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We consider how to merge a limited amount of data from a randomized controlled trial (RCT) into a much larger set of data from an observational data base (ODB), to estimate an average causal treatment effect. Our methods are based on stratification. The strata are defined in terms of effect moderators as well as propensity scores estimated in the ODB. Data from the RCT are placed into the strata they would have occupied, had they been in the ODB instead. We assume that treatment differences are comparable in the two data sources. Our first "spiked-in" method simply inserts the RCT data into their corresponding ODB strata. We also consider a data-driven convex combination of the ODB and RCT treatment effect estimates within each stratum. Using the delta method and simulations, we identify a bias problem with the spiked-in estimator that is ameliorated by the convex combination estimator. We apply our methods to data from the Women's Health Initiative, a study of thousands of postmenopausal women which has both observational and experimental data on hormone therapy (HT). Using half of the RCT to define a gold standard, we find that a version of the spiked-in estimator yields lower-MSE estimates of the causal impact of HT on coronary heart disease than would be achieved using either a small RCT or the observational component on its own.

Applied economists often use a non-linear function to estimate percentage-change effects of dummy variables in semi-logarithmic models. Delta-method-based inference on these marginal effects is que...

Under constant-stress accelerated life test, the general progressive type-II censoring sample and the two parameters following the linear Arrhenius model, the point estimation and interval estimation of the two parameters log-normal distribution were discussed. The unknown parameters of the model as well as reliability and hazard rate functions are estimated by using Maximum likelihood (ML) and Bayesian methods. The maximum-likelihood estimates are derived by the Newton–Raphson method and the corresponding asymptotic variance is derived by the Fisher information matrix. Since the Bayesian estimates (BEs) of the unknown parameters cannot be expressed explicitly, the approximate BEs of the unknown parameters. The approximate highest posterior density confidence intervals are calculated. The practicality of the proposed method is illustrated by simulation study and real data application analysis.

Conducting accelerated degradation tests (ADTs) is an effective way to assess reliability of highly reliable products. In the existing literature, most works deal with planning ADT with a single stress variable; however, the situation of more than one stress variable is commonly seen in engineering practice. To fill the gap, in this article, we provide an analytical approach to address the design issue when two stress variables are present. By using a linear mixed-effects model to describe the accelerated degradation process, we demonstrate that the design problem can be solved by, first, finding the optimal setting of test conditions and allocation of test units for a “single-variable” case, and then the initial solution is transformed to the test plan for the case of two stress variables. The transformation is done by maintaining the same value of the asymptotic variance of the estimated p th quantile lifetime, along with the consideration of reducing the asymptotic variance of model parameters estimation. We also discuss how to find compromise plans that satisfy practical demands. Finally, the proposed framework is illustrated using a real-world example.

The scheme of doubly type-II censored sampling is an important method of obtaining data in lifetime studies. Statistical analysis of life- time distributions under this censoring scheme is based on precise lifetime data. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. This paper deals with the prob- lem of estimating the scale parameter of Rayleigh distribution under doubly type-II censoring scheme when the lifetime observations are fuzzy and are assumed to be related to underlying crisp realization of a random sample. We propose a new method to determine the maximum likelihood estimate of the parameter of interest. The asymptotic variance of the ML estimate is then derived by using the missing information principle. Their performance is then assessed through Monte Carlo simulations. Finally, an illustrative example with real data concerning 25 ball bearings in a life test is presented.

In this paper, after closed loop system identification is reviewed, asymptotic analysis and finite sample analysis for closed loop system identification are studied respectively, corresponding to the infinite data and finite data. More specifically, within the framework of infinite data, the cost function is modified to its simplified form, and one optimal feedback controller is obtained based on our own derivations. The simplified cost function and optimal feedback controller are benefit for practical application. Furthermore, the asymptotic variance of that optimal feedback controller is also yielded from the point of asymptotic analysis. In the case of finite data, finite sample properties are constructed for closed loop system identification, then one difference between the sampled identification criterion and its corresponding expected criterion is derived as an explicit form, which can bound one guaranteed interval for the sampled identification criterion. Finally, one simulation example is used to prove the efficiency of our proposed theories.

