The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences

Let {X _j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X _j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.     

A unified approach to multilevel sample selection models

Abstract

We propose a unified approach for multilevel sample selection models using a generalized result on skew distributions arising from selection. If the underlying distributional assumption is normal, then the resulting density for the outcome is the continuous component of the sample selection density and has links with the closed skew-normal distribution (CSN). The CSN distribution provides a framework which simplifies the derivation of the conditional expectation of the observed data. This generalizes the Heckman’s two-step method to a multilevel sample selection model. Finite-sample performance of the maximum likelihood estimator of this model is studied through a Monte Carlo simulation.     

An improved estimator of omission rate for census count: With particular reference to India

ABSTRACT

Every large census operation should undergo evaluation programs to find the sources and extent of inherent coverage errors. In this article, we briefly discuss the statistical methodology to estimate the omission rate in Indian census using dual-system estimation (DSE) technique. We have explicitly studied the correlation bias factor involved in the estimate, its extent, and consequences. A new potential source of bias in the estimate is identified and discussed. During the survey, more efficient enumerators compared to the census operations are appointed, and this fact may inflate the dependency between two lists and lead to a significant bias. Some examples are given to demonstrate this argument in various plausible situations. We have suggested one simple and flexible approach which can control this bias. Our proposed estimator can efficiently overcome the potential bias by achieving the desired degree of accuracy (almost unbiased) with relatively higher efficiency. Overall improvements in the results are explored through simulation study on different populations.     

Age-period-cohort analysis: an illustration of the problems in assessing interaction in one observation per cell data

This paper discusses the specific problems of age-period-cohort (A-P-C) analysis within the general framework of interaction assessment for two-way cross-classified data with one observation per cell. The A-P-C multiple classification model containing the effects of age groups (rows), periods of observation (columns), and birth cohorts (diagonals of the two-way table) is characterized as one of a special class of models involving interaction terms assumed to have very specific forms. The so-called A-P-C identification problem, which results from the use of a particular interaction structure for detecting cohort effects, is shown to manifest itself in the form of an exact linear dependency among the columns of the design matrix. The precise relationship holding among these columns is derived, as is an explicit formula for the bias in the parameter estimates resulting from an incorrect specification of an assumed restriction on the parameters required to solve the normal equations. Current methods for modeling A-P-C data are critically reviewed, an illustrative numerical example is presented, and one potentially promising analysis strategy is discussed. However, gien the large number of possible sources for error in A-P-C analyses, it is strongly recommended that the results of such analyses be interpreted with a great deal of caution.     

An approximation for analyzing a broad class of implicitly and explicitly defined estimators

An approximation is presented that can be used to gain insight into the characteristics – such as outlier sensitivity, bias, and variability – of a wide class of estimators, including maximum likelihood and least squares. The approximation relies on a convenient form for an arbitrary order Taylor expansion in a multivariate setting. The implicit function theorem can be used to construct the expansion when the estimator is not defined in closed form. We present several finite-sample and asymptotic properties of such Taylor expansions, which are useful in characterizing the difference between the estimator and the expansion.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Peculiar bias properties of the ols estimator when applied to a dynamic model with autocorrelated disturbances

This study reveals that contrary to the conventional wisdom among econometricians, the bias of the OLS estimator can be quite small when the estimator is applied to a geometrically distributed lag model, y_t<ce:glyph name="dbnd6"/> α + βx _t+ λy _t-₁. + u_t, with autocorrelated disturbances, be they AR(1), MA(1), MA(2), AR(2), and ARMA(1,1). This happens when λ is large and x_tis smoothly trended (e.g., a real GNP series). In fact, the bias of the OLS estimator becomes zero at one parameter combination, and the OLS estimator performs well over a wide range around this parameter combination. By decomposing the disturbance term into two parts, the paper also explains why OLS shows such an unexpected property. These findings have both pedagogical and practical significance.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

Sliced inverse regression under linear constraints

We consider the semiparametric regression model introduced by Li (1991) and add to this model some linear constraints on the slope parameters. These constraints can be identifiability conditions or they may carry additional in¬formations on the slope parameters. Using a geometric argument, we develop a method to estimate the slope parameters. This link-free and distribution-free method splits in two steps: the first is a Sliced Inverse Regression (SIR); Canonical Analysis is used at the second step to transform the SIR estimates so that they satisfy the constraints. We establish yn-consistency and obtain the asymptotic distribution of the estimates.

This estimation method is applied to the general sample selection model which is very useful in Econometrics. A simulation study shows that the method performs well in the example considered.