A bias-corrected histogram estimator for line transect sampling

The classical histogram method has already been applied in line transect sampling to estimate the parameter f(0), which in turns is used to estimate the population abundance D or the population size N. It is well know that the bias convergence rate for histogram estimator of f(0) is o(h²) as h → 0, under the shoulder condition assumption. If the shoulder condition is not true, then the bias convergence rate is only o(h). This paper proposed two new estimators for f(0), which can be considered as modifications of the classical histogram estimator. The first estimator is derived when the shoulder condition is assumed to be valid and it reduces the bias convergence rate from o(h²) to o(h³). The other one is constructed without using the shoulder condition assumption and it reduces the bias convergence rate from o(h) to o(h²). The asymptotic properties of the proposed estimators are derived and formulas for bin width are also given. The finite properties based on a real data set and an extensive simulation study demonstrated the potential practical use of the proposed estimators.     

Efficient computation of the discrete autocorrelation wavelet inner product matrix

Discrete autocorrelation (a.c.) wavelets have recently been applied in the statistical analysis of locally stationary time series for local spectral modelling and estimation. This article proposes a fast recursive construction of the inner product matrix of discrete a.c. wavelets which is required by the statistical analysis. The recursion connects neighbouring elements on diagonals of the inner product matrix using a two-scale property of the a.c. wavelets. The recursive method is an (log (N)³) operation which compares favourably with the (N log N) operations required by the brute force approach. We conclude by describing an efficient construction of the inner product matrix in the (separable) two-dimensional case.     

Bandwidth selector for nonparametric recursive density estimation for spatial data defined by stochastic approximation method

Abstract

In this article we propose an automatic selection of the bandwidth of the recursive kernel density estimators for spatial data defined by the stochastic approximation algorithm. We showed that, using the selected bandwidth and the stepsize which minimize the MWISE (Mean Weighted Integrated Squared Error), the recursive estimator will be quite similar to the nonrecursive one in terms of estimation error and much better in terms of computational costs. In addition, we obtain the central limit theorem for the nonparametric recursive density estimator under some mild conditions.     

Implementation of recursive nonparametric kernel estimation and a monte carlo study on its finite sample properties

The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored by Ahmad and Lin (1976), and Singh and Ullah (1986). Their studies are mainly concerned with the estimator's asymptotic behaviour. However, they do not include much discussion on the strategy of computing the estimates. In this paper, we provide a convenient implementation of the recursive estimator and examine its finite sample properties through simulation studies. Our study has demonstrated that for relatively short length of recursive updating, the estimates are generally equivalent to their fixed window width counterparts However, we found that substantial recursive updating can seriously lower the estimator's efficiency even though it is a consistent estimator.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

Estimation and Inference for Linear Panel Data Models Under Misspecification When Both n and T are Large

This article considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, n, and the number of time periods, T, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarded as a pseudo-true parameter. We then establish the asymptotic distributional properties of the FE estimator around the pseudo-true parameter when n and T jointly go to infinity. Notably, we show that the FE estimator suffers from the incidental parameters bias of which the top order is O(T^{? 1}), and even after the incidental parameters bias is completely removed, the rate of convergence of the FE estimator depends on the degree of model misspecification and is either (nT)^{? 1/2} or n^{? 1/2}. Second, we establish asymptotically valid inference on the (pseudo-true) parameter. Specifically, we derive the asymptotic properties of the clustered covariance matrix (CCM) estimator and the cross-section bootstrap, and show that they are robust to model misspecification. This establishes a rigorous theoretical ground for the use of the CCM estimator and the cross-section bootstrap when model misspecification and the incidental parameters bias (in the coefficient estimate) are present. We conduct Monte Carlo simulations to evaluate the finite sample performance of the estimators and inference methods, together with a simple application to the unemployment dynamics in the U.S.     

A maximum-likelihood estimator with infinite error

The standard error of the maximum-likelihood estimator for 1/μ based on a random sample of size N from the normal distribution N(μ,σ²) is infinite. This could be considered to be a disadvantage.Another disadvantage is that the bias of the estimator is undefined if the integral is interpreted in the usual sense as a Lebesgue integral. It is shown here that the integral expression for the bias can be interpreted in the sense given by the Schwartz theory of generalized functions. Furthermore, an explicit closed form expression in terms of the complex error function is derived. It is also proven that unbiased estimation of 1/μ is impossible.Further results on the maximum-likelihood estimator are investigated, including closed form expressions for the generalized moments and corresponding complete asymptotic expansions. It is observed that the problem can be reduced to a one-parameter problem depending only on , and this holds also for more general location-scale problems. The parameter can be interpreted as a shape parameter for the distribution of the maximum-likelihood estimator.An alternative estimator is suggested motivated by the asymptotic expansion for the bias, and it is argued that the suggested estimator is an improvement. The method used for the construction of the estimator is simple and generalizes to other parametric families.The problem leads to a rediscovery of a generalized mathematical expectation introduced originally by Kolmogorov [1933. Foundations of the Theory of Probability, second ed. Chelsea Publishing Company (1956)]. A brief discussion of this, and some related integrals, is provided. It is in particular argued that the principal value expectation provides a reasonable location parameter in cases where it exists. This does not hold generally for expectations interpreted in the sense given by the Schwartz theory of generalized functions.     

Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences

The problem of estimation of the derivative of a probability density f is considered, using wavelet orthogonal bases. We consider an important kind of dependent random variables, the so-called mixing random variables and investigate the precise asymptotic expression for the mean integrated error of the wavelet estimators. We show that the mean integrated error of the proposed estimator attains the same rate as when the observations are independent, under certain week dependence conditions imposed to the {X _i }, defined in {Ω, N, P}.     

Implementation of recursive nonparametric kernel estimation and a monte carlo study on its finite sample properties

The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored by Ahmad and Lin (1976), and Singh and Ullah (1986). Their studies are mainly concerned with the estimator's asymptotic behaviour. However, they do not include much discussion on the strategy of computing the estimates. In this paper, we provide a convenient implementation of the recursive estimator and examine its finite sample properties through simulation studies. Our study has demonstrated that for relatively short length of recursive updating, the estimates are generally equivalent to their fixed window width counterparts However, we found that substantial recursive updating can seriously lower the estimator's efficiency even though it is a consistent estimator.     

Inference on finite population categorical response: nonparametric regression-based predictive approach

Suppose that a finite population consists of N distinct units. Associated with the ith unit is a polychotomous response vector, d _i , and a vector of auxiliary variable x _i . The values x _i ’s are known for the entire population but d _i ’s are known only for the units selected in the sample. The problem is to estimate the finite population proportion vector P. One of the fundamental questions in finite population sampling is how to make use of the complete auxiliary information effectively at the estimation stage. In this article a predictive estimator is proposed which incorporates the auxiliary information at the estimation stage by invoking a superpopulation model. However, the use of such estimators is often criticized since the working superpopulation model may not be correct. To protect the predictive estimator from the possible model failure, a nonparametric regression model is considered in the superpopulation. The asymptotic properties of the proposed estimator are derived and also a bootstrap-based hybrid re-sampling method for estimating the variance of the proposed estimator is developed. Results of a simulation study are reported on the performances of the predictive estimator and its re-sampling-based variance estimator from the model-based viewpoint. Finally, a data survey related to the opinions of 686 individuals on the cause of addiction is used for an empirical study to investigate the performance of the nonparametric predictive estimator from the design-based viewpoint.     

Weak Instrumental Variables Models for Longitudinal Data

This article considers the estimation and testing of a within-group two-stage least squares (TSLS) estimator for instruments with varying degrees of weakness in a longitudinal (panel) data model. We show that adding the repeated cross-sectional information into a regression model can improve the estimation in weak instruments. Moreover, the consistency and limiting distribution of the TSLS estimator are established when both N and T tend to infinity. Some asymptotically pivotal tests are extended to a longitudinal data model and their asymptotic properties are examined. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed estimators.     

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.     

Polynomial Histograms for Multivariate Density and Mode Estimation

Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n^?6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

Asymptotic normal tests for integration in panels with cross-dependent units

The asymptotically normal, regression-based LM integration test is adapted for panels with correlated units. The N different units may be integrated of different (fractional) orders under the null hypothesis. The paper first reviews conditions under which the test statistic is asymptotically (as T→∞) normal in a single unit. Then we adopt the framework of seemingly unrelated regression [SUR] for cross-correlated panels, and discuss a panel test statistic based on the feasible generalized least squares [GLS] estimator, which follows a χ ²(N) distribution. Third, a more powerful statistic is obtained by working under the assumption of equal deviations from the respective null in all units. Fourth, feasible GLS requires inversion of sample covariance matrices typically imposing T>N; in addition we discuss alternative covariance matrix estimators for T<N. The usefulness of our results is assessed in Monte Carlo experimentation.     

Estimation of a finite population distribution function based on a linear model with unknown heteroscedastic errors

The authors consider a finite population ρ = {(Y_k, x_k), k = 1,…,N} conforming to a linear superpopulation model with unknown heteroscedastic errors, the variances of which are values of a smooth enough function of the auxiliary variable X for their nonparametric estimation. They describe a method of the Chambers‐Dunstan type for estimation of the distribution of {Y_k, k = 1,…, N} from a sample drawn from without replacement, and determine the asymptotic distribution of its estimation error. They also consider estimation of its mean squared error in particular cases, evaluating both the analytical estimator derived by “plugging‐in” the asymptotic variance, and a bootstrap approach that is also applicable to estimation of parameters other than mean squared error. These proposed methods are compared with some common competitors in simulation studies.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

A remark on estimating the mean of a normal distribution with known coefficient of variation

Let X₁, X₂, …, X_n be iid N(μ, aμ²) (a>0) random variables with an unknown mean μ>0 and known coefficient of variation (CV) √a. The estimation of μ is revisited and it is shown that a modified version of an unbiased estimator of μ [cf. Khan RA. A note on estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1968;63:1039–1041] is more efficient. A certain linear minimum mean square estimator of Gleser and Healy [Estimating the mean of a normal distribution with known CV. J Am Stat Assoc. 1976;71:977–981] is also modified and improved. These improved estimators are being compared with the maximum likelihood estimator under squared-error loss function. Based on asymptotic consideration, a large sample confidence interval is also mentioned.