Balancing allocation of assigning units to two treatment groups to minimize the allocation differences is important in biomedical research. The complete randomization, rerandomization, and pairwise sequential randomization (PSR) procedures can be employed to balance the allocation. However, the first two do not allow a large number of covariates. In this article, we generalize the PSR procedure and propose a k-resolution sequential randomization (k-RSR) procedure by minimizing the Mahalanobis distance between both groups with equal group size. The proposed method can be used to achieve adequate balance and obtain a reasonable estimate of treatment effect. Compared to PSR, k-RSR is more likely to achieve the optimal value theoretically. Extensive simulation studies are conducted to show the superiorities of k-RSR and applications to the clinical synthetic data and GAW16 data further illustrate the methods.

When estimating the average causal effect in observational studies, researchers have to tackle both self-selection of treatment and outcome modeling. This is difficult because the parametric form of the outcome model is often unknown and there exists a large number of covariates. In this work, we present a semiparametric strategy for estimating the average causal effect by regressing on the propensity score. Furthermore, we show that regression error terms usually depend on the propensity score as well, which could cause heteroscedastic variances, and thus construct a refined estimator to improve the estimation efficiency. Both estimators are shown to be consistent and asymptotically normally distributed, with the latter one having a smaller asymptotic variance. The simulation studies indicate that our methods compare favorably with many competing estimators. Our methods are easy to implement and avoid hazardous impact due to extreme weights as often seen in weighting estimators. They can also be extended to handle subgroup effects with known structure. We apply the proposed methods to data from the Ohio Medicaid Assessment Survey 2012, estimating the effect of having health insurance on self-reported health status for a population with subsidized insurance plan choices under the Affordable Care Act.

High frequency financial data has become an essential component of the digital economy, yielding an increasing number of estimators. However, it is hard to reliably assess the uncertainty of such estimators. The Observed Asymptotic Variance (observed AVAR) is a non-parametric estimator for (squared) standard error in high frequency data. The device is related to observed information in likelihood theory, but in this case it is non-parametric and uses the high-frequency data structure. An earlier paper has developed the estimator in the case where edge effects are small to moderate. In practical data, it is often more realistic to assume that edge effects can be large, and this is the problem that we tackle in the current paper. We here find a regression approach to observed AVAR which is highly robust to large edges. This approach covers most high frequency estimators.

The median lethal dose (LD50) is commonly used to indicate acute toxicity of an insecticide to an insect species. Approximate confidence intervals for LD50s are often calculated using the Fieller and delta methods. It is often necessary to compare the relative potencies of several insecticides with a population or of one insecticide with different populations. Comparing the LD50s using probit/logit–log(dose) regressions with parallel slopes can be implemented in many software packages, but for the cases with arbitrary slopes are not generally available. We used the glm function in R to calculate and compare lethal doses without assuming equal slopes. Bioassay datasets from the literature fitted using the logit model gave the 95% confidence limits (95% CLs) for the lethal doses using Fieller’s theorem and incorporating a heterogeneity factor identical to the 95% CLs determined using the PoloPlus software. The delta method gave 95% CLs identical to the 95% CLs determined using the R drc package. The same datasets fitted using the probit model gave 95% CLs similar to the 95% CLs determined using PoloPlus and the drc package. The natural response rates for the control group were included using Abbott’s equation. When the potency ratio method and the z-test were used to identify differences between two lethal doses, and when the χ2 and log likelihood ratio tests were used to determine whether the regression lines were parallel, the conclusions were the same as those gave by PoloPlus and the drc package.

