KERNEL DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

The performance of kernel density estimation, in terms of mean integrated squared error, is investigated in the opposite of the usual situation, namely when the bandwidth is large. This affords noteworthy insights including the special role taken by the normal density function as kernel and a tie-in with ‘semiparametric’ approaches to density estimation.     

ESTIMATION OF A DENSITY FUNCTION USING ORDER STATISTICS1

A non-parametric estimator of a density at a particular quantile is based on sample quantiles. The optimal (to minimize M.S.E.) choice of these quantiles is considered and a method of removing the bias is suggested.     

CORRECTING FOR KURTOSIS IN DENSITY ESTIMATION

Using a global window width kernel estimator to estimate an approximately symmetric probability density with high kurtosis usually leads to poor estimation because good estimation of the peak of the distribution leads to unsatisfactory estimation of the tails and vice versa. The technique proposed corrects for kurtosis via a transformation of the data before using a global window width kernel estimator. The transformation depends on a “generalised smoothing parameter” consisting of two real-valued parameters and a window width parameter which can be selected either by a simple graphical method or, for a completely data-driven implementation, by minimising an estimate of mean integrated squared error. Examples of real and simulated data demonstrate the effectiveness of this approach, which appears suitable for a wide range of symmetric, unimodal densities. Its performance is similar to ordinary kernel estimation in situations where the latter is effective, e.g. Gaussian densities. For densities like the Cauchy where ordinary kernel estimation is not satisfactory, our methodology offers a substantial improvement.     

ON SOME OPTIMALITY PROPERTIES OF FISHER-RAO SCORE FUNCTION IN TESTING AND ESTIMATION

The score function is associated with some optimality features in statistical inference. This review article looks on the central role of the score in testing and estimation. The maximization of the power in testing and the quest for efficiency in estimation lead to score as a guiding principle. In hypothesis testing, the locally most powerful test statistic is the score test or a transformation of it. In estimation, the optimal estimating function is the score. The same link can be made in the case of nuisance parameters: the optimal test function should have maximum correlation with the score of the parameter of primary interest. We complement this result by showing that the same criterion should be satisfied in the estimation problem as well.     

TESTING THE EQUALITY OF PARAMETERS IN THE DISTRIBUTIONS WHEN THE RANGE DEPENDS UPON THE PARAMETER

The authors derive the null and non-null distributions of the test statistic v=y_min/y_max (where y_min= min x_ij, y_max= max x_ij, J=1,2, …, k) connected with testing the equality of scale parameters θ₁, θ₂, …θ_k in certain, class of density functions given by      

THE POTENTIAL FOR BIAS IN PRINCIPAL CAUSAL EFFECT ESTIMATION WHEN TREATMENT RECEIVED DEPENDS ON A KEY COVARIATE

Motivated by a potential-outcomes perspective, the idea of principal stratification has been widely recognized for its relevance in settings susceptible to posttreatment selection bias such as randomized clinical trials where treatment received can differ from treatment assigned. In one such setting, we address subtleties involved in inference for causal effects when using a key covariate to predict membership in latent principal strata. We show that when treatment received can differ from treatment assigned in both study arms, incorporating a stratum-predictive covariate can make estimates of the "complier average causal effect" (CACE) derive from observations in the two treatment arms with different covariate distributions. Adopting a Bayesian perspective and using Markov chain Monte Carlo for computation, we develop posterior checks that characterize the extent to which incorporating the pretreatment covariate endangers estimation of the CACE. We apply the method to analyze a clinical trial comparing two treatments for jaw fractures in which the study protocol allowed surgeons to overrule both possible randomized treatment assignments based on their clinical judgment and the data contained a key covariate (injury severity) predictive of treatment received.     

DENSITY ESTIMATION FOR CLUSTERED DATA

The commonly used survey technique of clustering introduces dependence into sample data. Such data is frequently used in economic analysis, though the dependence induced by the sample structure of the data is often ignored. In this paper, the effect of clustering on the non-parametric, kernel estimate of the density, f(x), is examined. The window width commonly used for density estimation for the case of i.i.d. data is shown to no longer be optimal. A new optimal bandwidth using a higher-order kernel is proposed and is shown to give a smaller integrated mean squared error than two window widths which are widely used for the case of i.i.d. data. Several illustrations from simulation are provided.     

A HEURISTIC TECHNIQUE FOR TESTING FOR THE PRESENCE OF INTERACTION IN NONREPLICATED FACTORIAL EXPERIMENTS

Abstract  This paper suggests univariate and multivariate techniques for investigating interaction in nonreplicated factorial experiments. The tests can detect certain types of interaction, but they are not powerful against all possible alternative hypotheses. The two-way factorial experiment is discussed in some detail and an example is used to demonstrate the procedure. The procedure is compared to other tests for interaction. These comparisons show that the procedure can detect some types of interaction which other tests cannot. Likewise other tests can detect interaction this procedure cannot.     

