Key Statistic Research Articles

On the Estimation of Survival Rate Using Ranked Set Sampling Design

The Nigerian pensioners have high expectations from the government to ensure an effective implementation of pension regulations existing in the country. These expectations arise from the need to have sustainable standard of living in retirement and their benefits paid as at when due. However, the government does not know how many retirees will be alive at a particular period of time to draw fund from the pension account. This calls for regular verification of pensioners by Pension Transitional Arrangements Directorate (PTAD) at national level, office of the accountant general for State level and Local Government Pensions Board for Unified Local Government Council. This process exposed pensioners to stress related problems and some pensioners even die during such verification exercise. Again, it gives rise to varying set of problems that limit the capacity of the stakeholders within Nigeria pension industry to meet pensioner’s expectations. Survival rate of these retirees is key important statistic to the government for adequate preparation for their terminal benefits and to provide other social packages for this vulnerable group of people. However, this statistic is lacking in the State. Consequently, this work intends to address this problem by using the theory of Ranked Set Sampling (RSS) for Survival analysis to estimate the Survival rate of the retirees which will help the government in making adequate budgetary provisions to the Nigeria Pension Industry. Analysis and evaluation are presented.

Surrogate endpoint meta-regression: Useful statistics for regulators and trialists

ObjectivesThe main purpose of using a surrogate endpoint is to estimate the treatment effect on the true endpoint sooner than with a true endpoint. Based on a meta-regression of historical randomized trials with surrogate and true endpoints, we discuss statistics for applying and evaluating surrogate endpoints. MethodsWe computed statistics from two types of linear meta-regressions for trial-level data: simple random effects and novel random effects with correlations among estimated treatment effects in trials with more than 2 arms. A key statistic is the estimated intercept of the meta-regression line. An intercept that is small or not statistically significant increases confidence when extrapolating to a new treatment because of consistency with a single causal pathway and invariance to labeling of treatments as controls. For a regulator applying the meta-regression to a new treatment, a useful statistic is the 95% prediction interval. For a clinical trialist planning a trial of a new treatment, useful statistics are the surrogate threshold effect proportion, the sample size multiplier adjusted for dropouts, and the novel true endpoint advantage. ResultsWe illustrate these statistics with surrogate endpoint meta-regressions involving anti-hypertension treatment, breast cancer screening, and colorectal cancer treatment. ConclusionRegulators and trialists should consider using these statistics when applying and evaluating surrogate endpoints.

A short term credibility index for central banks under inflation targeting: An application to Brazil

Our goal is to provide econometric tools that could act as an almost real-time warning-system for central banks working under an Inflation-Targeting Regime. In any given month, it computes the probability that inflation will remain within the tolerance bounds set in advance by the Regime. So, our short-term index gives a proper response time for Central Banks, something long-term indices prevalent in the literature do not provide. Although we showcase Brazil in our application, our method could be broadly applied to other countries that operate under an Inflation-Targeting Regime. Our key statistic is the probability that inflation will be within the bounds in the next 1- 3- and 6-months ahead. It is based on predictive densities obtained from a mixed causal-noncausal autoregressive (MAR) model. We polish the accuracy of our key statistic using the receiver operating characteristic curve (ROC curve), something new in this literature.

Bootstrapping Long-Run Covariance of Stationary Functional Time Series

A key summary statistic in a stationary functional time series is the long-run covariance function that measures serial dependence. It can be consistently estimated via a kernel sandwich estimator, which is the core of dynamic functional principal component regression for forecasting functional time series. To measure the uncertainty of the long-run covariance estimation, we consider sieve and functional autoregressive (FAR) bootstrap methods to generate pseudo-functional time series and study variability associated with the long-run covariance. The sieve bootstrap method is nonparametric (i.e., model-free), while the FAR bootstrap method is semi-parametric. The sieve bootstrap method relies on functional principal component analysis to decompose a functional time series into a set of estimated functional principal components and their associated scores. The scores can be bootstrapped via a vector autoregressive representation. The bootstrapped functional time series are obtained by multiplying the bootstrapped scores by the estimated functional principal components. The FAR bootstrap method relies on the FAR of order 1 to model the conditional mean of a functional time series, while residual functions can be bootstrapped via independent and identically distributed resampling. Through a series of Monte Carlo simulations, we evaluate and compare the finite-sample accuracy between the sieve and FAR bootstrap methods for quantifying the estimation uncertainty of the long-run covariance of a stationary functional time series.

