Open AccessArticle

Calibration Estimation of Cumulative Distribution Function Using Robust Measures

Hareem Abbasi

¹,

Muhammad Hanif

¹,

Usman Shahzad

^1,*

Walid Emam

Yusra Tashkandy

Soofia Iftikhar

³ and

Shabnam Shahzadi

⁴

Department of Mathematics and Statistics, PMAS-Arid Agriculture University, Rawalpindi 46300, Pakistan

Department of Statistics and Operations Research, Faculty of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar 25120, Pakistan

⁴

Department of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, China

Author to whom correspondence should be addressed.

Symmetry 2023, 15(6), 1157; https://doi.org/10.3390/sym15061157

Submission received: 3 May 2023 / Revised: 17 May 2023 / Accepted: 23 May 2023 / Published: 26 May 2023

(This article belongs to the Section Mathematics)

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Versions Notes

Abstract

Outliers are observations that are significantly different from the other observations in a dataset. These types of observations are asymmetric in nature due to a lack of symmetry. The estimation of the cumulative distribution function (CDF) is an important statistical measure commonly discussed for symmetric datasets. However, the estimation of the CDF in the case of the asymmetric nature of the dataset is not a much-explored topic. In this article, we use calibration methodology with auxiliary information for modifying the traditional stratification weight, and hence, we obtain efficient estimates of the CDF using robust measures, i.e., mid-range and tri-mean, under the different distance functions. A simulation study is carried out to see the performance of proposed and existing estimators using asymmetric real-life datasets.

Keywords:

calibration estimation; auxiliary data; cumulative distribution function; simulation study

1. Introduction

Finding the percentage of research variables

Y

that are less than or equal to a specific value is important, and this leads to the estimation method of the countable population CDF. In some cases, it is thought necessary to estimate the CDF. For instance, a soil scientist would be curious to discover how many people in a developing nation are living in poverty. We are usually concerned with the percentage of

y_{i}

values in the population. In certain situations, the need for a CDF is more important. Users of sample survey data frequently need to calculate the population CDF or, alternatively, the percentage of population elements whose values are less than or equal to a certain value

t_{y}

. For instance, we might be interested in the percentage of agricultural land where pesticide poisoning effects are less than zero or the percentage of filtration facilities where arsenic is present in portable water that is less than zero. Such a percentage is a specific value of the population’s CDF.

F_{Y} (t_{y}) = \frac{1}{M} \sum_{i = 0}^{M} Ι (y_{i} \leq t_{y})

where

Ι (y_{i} \leq t_{y}) = 1

for

y_{i} \leq t_{y}

and

Ι (y_{i} \leq t_{y}) = 0

for

y_{i} > t_{y}

. In surveys, we can frequently only measure the research variable for those items in a sample; hence, the typical estimation methods of the CDF depend solely on the choice of the sampling design and the sampled percentage of the population.

F_{Y} (t_{y})

can be estimated by

{\hat{F}}_{y} (t_{y}) = \frac{1}{m} \sum_{i = 0}^{m} Ι (y_{i} \leq t_{y})

Many researchers have calculated the CDF using data from one or more additional variables. First of all, Reference [1] proposed a method for estimating the countable population CDF. Reference [2] obtained ratio and difference estimation methods for a population CDF under a general sample design using supplementary population variables. They demonstrated the benefits of the design-based estimation method over the model-based estimation method in the case of model misspecifications, especially for large samples. Reference [3] developed a traditional as well as a prediction technique for estimating the CDF from survey data. Reference [4] proposes an estimator for the finite population CDF using the model-calibration pseudo-empirical likelihood technique. Reference [5] considers the issue of estimating the CDF and quantiles for a countable population using supplementary data. Reference [6] develops a generalized family of estimation methods for estimating the CDF using auxiliary variables. Reference [7] develops an efficient approach for the estimation of process variability by using the exponential technique. Reference [8] developed two new families for the estimation of the countable population CDF in the case of non-response under simple random sampling. They studied two different types of non-response situations: (i) non-response on both the research and supplementary data; and (ii) non-response just on the research data. The developed estimation methods are compared to existing estimation methods, both theoretically and numerically. Reference [9] developed a new family of estimation methods for the finite CDF using the stratified random sampling (StRS) method. Reference [10] also proposed a generalized class of exponential factor type estimation methods for estimating the countable population CDF with supplementary information in the form of the average and rank of the supplementary information.