The accuracy of different downscaling methods in projecting future precipitation and air temperature from general circulation models (GCMs) has rarely been addressed with regards to the Tibetan Plateau, and this information is important for future water resource management in the region. The performance of automated statistical downscaling (ASD) and Delta downscaling methods in predicting precipitation and air temperature was evaluated at 19 meteorological stations in the Qilian Mountains and Hexi Corridor (QM-HC) by comparing with in situ observations from 2006-2015. These comparisons, based on Representative Concentration Pathway 4.5 (RCP4.5), suggest that the difference in annual precipitation between the ASD model and the Delta method is 17 mm. Testing different weights of the 2 downscaling methods indicates that combining the 2 methods results in lower uncertainty. The downscaling of annual precipitation projected by weighting the results of the 2 methods suggested that, based on RCP4.5, precipitation will not increase significantly from 2021-2100 compared to the past (1961-2005) and will fluctuate steadily in the coming decades. These projections are in contrast with previous projections of a significant increase. Air temperature is projected to increase by approximately 0.2°C decade-1 from 2021-2100 according to the weighted average of the ASD model and Delta method based on RCP4.5. This study indicates that management measures based on projected increased precipitation should be carefully reconsidered in different regions.

In this paper we consider clinical trials with two treatments and a non-normally distributed response variable. In addition, we focus on ap- plications which include only discrete covariates and their interactions. For such applications, the semi-parametric Area Under the ROC Curve (AUC) regression model proposed by Dodd and Pepe (2003) can be used. However, because a logistic regression procedure is used to obtain parameter estimates and a bootstrapping method is needed for computing parameter standard errors, their method may be cumbersome to implement. In this paper we propose to use a set of AUC estimates to obtain parameter estimates and combine DeLong's method and the delta method for computing parameter standard errors. Our new method avoids heavy computation associated with the Dodd and Pepe's method and hence is easy to implement. We conduct simulation studies to show that the two methods yield similar results. Fi- nally, we illustrate our new method using data from urinary incontinence clinical trials.

ObjectiveTo develop an index to evaluate the bias attributable to composite outcomes (BACO) in comparative clinical studies.Study design and SettingI defined the BACO index as the ratio of the logarithm of the association measure (e.g., relative risk) of the composite outcome to that of its most relevant component endpoint (e.g., mortality). Methods to calculate the confidence intervals and test the null hypotheses (BACO index = 1) were described and applied in systematically selected clinical trials. Two other preselected trials were included as “positive controls” for being examples of primary composite outcomes disregarded due to inconsistency with the treatment effect on mortality.ResultsThe BACO index values different from one were classified according to whether the use of composite outcomes overestimated (BACO index >1), underestimated (BACO index between zero and ConclusionBACO index can warn that the composite outcome association is stronger, weaker, or even opposite than that of its most critical component.What is new?BACO index is the ratio of the logarithm of the association measure (e.g., RR) of the composite outcome to the logarithm of the association measure of the component endpoint that represents the study target (e.g., mortality).In comparative studies, the BACO index can be used to evaluate the correspondence between the effect on a composite outcome and that on its most critical component.This index could help to preset rules to make decisions for interpretation of clinical studies.A significant BACO should lead to the caution that the association of the composite is stronger (BACO index >1), weaker (BACO index between zero and

A randomly truncated sample appears when the independent variables T and L are observable if L ＜ T. The truncated version Kaplan-Meier estimator is known to be the standard estimation method for the marginal distribution of T or L. The inverse probability weighted (IPW) estimator was suggested as an alternative and its agreement to the truncated version Kaplan-Meier estimator has been proved. This paper centers on the weak convergence of IPW estimators and variance decomposition. The paper shows that the asymptotic variance of an IPW estimator can be decomposed into two sources. The variation for the IPW estimator using known weight functions is the primary source, and the variation due to estimated weights should be included as well. Variance decomposition establishes the connection between a truncated sample and a biased sample with know probabilities of selection. A simulation study was conducted to investigate the practical performance of the proposed variance estimators, as well as the relative magnitude of two sources of variation for various truncation rates. A blood transfusion data set is analyzed to illustrate the nonparametric inference discussed in the paper.

We consider a general problem of the confidence interval for a cross-product ratio ρ=p1(1-p2)/p2(1-p1) according to data from two independent samples. Each sample may be obtained in the framework of direct Binomial sampling scheme. Asymptotic confidence intervals are constructed in accordance with direct Binomial sampling scheme, with parameter estimators demonstrating exponentially decreasing bias. Our goal is to investigate the cases when the normal approximations (which are relatively simple) for estimators of the cross-product ratio are reliable for the construction of confidence intervals. We use the closeness of the confidence coefficient to the nominal confidence level as our main evaluation criterion, and use the Monte-Carlo method to investigate the key probability characteristics of intervals corresponding to direct Binomial sampling schemes. We present estimations of the coverage probability, expectation and standard deviation of interval widths in tables and provide some recommendations for applying each obtained interval.

