Percentile Estimates Research Articles

Migraine and work - beyond absenteeism: Migraine severity and occupational burnout - A cohort study.

Migraine, a condition affecting 12% of the population, is a prevalent cause of disability, significantly impacts individuals during their most productive working years. Several studies have established that a migraine patient's job performance is often limited by absenteeism and presenteeism. The present study aimed to investigate the impact of migraines on occupational burnout, which affects up to 40% of workers. A subset of participants from the Negev Migraine Cohort, including both migraine patients and non-migraine controls, were asked to complete the study questionnaire. The main exposures of interest were migraine diagnosis and severity. The primary outcome was occupational burnout. Migraine severity and associated disability were evaluated using the Migraine Disability Assessment (MIDAS) score, psychiatric comorbidities using the Depression, Anxiety and Stress Scale - 21 Items (DASS-21) scale and occupational burnout using the Maslach Burnout Inventory (MBI-GS version) scale. Statistical analyses included multivariable quantile regression models to identify associations and adjust for potential confounders. In total, 675 migraine patients and 232 non-migraine participants participated in the study. Migraine patients exhibited higher rates of depression (mean DASS-21: 0.864 vs. 0.664, standardized mean difference (SMD) = 0.262), tended to work longer hours (median weekly hours: 40.0 vs. 36.0, SMD = 0.148) and expressed a preference for remote work (20.3% vs. 10.3%, SMD = 0.097). Migraine patients reported significantly higher levels of occupational burnout (mean burnout score: 3.46 vs. 2.82, SMD = 0.469). Controlling for depression, anxiety and stress, migraine diagnosis (25th percentile estimate = 0.67, p = 0.002, 75th percentile estimate = 0.92, p = 0.032) and migraine severity (estimates: 2.2-5.3, p < 0.001 for all) were associated with higher levels of occupational burnout. Migraine diagnosis and severity is associated with an occupational burnout, after controlling for various psychological and work-related factors. The findings underscore the need for workplace adjustments to support migraine patients' participation in the work market.

Evaluating the impact of anatomical and physiological variability on human equivalent doses using PBPK models.

Addressing human anatomical and physiological variability is a crucial component of human health risk assessment of chemicals. Experts have recommended probabilistic chemical risk assessment paradigms in which distributional adjustment factors are used to account for various sources of uncertainty and variability, including variability in the pharmacokinetic behavior of a given substance in different humans. In practice, convenient assumptions about the distribution forms of adjustment factors and human equivalent doses (HEDs) are often used. Parameters such as tissue volumes and blood flows are likewise often assumed to be lognormally or normally distributed without evaluating empirical data for consistency with these forms. In this work, we performed dosimetric extrapolations using physiologically based pharmacokinetic (PBPK) models for dichloromethane (DCM) and chloroform that incorporate uncertainty and variability to determine if the HEDs associated with such extrapolations are approximately lognormal and how they depend on the underlying distribution shapes chosen to represent model parameters. We accounted for uncertainty and variability in PBPK model parameters by randomly drawing their values from a variety of distribution types. We then performed reverse dosimetry to calculate HEDs based on animal points of departure for each set of sampled parameters. Corresponding samples of HEDs were tested to determine the impact of input parameter distributions on their central tendencies, extreme percentiles, and degree of conformance to lognormality. This work demonstrates that the measurable attributes of human variability should be considered more carefully and that generalized assumptions about parameter distribution shapes may lead to inaccurate estimates of extreme percentiles of HEDs.

The modified control charts for monitoring the error detection of process control under different estimators&amp;nbsp;

The reflected power function distribution (RPFD) has increasing importance in practical life due to its application in diversified fields of life. Organisations often face difficulty monitoring operations to identify and remove errors during production. That is why there is a need to introduce control charts that effectively monitor the processes, mainly when the number of errors follows RPFD and the manufacturing process is in control. The current study suggested memory-based control charts as a solution to the problem. The control charts are based on the estimation methods and play a remarkable role in enhancing the machine process reliability. The parameters of RPFD are evaluated through the percentile estimator (PE) and modified maximum likelihood estimator methods (MMLM). Further, we create memory-based control charts, i.e., hybrid exponentially weighted moving average (HEWMA) and extended exponentially weighted moving average (EEWMA), using the PE and MMLM. The findings reflect that HEWMA control charts based on PE provide a better result in estimating the defects. The implications of the study will be helpful for practitioners and policy makers from reliability engineering, management sciences, and statisticians.