In recent years, the calibration estimation method has become an important area of study in survey sampling. By using auxiliary data, the calibration estimation technique increases the accuracy of estimations by adjusting the original design weights. The calibration estimation method is a procedure for adjusting survey sampling weights in order to simulate population means, totals, etc. with the help of supplementary data. The pioneering article on calibration was written by Reference [11]. Reference [12] developed a calibration estimation method for mean estimation. Reference [13] proposed a calibration estimation method for estimating the population mean in StRS with various calibration conditions based on supplementary information. Reference [14] proposes a novel calibration estimation method for the population parameter of the study variable using newly calibrated weights for two supplementary variables under StRS. Reference [15] proposes a distance function. Using their developed distance function, a calibration estimation method of the population mean in StRS is obtained. References [16,17] extended the work by utilizing linear moments’ characteristics. Reference [18] developed two novel classes of ratio- and regression-type estimation methods of population variation under SRSWOR by integrating knowledge on nonconventional and robust dispersion measures of supplementary data. Reference [19] proposes a new robust calibration estimation method for estimating the population mean under StRS. Reference [12] methodology for CDF estimation, however, has not received much attention yet.

This article proposes a new calibration estimation method for the population CDF under StRS using new calibration conditions that include robust measures. The use of robust measures makes the calibration estimator of CDF more efficient. The rest of the article is organized as follows: In Section 2, an adapted estimator of CDF using robust measures is shown. In Section 3, the proposed CDF using robust measure estimators is developed. In Section 4, a numerical study is conducted. The article concludes in Section 5.

2. First Adapted Calibration Estimator of CDF Using Robust Measure

Outliers can be caused by a variety of factors, such as measurement errors, sampling bias, or extreme values. As they belong to an asymmetric nature. So, they can have a long tail on one side or the other, indicating that there are more extreme values in one direction than the other. Outliers can have a major impact on statistical analyses, as they can distort summary statistics and lead to misleading conclusions. So, in this article, we will use robust measures such as the mid-range and tri-mean to reduce the impact of outliers.

Let

ϑ = 1,2, \dots, M

be a finite population

M

of units, which is divided into

γ

homogeneous strata, where the size of

φ^{t h}

stratum is

M_{φ}

, for

φ = 1,2, . ., γ

in such a manner that

\sum_{φ = 1}^{γ} M_{φ} = M

. Assume that

(Y, X)

are the study and auxiliary variables, respectively. The stratum weights are defined as

W_{φ} = \frac{M_{φ}}{M}

. The mid-range is defined as

M_{R} = \frac{X_{1 (1)} + X_{1 (M)}}{2}

where

X_{1 (1)}

is the minimum value in a population of size

M

and

X_{1 (M)}

is the maximum value in a population of size

M

. The next measure included in this article is the tri-mean

(T_{M})

, which is the weighted average of the population median and two quartiles and is defined as:

T_{M} = \frac{Q_{1} + 2 Q_{2} + Q_{3}}{4}

and

S_{φ x}^{2} = \frac{\sum_{φ = 1}^{γ} {(x_{φ i} - {\overline{x}}_{φ})}^{2}}{M_{φ} - 1} .

They denote the population variance of the supplementary variable in

φ^{t h}

stratum.

Under this StRS, the traditional unbiased estimator of the CDF is given by

T_{o} = \sum_{φ = 1}^{γ} W_{φ} {\hat{F}}_{y φ} (t_{y})

where

{\hat{F}}_{y φ} (t_{y}) = \frac{1}{m} \sum_{i = 0}^{m} Ι (y_{i} \leq t_{y})

is the sample CDF estimate of

Y

in the

φ^{t h}

stratum.