GEO-FPT: a model of the galaxy bispectrum at mildly non-linear scales

We present GEO-FPT (Geometric Fitted Perturbation Theory), a new model for the galaxy bispectrum anisotropic signal in redshift space, with functional form rooted in perturbation theory. It also models the dependence of the bispectrum with the geometric properties of the triangles in Fourier space, and has a broader regime of validity thanstate-of-the-art theoretical models based on perturbation theory. We calibrate the free parameters of this model using high-resolution dark matter simulations and perform stringent tests to show that GEO-FPTdescribes the galaxy bispectrum accurately up to scales of k ≃ 0.12hMpc-1 for different cosmological models, as well as for biased tracers of the dark matter field, considering a survey volume of 100 (Gpc h-1)3. In particular, a joint analysis ofthe power spectrum and bispectrum anisotropic signals, taking into account their full covariance matrix, reveals that therelevant physical quantities —the BAO peak position (along and across the line-of-sight), and the growth of structure parameters times the amplitude of dark matter fluctuations, fσ 8 — are recovered in an unbiased way, with an accuracy better than 0.4% and 2% respectively (which is our 2σ statistical limit of the systematic error estimate).In addition, the bispectrum signal breaks the fσ 8 degeneracy without detectable bias: f and σ 8 are recovered with better than 2.7% and 3.8% accuracy respectively (which is our 2σ statistical limit of the systematic error estimate).GEO-FPT boosts the applicability of the bispectrum signal of galaxy surveys beyond the current limitation of k ≲ 0.08hMpc-1 and makes the bispectrum a key statistic to unlock the information content from the mildly non-linear regime in the on-going and forthcoming galaxy redshift surveys.

Should higher education be subsidized more?

SUMMARY We investigate in which countries fiscal externalities provide a justification for increasing subsidies to higher education. First, we show that the marginal fiscal recovery rate, i.e., the ratio of the change in total net fiscal revenues and the change in total subsidy costs caused by a small change in tuition subsidies, is the key statistic: if larger than a well-specified threshold value, then a small increase in subsidies is desirable from a welfare point of view. We also show that the marginal fiscal recovery rate depends on three statistics: the elasticity of participation with respect to subsidies, the success probability of the marginal student, and the ratio of the net fiscal revenue gain and the subsidy level of a degree in higher education. Second, we use this formula to approximate the marginal fiscal recovery rate in twenty OECD countries. The average marginal fiscal recovery rate is equal to 0.94, meaning that, on average, 0.94 euro is recovered of an increase in subsidies by one euro. This average hides substantial heterogeneity between countries. In six countries, the marginal fiscal recovery rate is larger than one, implying that an increase in subsidies to higher education is unambiguously desirable (i.e., a Pareto improvement) in these countries. In the other 14 countries, the marginal fiscal recovery rate is below one. Yet, if the degree of inequality aversion is not extreme, then increasing subsidies is also desirable in 12 additional countries. Third, to check the quality of our approximation of the marginal fiscal recovery rate, we simulate it for Belgium (region of Flanders) on the basis of a more detailed model. This simulation provides a somewhat lower, yet fairly similar result compared to the approximation.