3-Monochloropropandiol and glycidyl esters in heat-processed oil-based food products: Exposure and risk

Fatty acid esters of 3-monochloro-1,2-propanediol (3-MCPDE) and glycidol (GE) are potentially harmful heat-induced contaminants produced during food processing. In this study, 100 heat-treated oil-based food samples covering fried, smoked, grilled, and baked food groups were collected in Koforidua, Ghana and analysed for the two esters using gas chromatography-mass spectrometry. The dietary exposures were estimated by a probabilistic approach using Monte Carlo Simulation. Levels of 3-MCPDE and GE in the foods ranged from <limit of detection (LOD) to 1.28 mg/kg and <LOD to 1.20 mg/kg, respectively, and the highest mean levels were found in smoked foods with concentrations of 0.91 mg/kg (3-MCPDE) and 0.61 mg/kg (GE). The modal dietary exposure to 3-MCPDE and GE in the different food groups ranged from 0.01 to 1.18 µg/kg bw/day and 0.01 to 0.8 µg/kg bw/day, respectively. The 95th percentile estimates for 3-MCPDE exposure (33 %) and GE margin of exposure (MoE) (92 %) contravened the health-based guidance values established by the Joint Food and Agriculture Organization/World Health Organisation Expert Committee on Food Additives (JECFA) and the European Food Safety Authority (EFSA). This suggests health concerns for heavy consumers, especially females and children aged 3 to 9 years. The food groups that contributed to these unsafe exposures were mostly fried and smoked foods. Therefore, reducing the levels of 3-MCPDE and GE in these foods is recommended to lower the health risk.

Statistical Inference for Heavy‐Tailed Burr X Distribution with Applications

In this article, we present a new distribution, the so‐called heavy‐tailed Burr X (HTBX) distribution. It comes from the newly discovered heavy‐tailed (HT) family of distributions. A notable feature is that the associated probability density function can have a right‐skewed distribution that approximates symmetry, unimodality, and decreasing values, which makes it well suited for modeling various datasets. The mathematical properties of the new distribution are obtained by calculating the quantile function, ordinary moments, incomplete moments, moment generating function, conditional moment, mean deviation, Bonferroni curve, and Lorenz curve. Extensive simulation was performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood estimation, least squares estimation, weighted least squares estimation, Cramer–von Mises estimation, Anderson–Darling estimation, maximum product of spacing estimation, and percentile estimation. The simulation results showed the computational efficiency of these strategies and showed that the maximum likelihood strategy of estimation is the best strategy. The utility and importance of the newly proposed model are demonstrated by analyzing three real datasets. The HTBX distribution is compared to several well‐known extensions of the Burr distribution such as exponentiated Kavya‐Manoharan Burr X, Kavya‐Manoharan Burr X, Burr X, Kumaraswamy Rayleigh, Kumaraswamy Burr III, exponentiated Burr III, Burr III, Kumaraswamy Burr‐II, Rayleigh, and HT Rayleigh models by using different measures. The numerical results showed that the HTBX model fit the data better than the other competitive models.

Spectral Doppler Parameters of Fetal Main Branch Pulmonary Artery at 20 to 40 Weeks of Gestation: Reference Ranges and Percentile Calculators

The published reference ranges for Doppler parameters of the fetal pulmonary artery (PA) are usually derived from small sample sizes with no practical standard score or percentile ranking, which hinders systematic comparisons of Doppler figures across different gestational ages (GAs). This study aimed to establish comprehensive reference ranges and provide a percentile ranking solution for key spectral Doppler parameters. This is a cross-sectional study of 465 uncomplicated singleton pregnancies during 20 to 40weeks of gestation. Spectral waveforms of the fetal main branch PA were obtained with a pulsed-wave Doppler interrogation site within 5mm from the vascular origin. Fifteen spectral Doppler parameters were identified. Associations between these parameters with GA and fetal heart rate were assessed and used to develop percentile calculators via different statistical models. The root mean squared error of each model was calculated to determine the best performance solution. Acceptable spectral waveforms were obtained for 94.1% (438/465) of the fetuses. All Doppler parameters except pulsatility index, manually traced pulsatility index, peak systolic velocity, and time to systolic notch/acceleration time ratio were significantly correlated with GA, while acceleration time, ejection time, time to systolic notch, peak early-diastolic reversal flow, and peak early-diastolic reversal flow/peak systolic velocity ratio were additionally significantly correlated with fetal heart rate. Support vector machine models with radial basis kernel yield the best percentile estimation (root mean squared error of 2.17-4.08 and R2 of >0.98). Furthermore, the top 5% and bottom 5% outliers could be identified with positive predictive values of 0.71 to 0.97. An online user interface of percentile calculators is available at https://github.com/cmb-chula/fetoPAD. This study presents normal reference ranges and percentile calculators for 15 spectral Doppler parameters of the fetal main branch PA, some of which have not been published. The estimated percentiles enhance comparison and outlier detection of the spectral Doppler figures among fetuses at different GAs.

