Selection Of Probability Distribution Research Articles

Modelling Diameter at Breast Height Distribution for Eight Commercial Species in Natural-Origin Mixed Forests of Ontario, Canada

Diameter at breast height (DBH) is a unique attribute used to characterize forest growth and development for forest management planning and to understand forest ecology. Forest managers require an array of DBHs of forest stands, which can be reconstructed using selected probability distribution functions (PDFs). However, there is a lack of practices that fit PDFs of sub-dominating species grown in natural mixed forests. This study aimed to fit PDFs and develop predictive models for PDF parameters, so that the predicted distribution would represent dynamic forest structures and compositions in mixed forest stands. We fitted three of the simplest forms of PDFs, log-normal, gamma, and Weibull, for the DBH of eight tree species, namely balsam fir (Abies balsamea [L.] Mill.), eastern white pine (Pinus strobus L.), paper birch (Betula papyrifera Marshall), red maple (Acer rubrum L.), red pine (Pinus resinosa Aiton), trembling aspen (Populus tremuloides Michx), and white spruce (Picea glauca [Moench] Voss), all grown in natural-origin mixed forests in Ontario province, Canada. We estimated the parameters of the PDFs as a function of DBH mean and standard deviation for these species. Our results showed that log-normal fit the best among the three PDFs. We demonstrated that the predictive model could estimate the recovered parameters unbiasedly for all species, which can be used to reconstruct the DBH distributions of these tree species. In addition to prediction, the cross-validated R2 for the DBH mean ranged between 0.76 for red maple and 0.92 for red pine. However, the R2 for the regression of the standard deviation ranged between 0.00 for red pine and 0.69 for sugar maple, although it produced unbiased predictions and a small mean absolute bias. As these mean and standard deviations are regressed with dynamic covariates (such as stem density and stand basal area), in addition to climate and static geographic variables, the predicted DBH distribution can reflect change over time in response to management or any type of disturbance in the regime of the given geography. The predictive model-based DBH distributions can be applied to the design of appropriate silviculture systems for forest management planning.

Hydrological drought assessment using the standardized groundwater index and the standardized precipitation index in the Berg River Catchment, South Africa

Study regionBerg River Catchment (BRC), South Africa (SA). Study focusHydrological droughts threaten water security and climate Change and groundwater abstractions may exacerbate their impact. Thus, early detection and prediction of droughts is essential. The integration of the Standardised Groundwater Index (SGI) with the Standardised Precipitation Index (SPI), has been applied worldwide, it but not explored in South Africa (SA). The aim of this study was to evaluate the integration of the SGI and SPI for hydrological drought assessment within the BRC, SA. New hydrological insights for the regionThe sensitivity of the SGI to selected Probability Distribution Functions (PDFs) was examined, resulting in the Gamma and Log-Normal PDFs being preferred in the BRC. SGI-Gamma and SPI-Gamma comparisons validated the SGI method and shed light on the key factors influencing groundwater-related droughts in the BRC. The following was revealed: Hydrological droughts in the BRC are influenced by climate factors, surface and groundwater interactions, and groundwater abstractions. When conducting hydrological drought assessments using the SGI and the SPI, the initial focus should be on catchment-scale assessments before expanding to national-scale evaluations. Groundwater in the BRC exhibit significant resilience to droughts induced by climate factors, and thus useful to mitigate the risk of future droughts. Finally, The SGI performed satisfactorily in the BRC with limited groundwater level data.

The Validity of the Poisson Distribution to Analyze Microbial Colony Counts on Agar Plates for Food Samples

Microbial colony counts of food samples in microbiological examinations are one of the most important items. The probability distributions for the colony counts per agar plate at the dilution of counting had not been intensively studied so far. Recently we analyzed the colony counts of food samples with several probability distributions using the Pearson's chi-square value by the "traditional" statistics as the index of fit [Fujikawa and Tsubaki, Food Hyg.Saf.Sc., 60, 88-95 (2019)]. As a result, the selected probability distributions depended on the samples. In this study we newly selected a probability distribution, namely a statistical model, suitable for the above data with the method of maximum likelihood from the probabilistic point of view. The Akaike's Information Criterion (AIC) was used as the index of fit. Consequently, the Poisson model were better than the negative binomial model for all of four food samples. The Poisson model was also better than the binomial for three of four microbial culture samples. With Baysian Information Criterion (BIC), the Poisson model was also better than these two models for all the samples. These results suggested that the Poisson distribution would be the best model to estimate the colony counts of food samples. The present study would be the first report on the statistical model selection for the colony counts of food samples with AIC and BIC.

Application of the Monte-Carlo Method to Assess the Operational Reliability of a Household-Constructed Wetland with Vertical Flow: A Case Study in Poland

The objective of this study was to model the operation of a vertical-flow constructed wetland (VF-CW) for domestic wastewater, using Monte-Carlo simulations and selected probability distributions of various random variables. The analysis was based on collected wastewater quality data, including the values of the pollutant indicators BOD5 (biochemical oxygen demand), CODCr (chemical oxygen demand), and TSS (total suspended solids), in the 2017–2020 period. Anderson–Darling (A–D) statistics were applied to assess the fit of the theoretical distributions to the empirical distributions of the random variables under study. The selection of the best-fitting statistical distributions was determined using the percentage deviation (PBIAS) criterion. Based on the analyses that were performed, the best-fitting statistical distributions for the pollution indicators of the raw wastewater were the generalised extreme value distribution for BOD5, the Gaussian distribution for CODCr, and the log-normal distribution for TSS. For treated effluent, the log-normal distribution was the best fit for BOD5 and CODCr; the semi-normal distribution, for TSS. The new data generated using the Monte-Carlo method allowed the reliability of the VF-CW operation to be assessed by determining the reliability indices, i.e., the average efficiency of the removal of pollutants (η), the technological efficiency index (R), the reliability index (CR), and the risk index of the negative control of the sewage treatment plant operation (Re). The obtained results indicate that only in the case of CODCr, the analysed treatment facility may fail to meet the requirements related to the reduction of organic pollutants to the required level, which is evidenced by the values of the indicators CR = 1.10, R = 0.49, and η = 0.82. In addition, the risk index of the negative operation of the facility (Re) assumes a value of 1, which indicates that during the period of its operation, the VF-CW system will not operate with the required efficiency in relation to this indicator. The novelty of this work is the implementation of the indicated mathematical simulation methods for analysing the reliability of the operation of the domestic wastewater treatment facility.

Implementation of the Mean Time to Failure Indicator in the Control of the Logistical Support of the Operation Process

The focus of this paper is to identify a method for defining the needs of logistical operational support based on the mean time to failure (MTTF) factor. The research was based on a helicopter intended for flight training. The MTTF indicator for selected equipment was determined based on failure data from previous flight operations. As the basic operational data for the developed method, the time from the beginning of the operation or the flight time from the last damage and the method of restoring airworthiness were selected. The MTTF and replacement index for the device were determined. The next step was to determine the index, based on selected probability distributions. The results were analyzed and presented in graphical form, and conclusions were drawn. Based on the MTTF index and replacement index, the logistics needs of selected devices were determined. The obtained results were compared with the actual exchanges of devices made in the year in question. The research proved that the MTTF reliability factor and the analysis of trends in value changes could be used to determine the needs for the logistical security of the operation process, particularly in relation to the equipment subject to accidental failures. This is important for maintaining high availability of an aircraft or other technical objects.

Stochastic models of tropical cyclone genesis in the Philippine Sea

Climate change has influenced the anomalies of extreme weather events, including tropical cyclones (TCs), strong wind, flood and landslide. The investigation of TC genesis is of great significance in evaluating TC activity in order to mitigate TC – induced risks. This study focuses on developing the stochastic models to simulate TC geneses over the Philippine Sea, considering seasonal effects. The historical TC data, including TC number and genesis positions, were obtained from the Japanese Meteorology Agency (JMA) over the period 1951 – 2020. To select the optimal probability distributions to fit the observed TC data, the good-of-fit tests were performed throughout three metrics, namely Mean Absolute Error; Normalized Mean Squared Error and Correlation Distance. Moreover, the Chi-square test and the Kolmogorov – Smirnov test were utilized to evaluate the homogeneity of the selected probability distributions to the observed TC data. The statistical analysis on TC characteristics presented that most TC geneses occurred in the latitudinal belt 8°N - 30°N from July to October, called the active season (AS). Meanwhile, there was a low TC genesis frequency distributed in the region south of 15°N from November to June, called the inactive TC season (IAS). The stochastic models showed the distributions of the observed TC number followed Negative binomial (NB) distribution for the two seasons. Generalized extreme value (Gev) distribution fitted TC genesis longitude well for the inactive season and fitted TC genesis latitude well for the active season. In the active season, the kernel distribution was appropriate for genesis longitude, whereas in the inactive season, it was suitable for genesis latitude. Monte Carlo simulations for 1000 years indicated the similarities in the geographical distributions of the simulated TC geneses and the historical ones for the inactive and active seasons. Data Availability StatementThe Best Track data can be accessed from the Japanese Meteorology Agency. Data of the Philippine Sea boundary can be accessed from Marine Regions, governed by the Flanders Marine Institute.

Runoff Reduction Effects at Installation of LID Facilities under Different Climate Change Scenarios

In order to prepare for floods and droughts that occur as a result of climate change, various studies in water-related fields are being carried out in various countries around the world. Among them, special attention is being paid to the low-impact development (LID) technique. This study measured the annual maximum daily rainfall data from 2019 to 2100 by climate change scenario, which is the annual maximum daily rainfall series observed for rainfall stations, and tested the hydrological data using statistical analysis. After determining whether the data could be analyzed, the probability distribution was selected, and the parameters of the selected probability distribution were calculated using the L-moment method for each rainfall station. The probabilities of rainfall data were derived using GEV distribution, and the United States Environmental Protection Agency Storm Water Management Model (SWMM), a runoff simulation program, was used to compare and analyze the runoff reduction rate before and after the installation of a permeable pavement as an LID facility. The results of the analysis showed that representative concentration pathway (RCP) 4.5 and RCP 8.5 had the effect of reducing the runoff for more than 100 years at a 30% reduction rate compared with before installation.

The impacts of climate change on rainfall modeling in the Pantanal of Mato Grosso do Sul

The most significant and influential meteorological element in environmental conditions and human activities is precipitation. The objective of this study was to adjust eight probability distributions to monthly, seasonal and annual rainfall data in the Pantanal of Mato Grosso do Sul, Brazil, using a time series of data (1983-2013) by the National Meteorological Water Agency (ANA). The performance evaluation of different probability distribution models was assessed by the quality of fit of the selected probability distributions for precipitation data. Quality tests as chi-square, Kolmogorov-Smirnov (KS) and Anderson-Darling (AD), the information criteria as Akaike (AIC) and the Bayesian criterion (BIC) were used. Then the mean root square error (RMSE) and the coefficient of determination (R2) were applied. The analyzes were made monthly, annually and by seasons. The 3-parameter Lognormal distribution performs the best for all twelve months and provides the best-fit to the monthly rainfall data. Thus characterizing a dry period that runs from May to September and a rainy period between the months of October and April, it was observed that the 3-parameter Lognormal distribution has best adjustment for spring and summer, and for winter and autumn the 2-parameter Gamma and 3-parameter Gamma distribution performed better. For annual observations, the function that best fits is 3-parameter Weibull distribution.

Tsunami evacuation risk assessment and probabilistic sensitivity analysis using augmented sample-based approach

Tsunami evacuation is an effective way to save lives from the near-field earthquake-induced tsunami. To accurately assess tsunami evacuation risk, various uncertainties in evacuation need to be considered. For risk mitigation, it is also important to identify critical parameters (or risk factors) that contribute more to the evacuation risk to guide more effective tsunami evacuation. Probabilistic sensitivity analysis can be used for the latter. However, both risk assessment and sensitivity analysis require a large number of model evaluations and entail significant computational challenges, especially for the expensive evacuation model. This paper proposes an efficient augmented sample-based approach to address the above challenges. It only requires one set of samples/simulations (hence the high efficiency) to estimate the evacuation risk and calculate the sensitivity measures for all uncertain parameters. The approach is applied to estimate the tsunami evacuation risk for Seaside, Oregon, where a novel agent-based tsunami evacuation model is used to simulate the evacuation process more realistically. Various uncertainties in the evacuation process are explicitly quantified by properly selected probability distribution models. Besides the evacuation risk, critical risk factors are identified using probabilistic sensitivity analysis. The results provide important insights on tsunami evacuation and critical information for guiding effective evacuation risk mitigation.

Patterns of Recurrence of Phonotraumatic Vocal Fold Lesions Suggest Distinct Mechanisms of Injury.

To examine patterns of recurrence of benign phonotraumatic vocal fold lesions over time for insights into pathophysiology. Case series with mathematical modeling. Medical records and stroboscopic exams of adults who underwent microlaryngoscopic resection of phonotraumatic vocal fold lesions over a 13-year period were reviewed for time to recurrence after surgery. Uniform and log-normal probability distributions were fitted to the time to recurrence curves for vocal fold polyps, midfold masses, and pseudocysts. Model fits were compared using the Akaike information criterion corrected, a standard measure of the goodness of fit. Stochastic simulations were used to verify that the mechanistic hypotheses were concordant with the selected probability distributions and empiric data. Of 567 patients who underwent microlaryngoscopic resection, 65 had a recurrence (16 polyps, 14 midfold masses, and 35 pseudocysts). Midfold mass and pseudocyst recurrences were predominantly seen in younger women. Polyps were best fit by a uniform distribution rather than log-normal, whereas midfold masses and pseudocysts were better fit by log-normal rather than uniform. Stochastic simulations suggest that polyps recur sporadically according to a paroxysmal-developmental model, whereas midfold mass and pseudocyst recurrences follow a force-multiplication, damage-accumulation process. Vocal fold polyps are acute lesions evenly distributed by age and gender that recur uniformly over time, suggesting they arise from sudden tissue reactions to phonotraumatic stress. Pseudocysts and midfold fibrous masses are chronic lesions predominantly found in young women that recur with log-normal distribution over time, suggesting gradual damage accumulation in larynges predisposed to enhanced phonotrauma. 4 Laryngoscope, 131:2523-2529, 2021.

Investigation of the Wastewater Treatment Plant Processes Efficiency Using Statistical Tools

The paper presents modelling of wastewater treatment plant (WWTP) operation work efficiency using a two-stage method based on selected probability distributions and the Monte Carlo method. Calculations were carried out in terms of sewage susceptibility to biodegradability. Pollutant indicators in raw sewage and in sewage after mechanical treatment and biological treatment were analysed: BOD5, COD, total suspended solids (TSS), total nitrogen (TN) and total phosphorus (TP). The compatibility of theoretical and empirical distributions was assessed using the Anderson–Darling test. The best-fitted statistical distributions were selected using Akaike criterion. Performed calculations made it possible to state that out of all proposed methods, the Gaussian mixture model (GMM) for distribution proved to be the best-fitted. Obtained simulation results proved that the statistical tools used in this paper describe the changes of pollutant indicators correctly. The calculations allowed us to state that the proposed calculation method can be an effective tool for predicting the course of subsequent sewage treatment stages. Modelling results can be used to make a reliable assessment of sewage susceptibility to biodegradability expressed by the BOD5/COD, BOD5/TN and BOD5/TP ratios. New data generated this way can be helpful for the assessment of WWTP operation work and for preparing different possible scenarios for their operation.

Accuracy of eight probability distribution functions for modeling wind speed data in Djibouti

In this paper, the regional assessment on the performance of eight different probability distributions functions is investigated and compared for estimating wind speed distributions in Republic of Djibouti; the stations are located in the rural areas which are Ghoubet and Bada Wein, and urban areas which are University of Djibouti and International Airport of Djibouti. To achieve this aim, the statistical test for ranking the selected probability distributions functions, is evaluated based on the coefficient of determination, the root mean square error and the index of agreement. It has been shown from the statistical results that Weibull, Rayleigh and Gamma distributions can generally considered as the appropriate distributions and are generally provide the best fit for all stations; however Nakagami distribution gave the best results for Ghoubet rural station compared to the others used distributions.

Simple to Use Microsoft Excel Template for Estimating the Parameters of Some Selected Probability Distribution Model by Method of L-Moment

The focus of this research was to design a simple to use Microsoft excel algorithm that will aid in the estimation of the parameters of generalized extreme value probability distribution (GEV), generalized logistics probability distribution (GLO) and generalized pareto probability distribution (GPA), calculate the predicted rainfall/discharge based on L-moment procedures and compute the quantile estimates at various return periods.The algorithm was design based on the underlying mathematics of L-moment and has the capacity to handle forty (40) year’s annual maximum series of either rainfall or discharge data which must first be ranked in ascending order of magnitude. Basic descriptive statistics such as the sample mean, variance, standard deviation, skewness, kurtosis, coefficient of variation have been built into the algorithm. Other exciting features include; the computation of Probability weighted moment parameters (b0, b1, b2 and b3), L-Moment values (ƛ1, ƛ2, ƛ3 and ƛ4) , L-Moment ratio values (Ʈ2, Ʈ3 and Ʈ4), and goodness of fit statistics (RRMSE, RMSE, MAE, MADI and PPCC). Others include; the shape (k), scale (α) and location (ξ) parameters of GEV, GPA and GLO probability distributions. To test the performance of the algorithm, forty (40) year’s annual maximum rainfall data from Benin City was used. Basic time series analysis such as test of normality, test of homogeneity and outlier detection was conducted to ensure that the data used are adequate and suitable.Results obtained revealed that generalized logistics probability distribution GLO was the best fit distribution model for analyzing the annual maximum rainfall series at the study site. The predicted rainfall quantile magnitude (Qt) based on the GLO model ranges from 425.877mm at 2years return period to 762.759mm at 200years return period. The coefficient of determination (r 2 ) for the observed versus predicted rainfall based on the best fit model was observed to be 0.9793. It was thereafter concluded that L-moments and L –moment ratios are useful summary statistics for analyzing rainfall data. Keywords: L-moments, probability distribution, normality test, goodness of fit statistics, coefficient of variation. DOI : 10.7176/CER/11-9-05 Publication date :October 31 st 2019

Frequency Analysis of Storm-Surge-Induced Flooding for the Huangpu River in Shanghai, China

Shanghai, as a coastal city, is vulnerable to various types of flooding. The floodwalls along the Huangpu River provide protection against typhoon-induced flooding. However, there is limited insight into the actual safety level of the flood defences in Shanghai, and recent failures have highlighted their vulnerability. Therefore, the aims of this paper are to derive a series of new flood frequency curves for the Huangpu River, and to evaluate the level of protection of the floodwall system. This paper analysed over 100 years (1912–2013) of annual maximum water levels for three stations at near-sea, mid-stream and inland locations along the Huangpu River. Best-fit curves were determined for a number of selected probability distributions using statistical performance indicators. As a result, new flood frequency curves of the water levels for different storm surge return periods were produced. The results showed that generalised extreme value (GEV) was identified as the most suitable probability distribution for the datasets. Analysis showed that the current design water levels correspond to exceedance probabilities of 1/500 per year at the near-sea and mid-stream stations, and no more than 1/50 per year at the inland station of the Huangpu River, whereas the intended safety standard is 1/1000 per year. A comparison of the findings with a dataset of the crest heights of the floodwalls showed that the current protection level of the floodwalls along the Huangpu River is expected to be around 1/50 per year in terms of overtopping for the lowest sections. The results of this study can be utilized to provide future recommendations for flood risk management in Shanghai.

Frequency analysis of low flows in intermittent and non-intermittent rivers from hydrological basins in Turkey

Abstract This study attempts to find out the best-fit probability distribution function to low flows using the up-to-date data of intermittent and non-intermittent rivers in four hydrological basins from different regions in Turkey. Frequency analysis of D = 1-, 7-, 14-, 30-, 90- and 273-day low flows calculated from the daily flow time series of each stream gauge was performed. Weibull (W2), Gamma (G2), Generalized Extreme Value (GEV) and Log-Normal (LN2) are selected among the 2-parameter probability distribution functions together with the Weibull (W3), Gamma (G3) and Log-Normal (LN3) from the 3-parameter probability distribution function family. Selected probability distribution functions are checked for their suitability to fit each D-day low flow sequence. LN3 mostly conforms to low flows by being the best-fit among the selected probability distribution functions in three out of four hydrological basins while W3 fits low flows in one basin. With the use of the best-fit probability distribution function, the low flow-duration-frequency curves are determined, which have the ability to provide the end-users with any D-day low flow discharge of any given return period.

Prediction of urban residential end-use water demands by integrating known and unknown water demand drivers at multiple scales II: Model application and validation

Detailed prediction of end-use water demand at multiple spatial and temporal scales is essential for planning urban water supply using multiple water sources based on fit-for-purpose criteria. This paper presents the application of a stochastic model to predict urban residential end-use water demands at multiple spatial and temporal scales. The model includes an improved representation of spatial and temporal variability of urban residential water use by considering the effect of a significant number of water demand drivers such as household size, dwelling type, appliance efficiency, availability of water end-uses/appliances at dwellings, presence of children, presence of people at home, diurnal behavioral patterns and temperature. A stochastic approach is used to describe the variability of residential water demand that is not captured by these known explanatory variables. The model is validated against quarterly meter readings and hourly water use data. The validation of household water demand at a quarterly scale with billing data shows Correlation coefficients (R2) ranging between 90% and 96% and Nash-Sutcliffe coefficients ranging between 0.70 and 0.92 for the four seasons analyzed which, verifies the predictive capacity of the model. The model validation also demonstrates the statistical stability of the selected probability distributions used in modeling the unexplained behavior of urban residential water consumers. The hourly scale validation also demonstrates a satisfactory predictive capacity in predicting household water demand. This also evidences the effectiveness of the modeling approach to predict urban residential water demand at multiple temporal scales.

Qualitative probabilistic inference under varied entropy levels

In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to yield probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems.

Hyperspectral Image Analysis by Spectral–Spatial Processing and Anticipative Hybrid Extreme Rotation Forest Classification

Recent classification-oriented proposals to thematic maps building from hyperspectral images have used both semisupervised approaches and spatial information for correction of spectral classification. Semisupervised approaches enrich the training data set adding similar samples to each class, whereas spatial correction is based on the natural assumption of thematic class spatial compactness. In this paper, we propose and validate the following innovations: 1) a new spectral classifier, which is called anticipative hybrid extreme rotation forest (AHERF); 2) a spatial-spectral semisupervised approach; and 3) a final spatial classification correction step. The novel heterogeneous ensemble learning approach AHERF starts with a model selection phase, using a small subsample of the training data, in order to define a ranking-based selection probability distribution of the classifier architectures that will be used in the ensemble, so that the architecture best adapted to the data domain will be used more frequently to train individual classifiers in the ensemble. After this initial phase, AHERF trains a heterogeneous ensemble applying random rotations to bootstrapped samples of the remaining training data, aiming to obtain diversified and data-domain adapted individual classifiers. The natural assumption that spatially close pixels will most likely have highly correlated values is exploited in two phases of the process pipeline. First, semisupervised label assignment is supported by spectral similarity and spatial proximity. Unsupervised spectral similarity is detected by latent class discovery. In this paper, we use a clustering algorithm (i.e., k-means). Second, maximizing class spatial compactness removes classification errors that appear as speckle noise in the classification image. The whole approach aims to use minimal sets of labeled pixels for training, which we call the seed training data set. Testing results are computed over the entire image ground truth. For comparison, we provide results in several steps: 1) of classification by AHERF and competing classifiers built by semisupervised training and 2) after spatial correction. We validate the approach on several conventional benchmarking images, achieving results which are comparable with state-of-the-art approaches.

Comparison of the effectiveness of forecasts obtained by means of selected probability functions with respect to forecast error distributions

ABSTRACTThe forecasting of sales in a company is one of the crucial challenges that must be faced. Nowadays, there is a large spectrum of methods that enable making reliable forecasts. However, sometimes the nature of time series excludes many well-known and widely used forecasting methods (e.g., econometric models). Therefore, the authors decided to forecast on the basis of a seasonally adjusted median of selected probability distributions. The obtained forecasts were verified by means of distributions of the Theil U2 coefficient and unbiasedness coefficient.

Combining Inverse Probability Weighting and Multiple Imputation to Improve Robustness of Estimation

Inverse probability weighting (IPW) and multiple imputation are two widely adopted approaches dealing with missing data. The former models the selection probability, and the latter models data distribution. Consistent estimation requires correct specification of corresponding models. Although the augmented IPW method provides an extra layer of protection on consistency, it is usually not sufficient in practice as the true data‐generating process is unknown. This paper proposes a method combining the two approaches in the same spirit of calibration in sampling survey literature. Multiple models for both the selection probability and data distribution can be simultaneously accounted for, and the resulting estimator is consistent if any model is correctly specified. The proposed method is within the framework of estimating equations and is general enough to cover regression analysis with missing outcomes and/or missing covariates. Results on both theoretical and numerical investigation are provided.