Monotonic minimax estimators of a 2×2 covariance matrix

Let S : 2 × 2 have a nonsingular Wishart distribution with unknown matrix σ and n degrees of freedom. For estimating σ two families of mimmax estimators, with respect to the entropy loss, are presented. These estimators are of the form σ(S) = Rø(L)R^t where R is orthogonal, L and Φ are diagonal, and RLR^T = S. Conditions under which the components of Φ and L follow the same order relation [i.e., writing Φ = diag(Φ₁,Φ₂) and L = diag(l₁,/₂) with l₁ ≥ l₂, we have Φ₁ ≥ Φ₂] are established. Comparisons with Stein's estimators and other orthogonally invariant estimators are discussed.     

New Shrinkage Parameters for the Liu-type Logistic Estimators

The binary logistic regression is a widely used statistical method when the dependent variable has two categories. In most of the situations of logistic regression, independent variables are collinear which is called the multicollinearity problem. It is known that multicollinearity affects the variance of maximum likelihood estimator (MLE) negatively. Therefore, this article introduces new shrinkage parameters for the Liu-type estimators in the Liu (2003) in the logistic regression model defined by Huang (2012) in order to decrease the variance and overcome the problem of multicollinearity. A Monte Carlo study is designed to show the goodness of the proposed estimators over MLE in the sense of mean squared error (MSE) and mean absolute error (MAE). Moreover, a real data case is given to demonstrate the advantages of the new shrinkage parameters.     

A RENEWAL THEOREM IN MULTIDIMENSIONAL TIME

Let Y_l, Y₂,… be i.i.d., positive, integer-valued random variables with means, μ. Let the sequences {Y_ij, j= 1,2,…}, i= 1,…, r be independent copies of {Y₁, Y₂,…}. For n={n₁,…, n_r.}, n₁≥1, let S_n=S?ⁿ¹_k1=1= 1 …S?^nr_kr=1 Y_ik1… Y_rkr. We show that S?^N_k=1S?^∞_k1=1…S?^∞_nr=1 P[[S_n= k] ? [μ^-r N log^r-1 (N)/(r-1)!] as N →∞.     

Asymptotic efficiency of model selection criteria: the nonzero mean gaussian ar(∞) case

Motivated by Shibata’s (1980) asymptotic efficiency results this paper dis-cusses the asymptotic efficiency of the order selected by a selection procedure for an infinite order autoregressive process with nonzero mean and unob servable errors that constitute a sequence of independent Gaussian random variables with mean zero and variance σ² The asymptotic efficiency is established for AIC–type selection criteria such as AIC’, FPE, and S_n(k). In addition, some asymptotic results about the estimators of the parameters of the process and the error–sequence are presented.     

Estimation of long-term trends and variation in avian survival probabilities using random effects models

We obtained banding and recovery data from the Bird Banding Laboratory (operated by the Biological Resources Division of the US Geological Survey) for adults from 129 avian species that had been continuously banded for > 24 years. Data were partitioned by gender, banding period (winter versus summer), and by states/provinces. Data sets were initially screened for adequacy based on specific criteria (e.g. minimum sample sizes). Fifty-nine data sets (11 waterfowl species, the Mourning Dove and Common Grackle) met our criteria of adequacy for further analysis. We estimated annual survival probabilities using the Brownie et al. recovery model {St, ft} in program MARK. Trends in annual survival and temporal process variation were estimated using random effects models based on shrinkage estimators. Waterfowl species had relatively little variation in annual survival probabilities (mean CV = 8.7% and 10% for males and females, respectively). The limited data for other species suggested similar low temporal variation for males, but higher temporal variation for females (CV = 40%). Evidence for long-term trends varied by species, banding period and sex, with no obvious spatial patterns for either positive or negative trends in survival probabilities. An exception was Mourning Doves banded in Illinois/Missouri and Arizona/New Mexico where both males (slope = -0.0122, se = 0.0019 and females (slope = -0.0109 to -0.0128, se = 0.0018 -0.0032) exhibited declining trends in survival probabilities. We believe our approach has application for large-scale monitoring. However, meaningful banding and recovery data for species other than waterfowl is very limited in North America.     

Some inadmissibility results for estimating ordered uniform scale parameters

Independent random samples (of possibly unequal sizes) are drawn from k (≥2) uniform populations having unknown scale parameters μ₁,…,μ_k. The problem of componentwise estimation of ordered parameters is investigated. The loss function is assumed to be squared error and the cases of known and unknown ordering among μ₁,…,μ_k. are dealt with separately. Sufficient conditions for an estimator to be inadmissible are provided and as a consequence, many natural estimators are shown to be inadmissible, Better estimators are provided.     

A class of fixed-width confidence intervals for a ranked normal mean

Suppose that we are given k(≥ 2) independent and normally distributed populations π₁, …, π_k where π_i has unknown mean μ_i and unknown variance σ² _i (i = 1, …, k). Let μ_[i] (i = 1, …, k) denote the i^th smallest one of μ₁, …, μ_k. A two-stage procedure is used to construct lower and upper confidence intervals for μ_[i] and then use these to obtain a class of two-sided confidence intervals on μ_[i] with fixed width. For i = k, the interval given by Chen and Dudewicz (1976) is a special case. Comparison is made between the class of two-sided intervals and a symmetric interval proposed by Chen and Dudewicz (1976) for the largest mean, and it is found that for large values of k at least one of the former intervals requires a smaller total sample size. The tables needed to actually apply the procedure are provided.     