Combined chronic dietary exposure to four nephrotoxic metals exceeds tolerable intake levels in the adult population of 10 European countries

A mixture risk assessment (MRA) for four metals relevant to chronic kidney disease (CKD) was performed. Dietary exposure to cadmium or lead alone exceeded the respective reference values in the majority of the 10 European countries included in our study. When the dietary exposure to those metals and inorganic mercury and inorganic arsenic was combined following a classical or personalised modified reference point index (mRPI) approach, not only high exposure (95th percentile) estimates but also the mean exceeded the tolerable intake of the mixture in all countries studied. Cadmium and lead contributed most to the combined exposure, followed by inorganic arsenic and inorganic mercury. The use of conversion factors for inorganic arsenic and inorganic mercury from total arsenic and total mercury concentration data was a source of uncertainty. Other uncertainties were related to the use of different principles to derive reference points. Yet, MRA at the target organ level, as performed in our study, could be used as a way to efficiently prioritise assessment groups for higher-tier MRA. Since the combined exposure to the four metals exceeded the tolerable intake, we recommend a refined MRA based on a common, specific nephrotoxic effect and relative potency factors (RPFs) based on a similar effect size.

Eliminate the Heterogeneous Variances Effect Using Quantile Regression

Quantile regression (QR) is a statistical method that addresses the issue of inconsistent data errors. QR utilizes minimum absolute deviation to reduce the absolute deviation by employing percentile estimators such as the median, 1st and 3rd quartiles, and 10th and 90th percentiles. This study focuses on the qth percentile estimators of QR. QR models not only detect varying effects of explanatory variables at different quantiles of the response variable but also provide more robust and accurate estimates compared to mean regression when normality assumptions are violated or when outliers and long tails are present. This study conducts several simulation studies to compare the suggested qth percentile estimators of QR under different sample sizes, explanatory variables, and qth percentiles. Additionally, the study examines the impact of error terms dependency on normal/non-normal distribution. The asymptotic properties of these estimators are also investigated. The study concludes with a discussion of the advantages and disadvantages of using these qth percentile estimators of QR.

Baseline Gastric Volume in Fasting Diabetic Patients Is Not Higher than That in Nondiabetic Patients: A Cross-sectional Noninferiority Study.

The physiology of diabetes mellitus can increase the risk of perioperative aspiration, but there is limited and contradictory evidence on the incidence of "full stomach" in fasting diabetic patients. The aim of this study is to assess the baseline gastric content (using gastric ultrasound) in diabetic and nondiabetic patients scheduled for elective surgery who have followed standard preoperative fasting instructions. This was a prospective, noninferiority study of 180 patients (84 diabetic and 96 nondiabetic patients). Bedside ultrasound was used for qualitative and quantitative assessment of the gastric antrum in the supine and right lateral decubitus positions. Fasting gastric volume was estimated based on the cross-sectional area of the gastric antrum and a validated model. The hypothesis was that diabetic patients would not have a higher baseline fasting gastric volume compared to nondiabetic patients, with a noninferiority margin of 0.4 ml/kg. Secondary aims included the comparison of the incidence of full stomach (solid content or more than 1.5 mL/kg of clear fluid), estimation of the 95th percentile of the gastric volume distribution in both groups, and examination of the association between gastric volume, glycemic control, and diabetic comorbidities. The baseline gastric volume was not higher in diabetic patients (0.81 ± 0.61 ml/kg) compared to nondiabetic patients (0.87 ± 0.53 ml/kg) with a mean difference of -0.07 ml/kg (95% CI, -0.24 to 0.10 ml/kg). A total of 13 (15.5%) diabetic and 11 (11.5%) nondiabetic patients presented more than 1.5 ml/kg of gastric volume (95% CI for difference, -7.1 to 15.2%). There was little correlation between the gastric volume and either the time since diagnosis or HbA1C. The data suggest that the baseline gastric volume in diabetic patients who have followed standard fasting instructions is not higher than that in nondiabetic patients.