Polygenic risk prediction: why and when out-of-sample prediction R2 can exceed SNP-based heritability

In polygenic score (PGS) analysis, the coefficient of determination (R2) is a key statistic to evaluate efficacy. R2 is the proportion of phenotypic variance explained by the PGS, calculated in a cohort that is independent of the genome-wide association study (GWAS) that provided estimates of allelic effect sizes. The SNP-based heritability (hSNP2, the proportion of total phenotypic variances attributable to all common SNPs) is the theoretical upper limit of the out-of-sample prediction R2. However, in real data analyses R2 has been reported to exceed hSNP2, which occurs in parallel with the observation that hSNP2 estimates tend to decline as the number of cohorts being meta-analyzed increases. Here, we quantify why and when these observations are expected. Using theory and simulation, we show that if heterogeneities in cohort-specific hSNP2 exist, or if genetic correlations between cohorts are less than one, hSNP2 estimates can decrease as the number of cohorts being meta-analyzed increases. We derive conditions when the out-of-sample prediction R2 will be greater than hSNP2 and show the validity of our derivations with real data from a binary trait (major depression) and a continuous trait (educational attainment). Our research calls for a better approach to integrating information from multiple cohorts to address issues of between-cohort heterogeneity.

Time to acute general surgical review: a retrospective study in a tertiary referral centre.

The Acute Surgical Unit (ASU) is a busy service receiving Emergency Department (ED) referrals for adult and paediatric general surgery care alongside trauma. The ASU model deviates from the traditional on-call model and has been shown to improve efficiency and patient outcomes. The primary aim was to evaluate time to surgical review ED presentation and general surgical referral. Secondary aims were to assess referral numbers, pathology and demographics at our institution. A retrospective observational analysis was conducted on all referral times from the ED to the ASU between 1 April and 30 September 2022. Patient demographics, triage and referral times, and diagnoses were collected from the electronic medical record. Time between referral, review and surgical admission were calculated. A total of 2044 referrals were collected during the study period, and 1951 (95.45%) were included for analysis. Average time from ED presentation to surgical referral was 4 hours and 54 min with average time to surgical review from referral taking 40 min. On average, total time from ED presentation to surgical admission was 5 h and 34 min. Trauma Responds took 6 min to review. Colorectal pathology was the most commonly referred disease type. The ASU model is efficient and effective within our health service. Overall delays in surgical care may be external to the general surgery unit, or before the patient is made known to the surgical team. Analysis of time to surgical review is a key statistic in the delivery of acute surgical care.

Tuberculosis mortality: quantifying agreement in clinical cause of death assessments

ObjectivesMortality is a key statistic for public health globally, and mortality reduction is a key target of ‘End TB’ strategy. However, cause of death in relation to tuberculosis (TB) may be controversial, and we aimed to evaluate classification in Australia. MethodsWe surveyed Australian clinicians and public health officers, presenting a variety of scenarios. Respondents were asked to classify each scenario with regards to whether TB was considered causative, contributory or not related to death. ResultsFifty‐nine individuals completed the survey. Respondents were experienced TB clinicians and public health officers (median 14 years of TB care experience), with a majority having recently been involved in death certification/classification. In most scenarios, there was substantial variation, particularly where death was related to TB medications, or if an alternative contributing process was recognised, such as cardiovascular complications. Variation in classification was not evidently associated with classification experience. ConclusionWe found significant variation in cause of death classification among experienced TB clinicians and public health officers, using representative TB death scenarios. Implications for public healthConsensus and transparency with regards to classification would assist in more uniform cause of death classification across jurisdictions and allow for better tracking of this critical performance measure.

Estimating the COVID-19 infection fatality ratio accounting for seroreversion using statistical modelling

BackgroundThe infection fatality ratio (IFR) is a key statistic for estimating the burden of coronavirus disease 2019 (COVID-19) and has been continuously debated throughout the COVID-19 pandemic. The age-specific IFR can be quantified using antibody surveys to estimate total infections, but requires consideration of delay-distributions from time from infection to seroconversion, time to death, and time to seroreversion (i.e. antibody waning) alongside serologic test sensitivity and specificity. Previous IFR estimates have not fully propagated uncertainty or accounted for these potential biases, particularly seroreversion.MethodsWe built a Bayesian statistical model that incorporates these factors and applied this model to simulated data and 10 serologic studies from different countries.ResultsWe demonstrate that seroreversion becomes a crucial factor as time accrues but is less important during first-wave, short-term dynamics. We additionally show that disaggregating surveys by regions with higher versus lower disease burden can inform serologic test specificity estimates. The overall IFR in each setting was estimated at 0.49–2.53%.ConclusionWe developed a robust statistical framework to account for full uncertainties in the parameters determining IFR. We provide code for others to apply these methods to further datasets and future epidemics.

Transitioning the Healthy Chicago Survey From a Telephone Mode to Self-administered by Mail Mode.

As response rates to health surveys conducted by telephone continue to decline and costs continue to increase, practitioners are increasingly considering a transition to self-administered mail contact modes. To compare empirical differences observed across adjacent administrations of the Healthy Chicago Survey (HCS) conducted by telephone versus self-administered via mail contact. Data from the 2016, 2018, and 2020 administrations of the HCS are contrasted, and demographic distributions are benchmarked against the American Community Survey to investigate differences that may be linked to the HCS' transition from a telephone to self-administered mail mode between 2018 and 2020. All survey data were collected from adult residents of Chicago, Illinois, between 2016 and 2020. Costs, response rates, key health statistics, demographic distributions, and measures of precision generated from the HCS. The mail mode led to a response rate increase of 6.8% to 38.2% at half the cost per complete. Mail respondents are more likely to be nonminority, female, and hold a college degree. Key health statistic differences are mixed, but design effects are larger in the mail mode, which we attribute to more detailed geographic stratification and weighting employed in 2020. The mail mode is a less costly data collection strategy for the HCS, but it comes with trade-offs. The quasi-random selection of an individual in the household exacerbates sociodemographic distribution disparities.

The DRAKE mission: finding the frequency of life in the Cosmos

ABSTRACT In the search for life in the Universe, exoplanets represent numerous natural experiments in planet formation, evolution, and the emergence of life. This raises the fascinating prospect of evaluating cosmic life on a statistical basis. One key statistic is the occurrence rate of life-bearing worlds, fL, the ‘frequency of life’ term in the famous Drake Equation. Measuring fL would give profound insight into how common life is and may help us to constrain origin-of-life theories. I propose fL as the goal for the DRAKE mission (Dedicated Research for Advancing Knowledge of Exobiology): a transit spectroscopy survey of M-dwarf habitable zone terrestrial planets. I investigate how the uncertainty on the observed value of fL scales with sample size. I determine that sampling error dominates over observational error and that the uncertainty is a function of the observed fL value. I show that even small sample sizes can provide significant constraints on fL, boding well for the transit spectroscopy approach. I perform a feasibility study of the DRAKE mission using a nominal instrument design and mission plan. Due to low observing efficiencies, DRAKE may need to be incorporated into a wider-ranging deep-space or lunar observatory. A 50-planet survey could constrain fL to ≤ 0.06 (at 95 per cent confidence) if the sample fL = 0, or 0.03–0.2 if the sample fL = 0.1. This can be achieved (on average) in 10 yr using a 17-m telescope with an unrestricted field-of-regard. DRAKE is a viable approach to attempting the first experimental measurement of fL.

1452A Cohort Study of Comorbidities of Diabetes in Regional Australia

Abstract Background Epidemic of diabetes mellitus and its complications poses a major global threat and the burden of type 2 DM (T2DM) is also a worldwide phenomenon. However, data to rationalize continuous health promotion at local context such as regional New South Wales are lacking, and this underscores the need for continued epidemiological study. This study investigated if poorly controlled T2DM coexists with comorbidities more than those with good glycaemic control. Methods This was a cohort study at a private General Practice in Orange NSW, Australia. The study involved 137 de-identified adults living with T2DM. HbA1c and other biochemical and haematological records were reviewed. Other data included clinical notes including comorbidities, and complications. The cohort was separated into 2 groups: well-controlled versus poor glycaemic control. The resulting data were analysed for frequency, using the statistical software for Microsoft Excel. Results There were 37/137 of the T2DM cohort with baseline HbA1c of ≤ 6.5%, implying 73% poor glycaemic control in the region, at first time of testing. According to the clinical notes, the well-controlled group showed 29.7% comorbidities, including 2.7% queried infection. Among the 100 cases with abnormal HbA1c, comorbidities were observed in 22% including 4% who had infections. Conclusions Our observation implies no less comorbidities in well-controlled diabetes. However, the higher percentage of infection among the poorly controlled group is confirmation of need for continuous health promotion among people living with diabetes. Key messages This primary health statistic suggests the need to monitor for comorbidities equally, whether poor or well-controlled diabetes

The social cost of carbon dioxide under climate-economy feedbacks and temperature variability

A key statistic describing climate change impacts is the ‘social cost of carbon dioxide’ (SCCO2), the projected cost to society of releasing an additional tonne of CO2. Cost-benefit integrated assessment models that estimate the SCCO2 lack robust representations of climate feedbacks, economy feedbacks, and climate extremes. We compare the PAGE-ICE model with the decade older PAGE09 and find that PAGE-ICE yields SCCO2 values about two times higher, because of its climate and economic updates. Climate feedbacks only account for a relatively minor increase compared to other updates. Extending PAGE-ICE with economy feedbacks demonstrates a manifold increase in the SCCO2 resulting from an empirically derived estimate of partially persistent economic damages. Both the economy feedbacks and other increases since PAGE09 are almost entirely due to higher damages in the Global South. Including an estimate of interannual temperature variability increases the width of the SCCO2 distribution, with particularly strong effects in the tails and a slight increase in the mean SCCO2. Our results highlight the large impacts of climate change if future adaptation does not exceed historical trends. Robust quantification of climate-economy feedbacks and climate extremes are demonstrated to be essential for estimating the SCCO2 and its uncertainty.

Quick sequential organ failure assessment score combined with other sepsis-related risk factors to predict in-hospital mortality: Post-hoc analysis of prospective multicenter study data.

This study aimed to assess the value of quick sequential organ failure assessment (qSOFA) combined with other risk factors in predicting in-hospital mortality in patients presenting to the emergency department with suspected infection. This post-hoc analysis of a prospective multicenter study dataset included 34 emergency departments across Japan (December 2017 to February 2018). We included adult patients (age ≥16 years) who presented to the emergency department with suspected infection. qSOFA was calculated and recorded by senior emergency physicians when they suspected an infection. Different types of sepsis-related risk factors (demographic, functional, and laboratory values) were chosen from prior studies. A logistic regression model was used to assess the predictive value of qSOFA for in-hospital mortality in models based on the following combination of predictors: 1) qSOFA-Only; 2) qSOFA+Age; 3) qSOFA+Clinical Frailty Scale (CFS); 4) qSOFA+Charlson Comorbidity Index (CCI); 5) qSOFA+lactate levels; 6) qSOFA+Age+CCI+CFS+lactate levels. We calculated the area under the receiver operating characteristic curve (AUC) and other key clinical statistics at Youden's index, where the sum of sensitivity and specificity is maximized. Following prior literature, an AUC >0.9 was deemed to indicate high accuracy; 0.7-0.9, moderate accuracy; 0.5-0.7, low accuracy; and 0.5, a chance result. Of the 951 patients included in the analysis, 151 (15.9%) died during hospitalization. The AUC for predicting in-hospital mortality was 0.627 (95% confidence interval [CI]: 0.580-0.673) for the qSOFA-Only model. Addition of other variables only marginally improved the model's AUC; the model that included all potentially relevant variables yielded an AUC of only 0.730 (95% CI: 0.687-0.774). Other key statistic values were similar among all models, with sensitivity and specificity of 0.55-0.65 and 0.60-0.75, respectively. In this post-hoc data analysis from a prospective multicenter study based in Japan, combining qSOFA with other sepsis-related risk factors only marginally improved the model's predictive value.

Global changes in soil organic carbon and implications for land degradation neutrality and climate stability

Soil organic carbon (SOC) is a critical indicator for healthy and fertile lands across the world. It is also the planet's largest terrestrial carbon pool, so any changes of this pool may have profound implications for both land productivity and climate stability. However, SOC changes have so far remained largely unexplored, although their understanding is essential for many international environmental policies. Here we investigate for the first time recent global SOC changes, based on some SOC stock interannual data that were processed for the 2001–2015 period on a planetary scale. We analysed the global SOC dynamics using the Mann-Kendall test and Sen's slope estimator, which are widely acknowledged to be reliable geostatistical tools for detecting various environmental trends from global to local scale. We explored SOC changes via three metrics (averages, quantities, areas) of negative and positive trends, but also of the balance between soil carbon trends, a key statistic for monitoring land quality stability and soil–atmosphere carbon fluxes in the global environmental policies. Globally, we estimated a net average decrease of −58.6 t C km2 yr−1, a total loss of ~3.1 Pg C, and an area affected by net SOC losses of ~1.9 million km2. Using this triple statistic, we found that 79% of countries worldwide have been affected by net declines of SOC after 2001, which suggests that halting land degradation and mitigating climate change through the SOC pathway are still far from being achieved by international policies.

Towards a better account and understanding of bus/rapid-transit interactions: The case of Los Angeles

Bus transfers (access) at rapid-transit stations represent more than one-third of total boardings in large, dispersed, polycentric cities in the U.S. Yet, many station-level access-mode statistics and person-trip surveys often ommitt bus access mode or confound it with walk access. The abscence of this key statistic in official survey reports results in an incomplete understanding of multimodal interactions and may mislead planners and decision-makers in crafting effective policies geared to increase rapid-transit ridership and performance. Study of eight large transit systems and further analysis of the Los Angeles case study, with its detailed 2012 bus-rail on-board survey, were conducted to improve our understanding and planning of bus/rapid-transit interactions in polycentric cities. In order to evaluate scope, share, and impact of bus feeder routes on station boardings, the author indexed bus access events in the original survey database to estimate the number of bus transfers (access) at station-level. The socio-economic profile of riders that accessed rapid transit via bus was also rendered and trip-origin spatial distributions by access mode was also identified and tabulated. In addition, the influence of different bus feeder services (e.g. local, rapid, express) on rapid-transit station boardings was assesed in multivariate generalized linear models. Descriptive and inferential statistics results indicate that in Los Angeles, >30% of patrons are estimated to access rapid-transit via a variety of bus services; bus network connectivity is highly significant in explaining rapid transit boardings with large effects comparable to key Transit Oriented Development (TOD) factors; and notable variance in the magnitude of effects exists depending on bus service type. Station-level boarding and bus-transfers predictive equations derived from multivariate regressions such as the ones illustrated in this paper, may be used for sketch planning purposes in direct-demand models. A more nuanced, periodic, and detailed compilation of statistics related to bus transfers at station-level (as well as that of other emerging and potentially disrupting access modes) would provide a better understanding and planning of large multi-modal transit systems in the U.S. It would also bring to light an important, but under-reported, set of users.

Two-point Statistics without Bins: A Continuous-function Generalization of the Correlation Function Estimator for Large-scale Structure

Abstract The two-point correlation function (2pcf) is the key statistic in structure formation; it measures the clustering of galaxies or other density field tracers. Estimators of the 2pcf, including the standard Landy–Szalay (LS) estimator, evaluate the 2pcf in hard-edged separation bins, which is scientifically inappropriate and results in a poor trade-off between bias and variance. We present a new 2pcf estimator, the Continuous-Function Estimator, which generalizes LS to a continuous representation and obviates binning in separation or any other pair property. Our estimator, inspired by the mathematics of least-squares fitting, replaces binned pair counts with projections onto basis functions; it outputs the best linear combination of basis functions to describe the 2pcf. The choice of basis can take into account the expected form of the 2pcf, as well as its dependence on pair properties other than separation. We show that the Continuous-Function Estimator with a cubic-spline basis better represents the shape of the 2pcf compared to LS. We also estimate directly the baryon acoustic scale, using a small number of physically motivated basis functions. Critically, this leads to a reduction in the number of mock catalogs required for covariance estimation, which is currently the limiting step in many 2pcf analyses. We discuss further applications of the Continuous-Function Estimator, including determination of the dependence of clustering on galaxy properties and searches for potential inhomogeneities or anisotropies in large-scale structure.

Should Higher Education Be Subsidized More?

Fiscal externalities arise if subsidies to higher education raise future net fiscal revenues. We investigate in which countries fiscal externalities provide a justification for increasing subsidies to higher education. First, we show that the marginal fiscal recovery rate, i.e. the ratio of the change in total net fiscal revenues and the change in total subsidy costs caused by a small change in tuition subsidies, is the key statistic: if larger than one, then a small increase in subsidies is unambiguously desirable. We also show that the marginal fiscal recovery rate depends on three sufficient statistics: the elasticity of participation with respect to subsidies, the success probability of the marginal student, and the ratio of the net fiscal revenue gain and the subsidy cost of a degree in tertiary education. Second, we use the sufficient statistics formula to approximate the marginal fiscal recovery rate in twenty OECD countries. The average marginal fiscal recovery rate is equal to 0.89, meaning that, on average, 0.89 euro is recovered of an increase in subsidies with one euro. This average hides substantial heterogeneity between countries. In six countries (Australia, Israel, the Netherlands, Ireland, the United Kingdom, and the United States), the marginal fiscal recovery rate is larger than one, implying that an increase in subsidies to higher education is unambiguously desirable in these countries. Third, to check the quality of our approximation, we also simulate the marginal fiscal recovery rate for one country (Belgium) on the basis of a more detailed model that allows for heterogeneity between students. Reassuringly, this simulation provides a roughly similar result than the approximation for this country. Moreover, the more detailed model allows for additional simulations (e.g., to compute a maximal tuition level) that are not feasible with the sufficient statistics formula.

Estimating high-dimensional covariance and precision matrices under general missing dependence

A sample covariance matrix S of completely observed data is the key statistic in a large variety of multivariate statistical procedures, such as structured covariance/precision matrix estimation, principal component analysis, and testing of equality of mean vectors. However, when the data are partially observed, the sample covariance matrix from the available data is biased and does not provide valid multivariate procedures. To correct the bias, a simple adjustment method called inverse probability weighting (IPW) has been used in previous research, yielding the IPW estimator. The estimator can play the role of S in the missing data context, thus replacing S in off-the-shelf multivariate procedures such as the graphical lasso algorithm. However, theoretical properties (e.g. concentration) of the IPW estimator have been only established in earlier work under very simple missing structures; every variable of each sample is independently subject to missingness with equal probability. We investigate the deviation of the IPW estimator when observations are partially observed under general missing dependency. We prove the optimal convergence rate Op( logp∕n) of the IPW estimator based on the element-wise maximum norm, even when two unrealistic assumptions (known mean and/or missing probabilities) frequently assumed to be known in the past work are relaxed. The optimal rate is especially crucial in estimating a precision matrix, because of the “meta-theorem” [26] that claims the rate of the IPW estimator governs that of the resulting precision matrix estimator. In the simulation study, we discuss one of practically important issues, non-positive semi-definiteness of the IPW estimator, and compare the estimator with imputation methods.