We study the selection of covariate adjustment sets for estimating the value of point exposure dynamic policies, also known as dynamic treatment regimes, assuming a non-parametric causal graphical model with hidden variables, in which at least one adjustment set is fully observable. We show that recently developed criteria, for graphs without hidden variables, to compare the asymptotic variance of non-parametric estimators of static policy values that control for certain adjustment sets, are also valid under dynamic policies and graphs with hidden variables. We show that there exist adjustment sets that are optimal minimal (minimum), in the sense of yielding estimators with the smallest variance among those that control for adjustment sets that are minimal (of minimum cardinality). Moreover, we show that if either no variables are hidden or if all the observable variables are ancestors of either treatment, outcome, or the variables that are used to decide treatment, a globally optimal adjustment set exists. We provide polynomial time algorithms to compute the globally optimal (when it exists), optimal minimal, and optimal minimum adjustment sets. Our results are based on the construction of an undirected graph in which vertex cuts between the treatment and outcome variables correspond to adjustment sets. In this undirected graph, a partial order between minimal vertex cuts can be defined that makes the set of minimal cuts a lattice. This partial order corresponds directly to the ordering of the asymptotic variances of the corresponding non-parametrically adjusted estimators., Comment: Fixed an error in Example 2

In the causal mediation framework, several parametric-regression–based approaches have been introduced in the last decade for estimating natural direct and indirect effects. For a binary outcome, a number of proposed estimators use a logistic model and rely on specific assumptions or approximations that may be delicate or not easy to verify in practice. To circumvent the challenges prompted by the rare outcome assumption in this context, an exact closed-form natural-effects estimator on the odds ratio scale was recently introduced for a binary mediator. In this work, we further push this exact approach and extend it for the estimation of natural effects on the risk ratio and risk difference scales. Explicit formulas for the delta method standard errors are provided. The performance of our proposed exact estimators is demonstrated in simulation scenarios featuring various levels of outcome rareness/commonness. The total effect decomposition property on the multiplicative scales is also examined. Using a SAS macro (SAS Institute, Inc., Cary, North Carolina) we developed, our approach is illustrated to assess the separate effects of exposure to inhaled corticosteroids and placental abruption on low birth weight mediated by prematurity. Our exact natural-effects estimators are found to work properly in both simulations and the real data example.

We use the jackknife to bias correct the log-periodogram regression(LPR) estimator of the fractional parameter in a stationary fractionally integrated model. The weights for the jackknife estimator are chosen in such a way that bias reduction is achieved without the usual increase in asymptotic variance, with the estimator viewed as `optimal' in this sense. The theoretical results are valid under both the non-overlapping and moving-block sub-sampling schemes that can be used in the jackknife technique, and do not require the assumption of Gaussianity for the data generating process. A Monte Carlo study explores the finite sample performance of different versions of the jackknife estimator, under a variety of scenarios. The simulation experiments reveal that when the weights are constructed using the parameter values of the true data generating process, a version of the optimal jackknife estimator almost always out-performs alternative semi-parametric bias-corrected estimators. A feasible version of the jackknife estimator, in which the weights are constructed using estimates of the unknown parameters, whilst not dominant overall, is still the least biased estimator in some cases. Even when misspecified short run dynamics are assumed in the construction of the weights, the feasible jackknife still shows significant reduction in bias under certain designs. As is not surprising, parametric maximum likelihood estimation out-performs all semi-parametric methods when the true values of the short memory parameters are known, but is dominated by the semi-parametric methods (in terms of bias) when the short memory parameters need to be estimated, and in particular when the model is misspecified., Comment: 57 pages

We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits.