Error detection through modified phase II process monitoring under different classical estimators

In real life, the distribution of the errors during any life testing of products or process does not meet the assumption of normality. Statistical process control (SPC) is defined as the use of statistical techniques to control a process or production method. SPC tools and procedures can help to monitor process behavior, discover problems in internal systems, and find solutions for production issues. To identify and remove the variation in different reliability processes and to monitor the reliability of machines where the number of errors follows skewed distributions, we develop control charts to keep the process in control. For such situations, we have modified the existing control charts such as Shewhart control chart, exponentially weighted moving average (EWMA), hybrid exponentially weighted moving average (HEWMA) and extended exponentially weighted moving average (EEWMA) control charts. The current study introduced classical estimator based modified control charts for phase-II monitoring by assuming that the errors occur during the process follow skewed distribution called Beta Lehmann 2 Power function distribution (BL2PFD). The proposal for these control charts is based on the percentile estimator. We have compared all these control charts using Monte Carlo simulation studies and real-life applications to compare the proposed control charts. This study shows that an EEWMA control chart based on PE performs better than Shewhart, EWMA and HEWMA control charts, when the underlying distribution of the errors in process monitoring follows BL2PFD. These findings can be useful for researchers and practitioners in dealing with production errors and optimizing the output.

Robust power spectral density estimation via a performance-weighted blend of order statistics

Loud transients introduce bias to background spectrum estimates based on the sample mean, like the Welch Overlapped Segment Averaging (WOSA) [1967]. The Schwock and Abadi [2021] Welch Percentile (SAWP) estimator avoids the loud transient bias by replacing the averaging of the WOSA with a scaled order statistic (OS) of the sample power spectrum. While the scaling ensures the SAWP is unbiased regardless of which OS is chosen, the 80th percentile minimizes the variance of the SAWP estimator in a scenario without loud transients. However, the 80th percentile SAWP is still vulnerable to bias and increased variance from more frequent loud transients. Also, the rate of occurrence of loud transients may change with time, requiring the SAWP to adapt which percentile is used. To approach this challenge, this talk proposes a Universal SAWP estimator which is a weighted sum across different fixed percentile SAWP estimators. At each iteration, the Universal SAWP updates the blend weights to promote the percentile with the lowest sample variance over recent observations. In computer simulations, the Universal SAWP quickly assigns higher weights to the lowest variance estimators as the occurrence of loud transients increases. Overall, the Universal SAWP achieves equal or lower variance as the best fixed percentile SAWP estimator. [Work supported by ONR Code 321US.]

Estimation of the Density and Cumulative Distribution Functions of the Topp-Leone Distribution

In this paper, we estimate the probability density function and the cumulative distribution function of the Topp-Leone distribution. We use the maximum likelihood estimation, uniformly minimum variance unbiased estimation, least squares estimation, weighted least squares estimation, Cramér-von Mises estimation, Anderson-Darling estimation and method of percentile estimation. The consistency of these methods is illustrated in a simulation study. Finally, a real data set is given to assess the performance of proposed methods.

Different estimation techniques for constant-partially accelerated life tests of chen distribution using complete data

The issue of various estimation techniques in constant partially accelerated life tests with complete data is the main subject of this research. The Chen distribution is regarded as an item’s lifetime under use conditions. To estimate the distribution parameters and the acceleration factor, maximum likelihood estimation, least square estimation, weighted least square estimation, Cramér Von–Mises estimation, Anderson–Darling estimation, right-tail Anderson–Darling estimation, percentile estimation, and maximum product of spacing estimation are presented for classical estimation. For illustrative purposes, two real data sets are analyzed. The investigation of the two real data sets reveals that the suggested techniques are practical and can be used to solve some engineering-related issues. In order to compare the results of the several estimation techniques that have been offered based on mean square error and absolute average bias, a simulation study is presented at the end. When adopting the smallest values for mean square error and absolute average bias, this study demonstrates that maximum product of spacing estimation is the technique that is most effective among the alternatives in most cases.