Comparison of Akaike information criterion and consistent Akaike information criterion for model selection and statistical inference from capture-recapture studies

SUMMARY We compare properties of parameter estimators under Akaike information criterion (AIC) and 'consistent' AIC (CAIC) model selection in a nested sequence of open population capture-recapture models. These models consist of product multinomials, where the cell probabilities are parameterized in terms of survival ( ) and capture ( p ) i i probabilities for each time interval i . The sequence of models is derived from 'treatment' effects that might be (1) absent, model H ; (2) only acute, model H ; or (3) acute and 0 2 p chronic, lasting several time intervals, model H . Using a 35 factorial design, 1000 3 repetitions were simulated for each of 243 cases. The true number of parameters ranged from 7 to 42, and the sample size ranged from approximately 470 to 55 000 per case. We focus on the quality of the inference about the model parameters and model structure that results from the two selection criteria. We use achieved confidence interval coverage as an integrating metric to judge what constitutes a 'properly parsimonious' model, and contrast the performance of these two model selection criteria for a wide range of models, sample sizes, parameter values and study interval lengths. AIC selection resulted in models in which the parameters were estimated with relatively little bias. However, these models exhibited asymptotic sampling variances that were somewhat too small, and achieved confidence interval coverage that was somewhat below the nominal level. In contrast, CAIC-selected models were too simple, the parameter estimators were often substantially biased, the asymptotic sampling variances were substantially too small and the achieved coverage was often substantially below the nominal level. An example case illustrates a pattern: with 20 capture occasions, 300 previously unmarked animals are released at each occasion, and the survival and capture probabilities in the control group on each occasion were 0.9 and 0.8 respectively using model H . There was a strong acute treatment effect 3 on the first survival ( ) and first capture probability ( p ), and smaller, chronic effects 1 2 on the second and third survival probabilities ( and ) as well as on the second capture 2 3 probability ( p ); the sample size for each repetition was approximately 55 000. CAIC 3 selection led to a model with exactly these effects in only nine of the 1000 repetitions, compared with 467 times under AIC selection. Under CAIC selection, even the two acute effects were detected only 555 times, compared with 998 for AIC selection. AIC selection exhibited a balance between underfitted and overfitted models (270 versus 263), while CAIC tended strongly to select underfitted models. CAIC-selected models were overly parsimonious and poor as a basis for statistical inferences about important model parameters or structure. We recommend the use of the AIC and not the CAIC for analysis and inference from capture-recapture data sets.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

Contributions to adaptive estimation

There are many statistics which can be used to characterize data sets and provide valuable information regarding the data distribution, even for large samples. Traditional measures, such as skewness and kurtosis, mentioned in introductory statistics courses, are rarely applied. A variety of other measures of tail length, skewness and tail weight have been proposed, which can be used to describe the underlying population distribution. Adaptive statistical procedures change the estimator of location, depending on sample characteristics. The success of these estimators depends on correctly classifying the underlying distribution model. Advocates of adaptive distribution testing propose to proceed by assuming (1) that an appropriate model, say Omega , is such that Omega { Omega , Omega , i i 1 2 … , Omega }, and (2) that the character of the model selection process is statistically k independent of the hypothesis testing. We review the development of adaptive linear estimators and adaptive maximum-likelihood estimators.     

Estimation of p{y p> max(y1, y2, …, yp-1)}in theexponential case

This paper deals with the derivation of (i) the MLE (ii) the MVUE (iii) a Bayes estimator of the probability in the title, for the case p = 2. Simulation studies are carried out to compare these estimators. The results suggest that the MLE and the Bayes estimator are biased and the Bayes estimator have the smallest MSE. In the general case, explicit expression for the probability in the title is derived and the MLE and Bayes estimator are obtained. A general method of deriving the MVUE is pointed out. Because of the simulation studies for p = 2 it is recommended that the Bayes or predictive estimator should be used.     

Estimate of variance of u-statistics

Let g(x₁,… , x_k) be a symmetric function with k arguments. Let U be a U-statistic based on a random sample of size n with kernel function g . In this paper, the problem of estimating var(U) is considered. Several estimators are compared by computer simulations and we conclude that two estimators, one is constructed as a U-statistic and the other is the bootstrap estimator, give good estimates for many U-statistics.     

Estimating the negative binomial dispersion parameter with highly stratified surveys

We investigate several estimators of the negative binomial (NB) dispersion parameter for highly stratified count data for which the statistical model has a separate mean parameter for each stratum. If the number of samples per stratum is small then the model is highly parameterized and the maximum likelihood estimator (MLE) of the NB dispersion parameter can be biased and inefficient. Some of the estimators we investigate include adjustments for the number of mean parameters to reduce bias. We extend other estimators that were developed for the iid case, to reduce bias when there are many mean parameters. We demonstrate using simulations that an adjusted double extended quasi-likelihood estimator we proposed gives much improved estimates compared to the MLE. Adjusted extended quasi-likelihood and adjusted maximum likelihood estimators also give much-improved results. We illustrate the various estimators with stratified random bottom trawl survey data for cod (Gadus morhua) off the south coast of Newfoundland, Canada.     

On improved estimation of a gamma shape parameter

This paper deals with improved estimation of a gamma shape parameter from a decision-theoretic point of view. First we study the second-order properties of three estimators – (i) the maximum-likelihood estimator (MLE), (ii) a bias corrected version of the MLE, and (iii) an improved version (in terms of mean squared error) of the MLE. It is shown that all the three estimators mentioned above are second-order inadmissible. Next, we obtain superior estimators which are second order better than the above three estimators. Simulation results are provided to study the relative risk improvement of each improved estimator over the MLE.     

Fixed Effects and Bias Due to the Incidental Parameters Problem in the Tobit Model

The maximum likelihood estimator (MLE) in nonlinear panel data models with fixed effects is widely understood (with a few exceptions) to be biased and inconsistent when T, the length of the panel, is small and fixed. However, there is surprisingly little theoretical or empirical evidence on the behavior of the estimator on which to base this conclusion. The received studies have focused almost exclusively on coefficient estimation in two binary choice models, the probit and logit models. In this note, we use Monte Carlo methods to examine the behavior of the MLE of the fixed effects tobit model. We find that the estimator's behavior is quite unlike that of the estimators of the binary choice models. Among our findings are that the location coefficients in the tobit model, unlike those in the probit and logit models, are unaffected by the “incidental parameters problem.” But, a surprising result related to the disturbance variance emerges instead - the finite sample bias appears here rather than in the slopes. This has implications for estimation of marginal effects and asymptotic standard errors, which are also examined in this paper. The effects are also examined for the probit and truncated regression models, extending the range of received results in the first of these beyond the widely cited biases in the coefficient estimators.     

Multi-level models can benefit from minimizing higher-order variations: an illustration using child malnutrition data

ABSTRACT

This study aims to measure the robustness of multi-level models designed for three anthropometric indices – height-for-age (HAZ), weight-for-age (WAZ) and weight-for-height (WHZ) Z-scores for estimating the childhood malnutrition indicators stunting, underweight and wasting in Bangladesh. The 2011 BDHS child malnutrition data have been used in developing multi-level models with and without incorporating specific contextual variables relating to lower administrative units extracted from the 2011 Bangladesh Population and Housing Census. The robustness of the models is examined through (i) testing significance of random effects corresponding to lower administrative units through selection criteria including conditional AIC, R-squared, and LRT; (ii) comparing multi-level model-based estimators to design-based estimators of child malnutrition indicators with their precision at division, district and sub-district levels; and (iii) assessing the impact of contextual variables in capturing higher-order administrative level variations. Findings reveal that the inclusion of important contextual variables helps capture variations at higher-level administrative units, and consequently assists in the selection of robust multi-level models which ultimately provide improved accuracy of estimated parameters. The findings support the application of lower administrative census information in developing a simpler multi-level model by minimizing higher-order variation.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

Location estimators based on linear combinations of modified order statistics

Let Xj,…, X be i.i.d random variables with common distribution function F(x-0), and let a(u) be a function defined on [0,1], For each t$$$R define the t-order statistics as: X. (t) = X. if there in k exist exactly (i- 1) X. !s such that |X.-t|greater|X.-t| and define the variable T (t) = n 2. , a(i/n) X, (t) . We consider estimates of 8 defined as solutions of the eauaticn T (8) =6 , and     

Pitman-closeness of some preliminary test and shrinkage estimators

For the model X ～ N_p: (θ,I)preliminary test estimator (PTE), shrinkage and positive-rule versions of the MLE (X) of θare mutually compared in the light of the Pitman closeness measure. The usual dominance properties of these estimators pertaining to the conventional quadratic loss criterion are shown to remain intact in the current context too. In an asymptotic setup, the conclusions hold for a much wider class of estimators pertaining to general parametric and nonparametric models.     

The estimation of the number of balls put in a sequence of urns

Let X₁,…,X_2n be independent and identically distributed copies of the non-negative integer valued random variable X distributed according to the unknown frequency function f(x). A total of 2n disjoint sequences of urns, each consisting of k urns, are given. X_j balls are placed in urn sequence j (1 ≤ j ≤ 2n). Each ball is placed in an urn of a given sequence with a certain known probability independently of the other balls. The variables X₁,…,X_2n are not observed; rather we observe whether certain pairs of urns are both empty or not. Our object is to estimate the mean μ of the number of balls X. Two different kinds of estimators of μ are investigated. One of the estimators studied is a method of moments type estimator while the other is motivated by the maximum likelihood principle. These estimators are compared on the basis of their asymptotic mean squared error as k tends to infinity. An application of these results to a problem in genetics involved with estimating codon substitution rates is discussed.