Poisson's equation plays a fundamental role as a tool for performance evaluation and optimization of Markov chains. For continuous-time birth-death chains with possibly unbounded transition and cost rates as addressed herein, when analytical solutions are unavailable its numerical solution can in theory be obtained by a simple forward recurrence. Yet, this may suffer from numerical instability, which can hide the structure of exact solutions. This paper presents three main contributions: (1) it establishes a structural result (convexity of the relative cost function) under mild conditions on transition and cost rates, which is relevant for proving structural properties of optimal policies in Markov decision models; (2) it elucidates the root cause, extent and prevalence of instability in numerical solutions by standard forward recurrence; and (3) it presents a novel forward-backward recurrence scheme to compute accurate numerical solutions. The results are applied to the accurate evaluation of the bias and the asymptotic variance, and are illustrated in an example., 30 pages, no figures

Maximum likelihood estimators are calculated for the parameters that define the compound Poisson process in the classical risk process with exponential claims. It is proved consistency and asymptotic normality for estimators obtained. Finally, with the help of invariance property of the maximum likelihood estimators, asymptotic normality and delta method, point and interval estimation of the ruin probability is performed. Se calculan los estimadores de máxima verosimilitud para los parámetros que definen al proceso de Poisson compuesto en el proceso de riesgo clásico con reclamaciones exponenciales. Se prueba consistencia y normalidad asintótica de los estimadores obtenidos. Finalmente, con ayuda de la propiedad de invarianza de los estimadores de máxima verosimilitud, la normalidad asintótica y el método delta, se realiza una estimación puntual y por intervalos de la probabilidad de ruina.

Summary In multicentre research, individual-level data are often protected against sharing across sites. To overcome the barrier of data sharing, many distributed algorithms, which only require sharing aggregated information, have been developed. The existing distributed algorithms usually assume the data are homogeneously distributed across sites. This assumption ignores the important fact that the data collected at different sites may come from various subpopulations and environments, which can lead to heterogeneity in the distribution of the data. Ignoring the heterogeneity may lead to erroneous statistical inference. We propose distributed algorithms which account for the heterogeneous distributions by allowing site-specific nuisance parameters. The proposed methods extend the surrogate likelihood approach (Wang et al. 2017; Jordan et al. 2018) to the heterogeneous setting by applying a novel density ratio tilting method to the efficient score function. The proposed algorithms maintain the same communication cost as existing communication-efficient algorithms. We establish a nonasymptotic risk bound for the proposed distributed estimator and its limiting distribution in the two-index asymptotic setting, which allows both sample size per site and the number of sites to go to infinity. In addition, we show that the asymptotic variance of the estimator attains the Cramér–Rao lower bound when the number of sites is smaller in rate than the sample size at each site. Finally, we use simulation studies and a real data application to demonstrate the validity and feasibility of the proposed methods.

This paper obtains asymptotic results for parametric inference using prediction-based estimating functions when the data are high frequency observations of a diffusion process with an infinite time horizon. Specifically, the data are observations of a diffusion process at $n$ equidistant time points $\Delta_n i$, and the asymptotic scenario is $\Delta_n \to 0$ and $n\Delta_n \to \infty$. For a useful and tractable classes of prediction-based estimating functions, existence of a consistent estimator is proved under standard weak regularity conditions on the diffusion process and the estimating function. Asymptotic normality of the estimator is established under the additional rate condition $n\Delta_n^3 \to 0$. The prediction-based estimating functions are approximate martingale estimating functions to a smaller order than what has previously been studied, and new non-standard asymptotic theory is needed. A Monte Carlo method for calculating the asymptotic variance of the estimators is proposed., Comment: 26 pages

In this paper, we consider a comparison between two estimators of the parameter $$p$$ of the discrete Laplace distribution. A new method of moments estimator (MME) is derived and the asymptotic normality of its distribution is proven by applying the classical Delta method. The new MME is compared with the already known maximum likelihood estimator (MLE). Note that no accuracy properties of the MLE have been investigated before. The accuracy and the asymptotic normality of both estimators are investigated theoretically and using Monte Carlo simulation studies. We show that the MLE possesses better accuracy properties than the MME.