A two‐parameter parsimonious bathtub model for the analysis of system failure times with illustrations to reliability data

AbstractIn this study, a two‐parameter, upper‐bounded probability distribution called the tau distribution is introduced and its applications in reliability engineering are presented. Each of the parameters of the tau distribution has a clear semantic meaning. Namely, one of them determines the upper bound of the distribution, while the value of the other parameter influences the shape of the cumulative distribution function. A remarkable property of this new probability distribution is that its probability density function, survival function, hazard rate function (HRF), and quantile function can all be expressed in terms of its cumulative distribution function. The HRF of the proposed probability distribution can exhibit an increasing trend and various bathtub shapes with or without a low and long‐flat phase (useful time phase), which makes this new distribution suitable for modeling a wide range of real‐world problems. The constraint maximum likelihood estimation, percentile estimation, approximate Bayesian computation, and approximate quantile estimation computation are proposed to calculate the unknown parameters of the model. The suitability of the estimation methods is verified with the aid of simulation and real‐world data results. The modeling capability of the tau distribution was compared with that of some well‐known two‐ and three‐parameter probability distributions using two data sets known from the literature of reliability engineering: time between failures data of a machining center, and time to failure of data acquisition system cards. Based on empirical results, the new distribution may be viewed as a viable competitor to the Weibull, Gamma, Chen, and modified Weibull distributions.

Spatio-temporal trends of the age-at-menarche percentiles among Portuguese women since 1920

BackgroundThis work aims to study the spatio-temporal evolution of a woman’s age at menarche in the central region of Portugal. One of the concerns of the study is early or late menarches; thus, we consider percentile regression to build the respective curves as opposed to the more traditional mean regression approach.MethodsWe analysed the data from N=452,348 women born in the period 1920–1973 who attended a free breast cancer screening program between 1990 and 2019. Distributional regression models inside the package GAMLSS in R were considered. These methods allowed us not only to model the location (mean) of the specific probability distribution of the age at menarche, but also allowed for the scale (variance) parameter of this distribution to depend on covariates. Additionally, a spatial random-effect was considered in order to capture the correlation at the regional level. The obtained clustered spatial effects were analysed to assess geographical differences among the percentiles of the age at menarche by year of birth.ResultsA decreasing trend in the age at menarche (about 1.5 years in 5 decades) and regional differences for all the considered percentiles were found. Women living in the north-central areas of the central region of Portugal tend to have menarche at older ages.ConclusionWe obtained percentile estimates for the age at menarche by year of birth and region of residence and demonstrated that these two explanatory variables have an impact on the explanation about the decreasing trend in age at a woman’s first menstruation.

Bayesian analysis of accelerated life test under constrained randomization

Reliability engineers typically prioritize the lower percentiles. Accurately assessing these lower percentiles enables engineers to delve deeper into early product failures, paving the way for enhanced product reliability. In the manufacturing realm, many accelerated life tests (ALTs) veer away from completely randomized designs (CRDs) due to constraints in time and budget. Within ALTs, alterations in stress can lead to shifts in the failure mechanism of products. To accurately discern product lifetime percentiles, there is an imperative need to account for these varying failure mechanisms and random effects. Our approach introduces a re-parameterization model encapsulating random effects and disparate failure mechanisms. In this model, a specific percentile is employed as a stand-in for the scale parameter, laying the groundwork for a regression model interlinking the percentile, acceleration stress, and random effect. Concurrently, a separate regression model is designed for shape parameters in relation to acceleration stresses. Leveraging the Bayesian method, we ascertain the estimated values for the model parameters. The model is applied to an ALT example focusing on glass capacitors. The simulations underline the model’s prowess in delivering a more precise estimation of lower lifetime percentiles. Additionally, the Bayesian method further refines the accuracy of the lifetime percentile estimations.

From Immunosenescence to Aging Types-Establishing Reference Intervals for Immune Age Biomarkers by Centile Estimation.