ON LIKELIHOOD ESTIMATION FOR DISCRETELY OBSERVED MARKOV JUMP PROCESSES

The parameter estimation problem for a Markov jump process sampled at equidistant time points is considered here. Unlike the diffusion case where a closed form of the likelihood function is usually unavailable, here an explicit expansion of the likelihood function of the sampled chain is provided. Under suitable ergodicity conditions on the jump process, the consistency and the asymptotic normality of the likelihood estimator are established as the observation period tends to infinity. Simulation experiments are conducted to demonstrate the computational facility of the method.     

ESTIMATION OF THE PARAMETER OF A POISSON DISTRIBUTION USING A LINEX LOSS FUNCTION

This paper considers estimation of the parameter of a Poisson distribution using Varian's (1975) asymmetric LINEX loss function L (δ) = b{exp(aδ) - aδ - 1}, where δ is the estimation error and b > 0, a 0. It is shown that for a < 0, the sample mean X¯ is admissible whereas for a > 0, X¯ is dominated by c*X¯, where c*= (n/a)log(1+a/n). Practical implications of this result are indicated. More general results, concerning the admissibility of estimators of the form cX¯+ d are also presented.     

ESTIMATION OF THE PARAMETERS OF A NORMAL DISTRIBUTION WHEN THE MEAN IS RESTRICTED TO AN INTERVAL1

Suppose it is known that the mean of a normal distribution is non-negative. Naturally one will use the sample mean truncated at zero as an estimator of the distribution mean. In this paper the properties of such an estimator are investigated.     

ESTIMATION OF A DISTRIBUTION FUNCTION DOMINATING STOCHASTICALLY A KNOWN DISTRIBUTION FUNCTION

This paper considers the problem of estimating a cumulative distribution function (cdf), when it is known a priori to dominate a known cdf. The estimator considered is obtained by adjusting the empirical cdf using the prior information. This adjusted estimator is shown to be consistent, its limiting distribution is found, and its mean squared error (MSE) is shown to be smaller than the MSE of the empirical cdf. Its asymptotic efficiency (compared to the empirical cdf) is also found.     

ESTIMATION OF THE NUMBER OF PEOPLE IN A DEMONSTRATION

It is of interest to estimate the size of a crowd in a demonstration. We propose a practical method to obtain an estimate of the size of the crowd and its standard error. This method has been implemented in practice and, compared with other counting methods, is found to be more efficient, more timely and have less scope for bias. The method described in this paper was motivated by the annual 1 July demonstrations in Hong Kong, and data from the 2006 demonstration are used as an example of the proposed method.     

ON THE ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM CENSORSHIP

ABSTRACT

The paper deals with an improvement of the well-known Kaplan–Meier estimator of survival function when the censoring mechanism is random and independent of the failure times. Small sample size properties of the new estimator, as well as the original Kaplan–Meier estimator are inspected by means of Monte Carlo simulations. It follows from the simulations that the proposed estimator prevails with respect to some basic statistical characteristics.     

ESTIMATION AND TESTING FOR PARTIALLY LINEAR SINGLE-INDEX MODELS

In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.     

A PROCEDURE FOR TESTING THE MEAN OF A NORMAL DISTRIBUTION1

A sequential procedure for comparing a null hypothesis concerning the unknown mean of a normal distribution whose variance is known, against a two-sided alternative hypothesis is presented. This procedure incorporates additional safeguards as required by the practicing experimenter. Furthermore, approximate formulae for the O.C. function and the A.S.N. function are derived.     

COMPARISON OF COSTS FOR MULTI-STAGE GROUP TESTING METHODS IN THE PHARMACEUTICAL INDUSTRY

The issue of whether to test components individually, or alternatively in groups, in order to detect certain chemical properties remains an important issue in the pharmaceutical industry. Economic viability is of paramount importance since, for multi-stage procedures, the cost of additional stages must be taken into consideration, along with the cost of testing mixtures of components. Optimum groups sizes are calculated for the two-stage, three-stage (both members of Li's family of algorithms) and the row-and-column procedures. The γ-two-stage design is investigated, which involves using a γ-separating design at the first stage, followed (if necessary) by a strongly separating design at the second stage. Finally, comparisons are made between the costs of single and multi-stage procedures, for both optimum and standard groups sizes, through the use of two different cost functions.     

LARGE AND MODERATE DEVIATIONS PRINCIPLES FOR KERNEL ESTIMATION OF A MULTIVARIATE DENSITY AND ITS PARTIAL DERIVATIVES

This paper studies the large deviations behaviour of the kernel estimator of a probability density f, by considering the case when the kernel takes negative values. It establishes large and moderate deviations principles for the kernel estimators of the partial derivatives of f. The estimators of the derivatives exhibit a quadratic behaviour for both the large and the moderate deviations scales, whereas for the density estimator there is a classical gap between the large deviations and the moderate deviations asymptotics.