2.1. First Adapted Calibration Estimator of CDF

Taking motivation from Reference [15], the first adapted estimators are as follows:

G_{{R M}_{(A_{1})}} = \sum_{φ = 1}^{γ} Ω_{A_{1} φ} {\hat{F}}_{y φ} (t_{y})

(1)

where

{\hat{F}}_{y φ} (t_{y})

is the sample CDF of the study variable in

φ^{t h}

stratum. Further,

Ω_{A_{1} φ}

is the calibrated weight; we will use the sum of weighted squared deviation of calibrated weights function as given below:

\sum_{φ = 1}^{γ} S_{φ x}^{2} {{(Q_{φ})}^{- 1} (Ω_{A_{1} φ} - W_{φ})}^{2}

(2)

and satisfy the calibration constraint

\sum_{φ = 1}^{γ} Ω_{A_{1} φ} {\hat{M}}_{R φ (x)} = \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}

(3)

Note that

W_{φ} = \frac{M_{φ}}{M}

denote the traditional stratum weight,

({\hat{M}}_{R φ (x)}, M_{R φ (x)})

are presenting the sample and population mid-range of the supplementary variable in the

φ^{t h}

stratum, and

Q_{φ}

is suitably chosen weights to decide different types of estimation methods. The Lagrange function is given by

\begin{array}{l} ∆ (Ω_{A_{1} φ}, W_{φ}) & = & \sum_{φ = 1}^{γ} S_{φ x}^{2} {(Q_{φ})}^{- 1} {(Ω_{A_{1} φ} - W_{φ})}^{2} \\ - 2 λ_{A} (\sum_{φ = 1}^{γ} Ω_{A_{1} φ} {\hat{M}}_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}) \end{array}

(4)

where

λ_{A}

are multipliers of Lagrange and setting

\frac{\partial ∆ (Ω_{A_{1} φ}, W_{φ})}{\partial Ω_{A_{1} φ}} = 0

, we obtain

Ω_{A_{1} φ} = W_{φ} + λ_{A} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}

(5)

Substituting Equation (5) in Equation (3) and solving for lambda, we have

λ_{A} = \frac{\sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} {\hat{M}}_{R φ (x)}}{\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}}

(6)

Substituting Equation (6) in Equation (5), we obtain the calibration weight as

Ω_{A_{1} φ} = W_{φ} + \frac{\sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} {\hat{M}}_{R φ (x)}}{\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}} {\hat{M}}_{R φ (x)} Q_{φ} {(S_{φ x}^{2})}^{- 1}

(7)

Substituting Equation (7) in Equation (1), we obtain the calibrated estimator of CDF as given below:

G_{{R M}_{(A_{1})}} = \sum_{φ = 1}^{γ} W_{φ} {\hat{F}}_{y φ} (t_{y}) + \frac{\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})}{\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}} \sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y})

2.2. Second Adapted Calibration Estimator of CDF

Taking motivation from Reference [15], the second adapted estimators are as follows:

G_{{R M}_{(A_{2})}} = \sum_{φ = 1}^{γ} Ω_{A_{2} φ} {\hat{F}}_{y φ} (t_{y})

(8)

where

{\hat{F}}_{y φ} (t_{y})

is the sample CDF of the study variable in the

φ^{t h}

stratum. Further,

Ω_{A_{2} φ}

is the calibrated weight; we will use the sum of weighted squared deviation of calibrated weights function as given below:

\sum_{φ = 1}^{γ} S_{φ x}^{2} {{(Q_{φ})}^{- 1} (Ω_{A_{2} φ} - W_{φ})}^{2}

(9)

Subject to calibration constraints defined by

\sum_{φ = 1}^{γ} Ω_{A_{2} φ} {\hat{M}}_{R φ (x)} = \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}

(10)