Immunological aging type definition requires establishing reference intervals from the distribution of immunosenescence biomarkers conditional on age. For 1605 individuals (18-97 years), we determined the comprehensive immune age index IMMAX from flow-cytometry-based blood cell sub-populations and identified age-specific centiles by fitting generalized additive models for location, scale, and shape. The centiles were uncorrelated with age and facilitated the categorization of individuals as immunologically slow or fast aging types. Using its 50th percentile as a reference, we rescaled the IMMAX to equivalent years of life (EYOL) and computed the immunological age gap as the difference between EYOL and chronological age. Applied to preliminary baseline and follow-up measurements from 53 participants of the Dortmund Vital Study (Clinical-Trials.gov Identifier: NCT05155397), the averaged changes in the IMMAX and EYOL conformed to the 5-year follow-up period, whereas no significant changes occurred concerning IMMAX centiles and age gap. This suggested that the participants immunologically adapted to aging and kept their relative positions within the cohort. Sex was non-significant. Methodical comparisons indicated that future confirmatory analyses with the completed follow-up examinations could rely on percentile curves estimated by simple linear quantile regression, while the selection of the immunosenescence biomarker will greatly influence the outcome, with IMMAX representing the preferable choice.

A tutorial on automatic post-stratification and weighting in conventional and regression-based norming of psychometric tests.

Norm scores are an essential source of information in individual diagnostics. Given the scope of the decisions this information may entail, establishing high-quality, representative norms is of tremendous importance in test construction. Representativeness is difficult to establish, though, especially with limited resources and when multiple stratification variables and their joint probabilities come into play. Sample stratification requires knowing which stratum an individual belongs to prior to data collection, but the required variables for the individual's classification, such as socio-economic status or demographic characteristics, are often collected within the survey or test data. Therefore, post-stratification techniques, like iterative proportional fitting (= raking), aim at simulating representativeness of normative samples and can thus enhance the overall quality of the norm scores. This tutorial describes the application of raking to normative samples, the calculation of weights, the application of these weights in percentile estimation, and the retrieval of continuous, regression-based norm models with the cNORM package on the R platform. We demonstrate this procedure using a large, non-representative dataset of vocabulary development in childhood and adolescence (N = 4542), using sex and ethnical background as stratification variables.

Anthropometric and Physical Profile Among the Different Age Groups of Tunisian Tennis Players

Background: The popularity of Tunisian tennis is on the rise, due to emerging players gaining global recognition, leading to growth and evolution in the sport. In order to optimize training and performance, coaches need valuable information on physical fitness and player profiling. Objectives: The study aimed to investigate the physical profile of Tunisian tennis players in different age groups ranging from 7 to 17 years. Methods: One hundred and one tennis players from the same team were assigned to six age groups (U9-U18); U9 (7.89 ± 0.32), U10 (8.88 ± 0.34), U11 (9.78 ± 0.43), U12 (10.81 ± 0.4), U14 (12.3 ± 0.73), and U18 (14.69 ± 0.95) and were assessed for standing long jump (SLJ), countermovement jump, medicine ball throw (MBT), various sprint distances, agility drills, 20m Shuttle run (20mSRT), and Sit and reach tests. Results: The results showed that U14 and U18 age groups demonstrated better performance than U9 and U10 in jump tests, while U18 and U14 outperformed only U9 in the MBT test. U11 showed better performance than U9 and U10 in sprints, sideways shuffle, and spider drill tests. U9 and U10 had lower performance than U12 in SLJ, 20m sprint, sideways shuffle, and zigzag tests, and U11 outperformed U9 in the SLJ test. U18 and U14 had higher VO2max than U10 and U11, while U12 was higher than U11 only. No difference in flexibility was reported. The centile estimates of physical performances among Tunisian tennis players provide a means to compare individual player test results with the standard performance levels of the group. Conclusions: In conclusion, this study found that 11–17-year-old players exhibited greater physical performance than 7–10-year-old players. However, no age effect was found for flexibility.

Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and Cumulative Estimation in High Dimensional Data.

The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is a fundamental tool in characterizing the convergence rate of the local linear estimation. Precisely, we use this tool to derive the uniform convergence rate of the local linear estimation of the conditional cumulative distribution function and the local linear estimation conditional quantile function. Second, a central limit theorem for the proposed estimators is established. These results are proved under general assumptions, allowing for the incomplete functional time series case to be covered. Specifically, we model the correlation using the ergodic assumption and assume that the response variable is collected with missing at random. Finally, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimators.