MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS OF THE TRUNCATED CAUCHY DISTRIBUTION

Truncated Cauchy distribution with four unknown parameters is considered and derivation and existence of the maximum likelihood estimates is investigated here. We provide a sufficient condition for the maximum likelihood estimate of the scale parameter to be finite, and also show that the condition is necessary for sufficiently large samples. Note that all the moments of the truncated Cauchy distribution exist which makes it much more attractive as a model when compared to the regular Cauchy. We also study, using simulations, the small sample properties of the maximum likelihood estimates.     

THE BIAS IN PERIODOGRAM ORDINATES AND THE ESTIMATION OF ARMA MODELS IN THE FREQUENCY DOMAIN

The paper has its origin in the finding that the frequency-domain estimation of ARh4A models can produce estimates which may be remarkably biased. Both of the frequency-domain estimation methods considered in the paper are based on the frequency-domain likelihood function, which depends on the periodogram ordinates of the time series. It is found that, as estimates of the spectrum ordinates, the corresponding periodogram ordinates may contain a rather remarkable bias, which again causes bias in the estimates of parameters produced by a frequency-domain estimation method of an ARMA model. The bias is reduced by tapering the observed time series. An example is given of estimation experiments for simulated time series from a pure autoregressive process of order two.     

AN EVALUATION OF NON-ITERATIVE METHODS FOR ESTIMATING THE LINEAR-BY-LINEAR PARAMETER OF ORDINAL LOG-LINEAR MODELS

Parameter estimation for association and log-linear models is an important aspect of the analysis of cross-classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear-by-linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non-iterative procedures for estimating the linear-by-linear parameter for ordinal log-linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non-iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered.     

ESTIMATION OF THE CONDITIONAL PROBABILITY OF TUBERCULOSIS GIVEN THE RESPONSE OF A MANTOUX TEST

For the planning of community tuberculosis, prophylaxis one must know the prevalence of tuberculosis (TB) infection as a function of response to the Mantoux intradermal tuberculin test; the standard test for indicating Mycobacterium TB infection. The skin induration size used to select individuals for prophylaxis must be chosen carefully, in view of the costs associated with carrying out and supervising such prophylaxis. The Mantoux test was used to obtain measurements on adolescents in metropolitan Victoria and on a small sample of adolescents with clinical TB. These data are employed to obtain estimates and to construct upper confidence bounds for the conditional probability of TB infection, given the level of Mantoux response. Two conservative methods are presented; one is ‘nonparametric’, the other ‘semiparametric’. The analyses indicate that for responses up to and including 13 mm, the probability of TB infection is less than.07 with ninety-five percent confidence.     

WEIGHTED AREA TECHNIQUES FOR THE ESTIMATION OF THE PARAMETERS OF A GROWTH CURVE1

A weighted area estimation technique which allows explicit estimation of the parameters of the growth curve l_t= L∞[1-exp(-K(t-t₀))] is introduced. Simulated data are used to compare an optimal weighted area method with that of maximum likelihood. Length at age data from fisheries field studies are used to compare the weighted area and maximum likelihood techniques with the “graphical” method of Ford-Walford. The weighted area methods achieve 80–90% efficiency on simulated data, and are shown to provide robust and sensible estimates on field data.     

ROBUST ANALYSIS OF THE BIFURCATING AUTOREGRESSIVE MODEL IN CELL LINEAGE STUDIES

A robust estimation procedure for the bifurcating autoregressive model in cell lineage studies is proposed. The method is illustrated by application to a real data set and is compared with least squares estimates in a small simulation study.     

AN ASYMPTOTIC EXPANSION OF THE EXPECTATION OF THE ESTIMATED ERROR RATE IN DISCRIMINANT ANALYSIS1

When a sample discriminant function is computed, it is desired to estimate the error rate using this function. This is often done by computing G(-D/2), where G is the cumulative normal distribution and D² is the estimated Mahalanobis' distance. In this paper an asymptotic expansion of the expectation of G(-D/2) is derived and is compared with existing Monte Carlo estimates. The asymptotic bias of G(-D/2) is derived also and the well-known practical result that G(-D/2) gives too favourable an estimate of the true error rate     

ESTIMATING A LOGIT MODEL WITH RANDOMIZED DATA: THE CASE OF COCAINE USE

In his influential book, Maddala (1983) suggests combining randomized response survey data with other personal information to estimate logit models predicting immoral, unpopular, or unlawful behaviour. This study is one of the first to implement this technique using real data. Models of college students' recent cocaine use are estimated with academic performance and socio-economic characteristics as determinants. Parameter estimates obtained from randomized response surveys are compared to those obtained using conventional, direct question surveys. The results indicate that randomized response estimates provide useful information on the degree to which inferences regarding the determinants of cocaine use are sensitive to survey type.     

THE BOOTSTRAP RISK OF LINEAR EMPIRICAL BAYES ESTIMATES

This paper discusses the bootstrap risk of the linear empirical Bayes estimate of the form θ=Ǎ+B̌x, where x is the current observation, and Ǎ and B̌ are generally functions of the estimates of the prior parameters. The standard error of this risk is developed and ‘computations’ of both the bootstrap risk and its standard error are made.     

SMALL‐AREA ESTIMATION OF POVERTY: THE AID INDUSTRY STANDARD AND ITS ALTERNATIVES

Small‐area estimation of poverty‐related variables is an increasingly important analytical tool in targeting the delivery of food and other aid in developing countries. We compare two methods for the estimation of small‐area means and proportions, namely empirical Bayes and composite estimation, with what has become the international standard method of Elbers, Lanjouw & Lanjouw (2003) . In addition to differences among the sets of estimates and associated estimated standard errors, we discuss data requirements, design and model selection issues and computational complexity. The Elbers, Lanjouw and Lanjouw (ELL) method is found to produce broadly similar estimates but to have much smaller estimated standard errors than the other methods. The question of whether these standard error estimates are downwardly biased is discussed. Although the question cannot yet be answered in full, as a precautionary measure it is strongly recommended that the ELL model be modified to include a small‐area‐level error component in addition to the cluster‐level and household‐level errors it currently contains. This recommendation is particularly important because the allocation of billions of dollars of aid funding is being determined and monitored via ELL. Under current aid distribution mechanisms, any downward bias in estimates of standard error may lead to allocations that are suboptimal because distinctions are made between estimated poverty levels at the small‐area level that are not significantly different statistically.     

THE ASYMPTOTIC EQUIVALENCE OF THE FISHER INFORMATION MATRICES FOR TYPE I AND TYPE II CENSORED DATA FROM LOCATION-SCALE FAMILIES

Type I and Type II censored data arise frequently in controlled laboratory studies concerning time to a particular event (e.g., death of an animal or failure of a physical device). Log-location-scale distributions (e.g., Weibull, lognormal, and loglogistic) are commonly used to model the resulting data. Maximum likelihood (ML) is generally used to obtain parameter estimates when the data are censored. The Fisher information matrix can be used to obtain large-sample approximate variances and covariances of the ML estimates or to estimate these variances and covariances from data. The derivations of the Fisher information matrix proceed differently for Type I (time censoring) and Type II (failure censoring) because the number of failures is random in Type I censoring, but length of the data collection period is random in Type II censoring. Under regularity conditions (met with the above-mentioned log-location-scale distributions), we outline the different derivations and show that the Fisher information matrices for Type I and Type II censoring are asymptotically equivalent.     

A BETA-BINOMIAL MODEL FOR ESTIMATING THE SIZE OF A HETEROGENEOUS POPULATION

This paper compares the properties of various estimators for a beta‐binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis‐Hastings algorithm provide reasonable posterior estimates for informative priors.     

THE MONTE CARLO EVALUATION OF ORTHANT PROBABILITIES FOR MULTIVARIATE NORMAL DISTRIBUTIONS

The calculation of multivariate normal orthant probabilities is practically impossible when the number of variates is greater than five or six, except in very special cases. A transformation of the integral is obtained which enables quite accurate Monte Carlo estimates to be obtained for a fairly high number of dimensions, particularly if control variates are used.     

THE ANALYSIS OF DISCRETE CHOICE EXPERIMENTS WITH CORRELATED ERROR STRUCTURE

In a stated preference discrete choice experiment each subject is typically presented with several choice sets, and each choice set contains a number of alternatives. The alternatives are defined in terms of their name (brand) and their attributes at specified levels. The task for the subject is to choose from each choice set the alternative with highest utility for them. The multinomial is an appropriate distribution for the responses to each choice set since each subject chooses one alternative, and the multinomial logit is a common model. If the responses to the several choice sets are independent, the likelihood function is simply the product of multinomials. The most common and generally preferred method of estimating the parameters of the model is maximum likelihood (that is, selecting as estimates those values that maximize the likelihood function). If the assumption of within-subject independence to successive choice tasks is violated (it is almost surely violated), the likelihood function is incorrect and maximum likelihood estimation is inappropriate. The most serious errors involve the estimation of the variance-covariance matrix of the model parameter estimates, and the corresponding variances of market shares and changes in market shares.

In this paper we present an alternative method of estimation of the model parameter coefficients that incorporates a first-order within-subject covariance structure. The method involves the familiar log-odds transformation and application of the multivariate delta method. Estimation of the model coefficients after the transformation is a straightforward generalized least squares regression, and the corresponding improved estimate of the variance-covariance matrix is in closed form. Estimates of market share (and change in market share) follow from a second application of the multivariate delta method. The method and comparison with maximum likelihood estimation are illustrated with several simulated and actual data examples.

Advantages of the proposed method are: 1) it incorporates the within-subject covariance structure; 2) it is completely data driven; 3) it requires no additional model assumptions; 4) assuming asymptotic normality, it provides a simple procedure for computing confidence regions on market shares and changes in market shares; and 5) it produces results that are asymptotically equivalent to those produced by maximum likelihood when the data are independent.     

ON ESTIMATING THE SPREAD OF A DISTRIBUTION ABOUT ITS REGRESSION SLOPES1

Nonparametric tolerance regions are used to estimate the spread of a distribution about known regression slopes. When the slopes are not known their sample estimates are employed.     

RECURSIVE ESTIMATION OF THE GENERAL LINEAR MODEL WITH DEPENDENT ERRORS AND MULTIPLE ADDITIONAL OBSERVATIONS1

The techniques for recursive estimation of the general linear model with dependent errors and known second order properties, is generalised to allow for simultaneous addition of an arbitrary number of additional observations. Computational formulae for recursive updating of parameter estimates are derived, together with a sequence of univariate recursive residuals for testing the constancy of the regression relation over time.     

PARAMETER ESTIMATION AND APPLICATIONS FOR A GENERALISATION OF THE BETA-BINOMIAL DISTRIBUTION

A three-parameter generalisation of the beta-binomial distribution (BBD) derived by Chandon (1976) is examined. We obtain the maximum likelihood estimates of the parameters and give the elements of the information matrix. To exhibit the applicability of the generalised distribution we show how it gives an improved fit over the BBD for magazine exposure and consumer purchasing data. Finally we derive an empirical Bayes estimate of a binomial proportion based on the generalised beta distribution used in this study.     

SAMPLING AND ESTIMATION IN THE PRESENCE OF CUT-OFF SAMPLING

Cut-off sampling consists of deliberately excluding a set of units from possible selection in a sample, for example if the contribution of the excluded units to the total is small or if the inclusion of these units in the sample involves high costs. If the characteristics of interest of the excluded units differ from those of the rest of the population, the use of naïve estimators may result in highly biased estimates. In this paper, we discuss the use of auxiliary information to reduce the bias by means of calibration and balanced sampling. We show that the use of the available auxiliary information related to both the variable of interest and the probability of being excluded enables us to reduce the potential bias. A short numerical study supports our findings.     

ON THE ROBUST ANALYSIS OF VARIANCE COMPONENTS MODELS FOR PEDIGREE DATA

Quantitative traits measured over pedigrees of individuals may be analysed using maximum likelihood estimation, assuming that the trait has a multivariate normal distribution. This approach is often used in the analysis of mixed linear models. In this paper a robust version of the log likelihood for multivariate normal data is used to construct M-estimators which are resistant to contamination by outliers. The robust estimators are found using a minimisation routine which retains the flexible parameterisations of the multivariate normal approach. Asymptotic properties of the estimators are derived, computation of the estimates and their use in outlier detection tests are discussed, and a small simulation study is conducted.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.