\sum_{φ = 1}^{γ} Ω_{A_{2} φ} {\hat{T}}_{M φ (x)} = \sum_{φ = 1}^{γ} W_{φ} T_{M φ (x)}

(11)

({\hat{M}}_{R φ (x)}, M_{R φ (x)}), ({\hat{T}}_{M φ (x)}, T_{M φ (x)})

are presenting the sample and population mid-range and tri-mean of the supplementary variable in the

φ^{t h}

stratum. The Lagrange function is given by

\begin{array}{l} ∆ (Ω_{A_{2} φ}, W_{φ}) & = & \sum_{φ = 1}^{γ} S_{φ x}^{2} {{(Q_{φ})}^{- 1} (Ω_{A_{2} φ} - W_{φ})}^{2} \\ - 2 λ_{A_{1}} (\sum_{φ = 1}^{γ} Ω_{A_{2} φ} {\hat{M}}_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}) \\ - 2 λ_{A_{2}} (\sum_{φ = 1}^{γ} Ω_{A_{2} φ} {\hat{T}}_{M φ (x)} - \sum_{φ = 1}^{γ} W_{φ} T_{M φ (x)}) \end{array}

(12)

where

λ_{A_{1}}

and

λ_{A_{2}}

are the Lagrange’s multipliers, setting

\frac{\partial ∆ (Ω_{A_{2} φ}, W_{φ})}{\partial Ω_{A_{2} φ}} = 0

, we obtain

2 {(Q_{φ})}^{- 1} S_{φ x}^{2} (Ω_{A_{2} φ} - W_{φ}) - 2 λ_{A_{1}} {\hat{M}}_{R φ (x)} - 2 λ_{A_{2}} {\hat{T}}_{M φ (x)} = 0

(13)

Thus, the calibration weight can be obtained as

Ω_{A_{2} φ} = W_{φ} + Q_{φ} {(S_{φ x}^{2})}^{- 1} (λ_{A_{1}} {\hat{M}}_{R φ (x)} + λ_{A_{2}} {\hat{T}}_{M φ (x)})

(14)

Substituting Equation (14) in Equations (10) and (11), respectively, we obtain

[\begin{matrix} (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) & (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) \\ (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) & (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) \end{matrix}] [\begin{matrix} λ_{A_{1}} \\ λ_{A_{2}} \end{matrix}] = [\begin{matrix} \sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}) \\ \sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}) \end{matrix}]

Solving the system of equations for lambdas, we obtain

λ_{A_{1}} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}))}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

and

λ_{A_{2}} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}))}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

Substituting these values into Equation (14), we obtain the weights as given by

\begin{matrix} Ω_{A_{2} φ} & = & W_{φ} + \frac{(Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}) [(\sum_{φ = 1}^{γ} {Q_{φ} (S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}))]}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}} \\ + \frac{(Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}) [(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}))]}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}} \end{matrix}

Writing these weights in Equation (8), we obtain the calibration estimator of CDF as

G_{{R M}_{(A_{2})}} = \sum_{φ = 1}^{γ} W_{φ} {\hat{F}}_{y φ} (t_{y}) + {\hat{β}}_{1 (R M)} (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) + {\hat{β}}_{2 (R M)} (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}))

where betas are given by

{\hat{β}}_{1 (R M)} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y})) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} {Q_{φ} {(S_{φ x}^{2})}^{- 1} \hat{T}}_{M φ (x)} {\hat{F}}_{y φ} (t_{y}))}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

and

{\hat{β}}_{2 (R M)} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)} {\hat{F}}_{y φ} (t_{y})) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y}))}{(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} {(S_{φ x}^{2})}^{- 1} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

3. Proposed Estimator

3.1. First Proposed Calibration Estimator of CDF

Taking inspiration from the first adapted estimator, we proposed the following CDF estimator:

G_{{R M}_{(P_{1})}} = \sum_{φ = 1}^{γ} Ω_{P_{1} φ} {\hat{F}}_{y φ} (t_{y})

(15)

where

{\hat{F}}_{y φ} (t_{y})

is the sample CDF of the study variable in

φ^{t h}

stratum. Further,

Ω_{P_{1} φ}

is the calibrated weight, we will use the chi-square distance function, as given below:

\sum_{φ = 1}^{γ} \frac{{(Ω_{P_{1} φ} - W_{φ})}^{2}}{Q_{φ} W_{φ}}

(16)

and satisfy the calibration constraint

\sum_{φ = 1}^{γ} Ω_{P_{1} φ} {\hat{M}}_{R φ (x)} = \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}

(17)

Note that

W_{φ} = \frac{M_{φ}}{M}

denote the traditional stratum weight, and

({\hat{M}}_{R φ (x)}, M_{R φ (x)})

are presenting the sample and population mid-range of the auxiliary variable in the

φ^{t h}

stratum. The Lagrange function is given by

∆ (Ω_{P_{1} φ}, W_{φ}) = \sum_{φ = 1}^{γ} \frac{{(Ω_{P_{1} φ} - W_{φ})}^{2}}{Q_{φ} W_{φ}} - 2 λ_{P} (\sum_{φ = 1}^{γ} Ω_{P_{1} φ} {\hat{M}}_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)})

(18)

where

λ_{P}

are multipliers of Lagrange, setting

\frac{\partial ∆ (Ω_{P_{1} φ}, W_{φ})}{\partial Ω_{P_{1} φ}} = 0

, we obtain

Ω_{P_{1} φ} = W_{φ} + λ_{P} {\hat{M}}_{R φ (x)} Q_{φ} W_{φ}

(19)

Substituting Equation (19) in Equation (17), and solving for lambda, we have

λ_{P} = \frac{\sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} {\hat{M}}_{R φ (x)}}{\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}}

(20)

Substituting Equation (20) in Equation (19), we obtain the calibration weight as

Ω_{P_{1} φ} = W_{φ} + \frac{\sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} {\hat{M}}_{R φ (x)}}{\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}

(21)

Substituting Equation (21) in Equation (15), we obtain the calibrated estimator of CDF, as given below

G_{{R M}_{(P 1)}} = \sum_{φ = 1}^{γ} W_{φ} {\hat{F}}_{y φ} (t_{y}) + \frac{\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})}{\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}} \sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y})

3.2. Second Proposed Calibration Estimator of CDF

Taking inspiration from the second adapted estimator, we proposed the following CDF estimator:

G_{{R M}_{(P_{2})}} = \sum_{φ = 1}^{γ} Ω_{P_{2} φ} {\hat{F}}_{y φ} (t_{y})

(22)

where

{\hat{F}}_{y φ} (t_{y})

is the sample CDF of the study variable in the

φ^{t h}

stratum. Further,

Ω_{P_{2} φ}

is the calibrated weight, we will use the chi-square distance function, as given below:

\sum_{φ = 1}^{γ} \frac{{(Ω_{P_{2} φ} - W_{φ})}^{2}}{Q_{φ} W_{φ}}

(23)

Subject to calibration constraints defined by

\sum_{φ = 1}^{γ} Ω_{P_{2} φ} {\hat{M}}_{R φ (x)} = \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}

(24)

\sum_{φ = 1}^{γ} Ω_{P_{2} φ} {\hat{T}}_{M φ (x)} = \sum_{φ = 1}^{γ} W_{φ} T_{M φ (x)}

(25)

where

({\hat{M}}_{R φ (x)}, M_{R φ (x)}), ({\hat{T}}_{M φ (x)}, T_{M φ (x)})

are presenting the sample and population mid-range and tri-mean of the supplementary variable in the

