situations. Finally, we use the data taken from a randomized encouragement design studying the effect of flu shots on the flu-related hospitalization rate to illustrate the use of the MLE and the asymptotic interval estimators for the OR developed here.     

Nonparametric estimation of the lower tail dependence λL in bivariate copulas

The lower tail dependence λ_L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ_L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ_L. They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ_L.     

Estimation of the intercept parameter for linear regression model with uncertain non-sample prior information

This paper considers alternative estimators of the intercept parameter of the linear regression model with normal error when uncertain non-sample prior information about the value of the slope parameter is available. The maximum likelihood, restricted, preliminary test and shrinkage estimators are considered. Based on their quadratic biases and mean square errors the relative performances of the estimators are investigated. Both analytical and graphical comparisons are explored. None of the estimators is found to be uniformly dominating the others. However, if the non-sample prior information regarding the value of the slope is not too far from its true value, the shrinkage estimator of the intercept parameter dominates the rest of the estimators.     

Divergence-based estimation and testing with misclassified data

The well-known chi-squared goodness-of-fit test for a multinomial distribution is generally biased when the observations are subject to misclassification. In Pardo and Zografos (2000) the problem was considered using a double sampling scheme and ø-divergence test statistics. A new problem appears if the null hypothesis is not simple because it is necessary to give estimators for the unknown parameters. In this paper the minimum ø-divergence estimators are considered and some of their properties are established. The proposed ø-divergence test statistics are obtained by calculating ø-divergences between probability density functions and by replacing parameters by their minimum ø-divergence estimators in the derived expressions. Asymptotic distributions of the new test statistics are also obtained. The testing procedure is illustrated with an example.     

Assessing accuracy of a continuous screening test in the presence of verification bias

Summary.  In studies to assess the accuracy of a screening test, often definitive disease assessment is too invasive or expensive to be ascertained on all the study subjects. Although it may be more ethical or cost effective to ascertain the true disease status with a higher rate in study subjects where the screening test or additional information is suggestive of disease, estimates of accuracy can be biased in a study with such a design. This bias is known as verification bias. Verification bias correction methods that accommodate screening tests with binary or ordinal responses have been developed; however, no verification bias correction methods exist for tests with continuous results. We propose and compare imputation and reweighting bias-corrected estimators of true and false positive rates, receiver operating characteristic curves and area under the receiver operating characteristic curve for continuous tests. Distribution theory and simulation studies are used to compare the proposed estimators with respect to bias, relative efficiency and robustness to model misspecification. The bias correction estimators proposed are applied to data from a study of screening tests for neonatal hearing loss.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000