φ^{t h}

stratum. The Lagrange function is given by

\begin{array}{l} ∆ (Ω_{P_{2} φ}, W_{φ}) & = & \sum_{φ = 1}^{γ} \frac{{(Ω_{P_{2} φ} - W_{φ})}^{2}}{Q_{φ} W_{φ}} - 2 λ_{P_{1}} (\sum_{φ = 1}^{γ} Ω_{P_{2} φ} {\hat{M}}_{R φ (x)} - \sum_{φ = 1}^{γ} W_{φ} M_{R φ (x)}) \\ - 2 λ_{P_{2}} (\sum_{φ = 1}^{γ} Ω_{P_{2} φ} {\hat{T}}_{M φ (x)} - \sum_{φ = 1}^{γ} W_{φ} T_{M φ (x)}) \end{array}

(26)

where

λ_{P_{1}}

and

λ_{P_{2}}

are the Lagrange’s multipliers, setting

\frac{\partial ∆ (Ω_{P_{2} φ}, W_{φ})}{\partial Ω_{P_{2} φ}} = 0

, we obtain

2 \frac{Ω_{P_{2} φ} - W_{φ}}{Q_{φ} W_{φ}} - 2 λ_{P_{1}} {\hat{M}}_{R φ (x)} - 2 λ_{P_{2}} {\hat{T}}_{M φ (x)} = 0

(27)

Thus, the calibration weight can be obtained as

Ω_{P_{2} φ} = W_{φ} + W_{φ} Q_{φ} (λ_{P_{1}} {\hat{M}}_{R φ (x)} + λ_{P_{2}} {\hat{T}}_{M φ (x)})

(28)

Substituting Equation (28) in Equations (24) and (25), respectively, we obtain

[\begin{matrix} (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) & (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) \\ (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) & (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) \end{matrix}] [\begin{matrix} λ_{P_{1}} \\ λ_{P_{2}} \end{matrix}] = [\begin{matrix} \sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}) \\ \sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}) \end{matrix}]

Solving the system of equations for lambdas, we obtain

λ_{P_{1}} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}))}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

and

λ_{P_{2}} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}))}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

Substituting these values into Equation (28), we obtain the weights as given by

\begin{array}{l} Ω_{P_{2} φ} & = & W_{φ} + \frac{({Q_{φ} W_{φ} \hat{M}}_{R φ (x)}) [(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)}))]}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}} \\ + \frac{(Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}) [(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)})) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)}))]}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}} \end{array}

Writing these weights in Equation (22), we obtain the calibration estimator of CDF as

\begin{array}{l} G_{{R M}_{(P_{2})}} & = & \sum_{φ = 1}^{γ} W_{φ} {\hat{F}}_{y φ} (t_{y}) + {\hat{β}}_{P_{1} (R M)} (\sum_{φ = 1}^{γ} W_{φ} (M_{R φ (x)} - {\hat{M}}_{R φ (x)})) \\ + {\hat{β}}_{P_{2} (R M)} (\sum_{φ = 1}^{γ} W_{φ} (T_{M φ (x)} - {\hat{T}}_{M φ (x)})) \end{array}

where betas are given by

{\hat{β}}_{P_{1} (R M)} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y})) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} Q_{φ} {W_{φ} \hat{T}}_{M φ (x)} {\hat{F}}_{y φ} (t_{y}))}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

and

{\hat{β}}_{P_{2} (R M)} = \frac{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)} {\hat{F}}_{y φ} (t_{y})) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{F}}_{y φ} (t_{y}))}{(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{T}}_{M φ (x)}^{2}) (\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)}^{2}) - {(\sum_{φ = 1}^{γ} Q_{φ} W_{φ} {\hat{M}}_{R φ (x)} {\hat{T}}_{M φ (x)})}^{2}}

4. Numerical Study

To study the performance of the developed calibration estimation methods of CDF using robust measures, we generated four different real-life datasets. The Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 show that these populations have outliers and therefore belong to an asymmetric nature. We compared the mean square error (MSE) of the proposed estimators with the adapted estimators to evaluate which estimators performed more efficiently. For MSE estimation, we perform the steps of the simulation study as given below:

Step-1: Select a random sample with size

n_{φ}

through StRS from stratum

φ

;

Step-2: Find the value of CDF estimates (say)

\hat{ξ} = G_{{R M}_{(A_{1})}}, G_{{R M}_{(A_{2})}}, G_{{R M}_{(P_{1})}}, G_{{R M}_{(P_{2})}}

;

Step-3: Replicate the above steps

G = 5000

times and attained

{\hat{ξ}}_{1}, {\hat{ξ}}_{2}, \dots, {\hat{ξ}}_{G}

;

Step-4: Compute the MSE as

M S E (\hat{ξ}) = \frac{1}{G} \sum_{i = 1}^{R} {(\hat{ξ} - F_{Y} (t_{y}))}^{2}

The bias MSEs and PREs are provided in Table 1, Table 2 and Table 3, respectively. It is interesting to notice that in the following part, we will compare the outcomes of all four populations using the

t = 0.25

quantile point.

4.1. Apple Data (Population 1 and 2)

To demonstrate the performance of the proposed estimation method in this article, we examine a dataset of apples used in References [16,20].

Population 1

We consider the following variables for population 1:

x =

The list of apple trees in 1999;

y =

The amount of apples produced in 1999.

The extreme values of each stratum are clearly shown in the scatter plots in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, and as a result, the data are appropriate for our suggested estimators.

Population 2

We consider the following variables for population 2:

x =

The amount of apples produced in 1998;

y =

The amount of apples produced in 1999.

4.2. COVID-19 Data (Populations 3 and 4)

To demonstrate the performance of the proposed estimation method in this article, we examine a COVID-19 dataset, used in Reference [21].

Population 3

We consider the following variables for population 3:

x =

Total cases per million;

y =

Total deaths per million.

The extreme values of each stratum are clearly shown in the scatter plots in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, and as a result, the data are appropriate for our suggested estimators.

Population 4

We consider the following variables for population 4:

x =

Total number of cases per million;

y =

Total number of recoveries per million.

4.3. Interpretation

Results of Table 2, indicate that:

For population 1, the first proposed estimator $G_{{R M}_{(P_{1})}} = 2.284482$ is better than first adapted estimator $G_{{R M}_{(A_{1})}} = 4.083395$ and the second proposed estimator $G_{{R M}_{(P_{2})}} = 0.9799003$ is better than second adapted estimator $G_{{R M}_{(A_{2})}} = 1.538583$ at quantile $(t = 0.25)$ ;
For population 2, the first proposed estimator $G_{{R M}_{(P_{1})}} = 4.106615$ is better than first adapted estimator $G_{{R M}_{(A_{1})}} = 6.378484$ and the second proposed estimator $G_{{R M}_{(P_{2})}} = 2.021053$ is better than second adapted estimator $G_{{R M}_{(A_{2})}} = 3.500668$ at quantile $(t = 0.25)$ ;
For population 3, the first proposed estimator $G_{{R M}_{(P_{1})}} = 0.5599689$ is better than first adapted estimator $G_{{R M}_{(A_{1})}} = 0.666069$ and the second proposed estimator $G_{{R M}_{(P_{2})}} = 0.56593$ is better than second adapted estimator $G_{{R M}_{(A_{2})}} =$ at quantile $(t = 0.25)$ ;
For population 4, the first proposed estimator $G_{{R M}_{(P_{1})}} = 43.68103$ is better than first adapted estimator $G_{{R M}_{(A_{1})}} = 65.88738$ and the second proposed estimator $G_{{R M}_{(P_{2})}} = 34.73049$ is better than second adapted estimator $G_{{R M}_{(A_{2})}} = 87.86289$ at quantile $(t = 0.25)$ .

The similar pattern of performance for PREs of the suggested estimation methods can be observed in Table 3.

Based on these results for all the estimators, we conclude that the proposed estimators has minimum bias, MSE, and maximum PRE values for all four populations compared to the adapted estimators.

5. Conclusions

There are a variety of calibration estimation methods that use one or two calibration constraints based on supplementary data. In this article, a new, improved calibration estimator of CDF using robust measures is developed under StRS. To evaluate the effectiveness of the developed calibration estimators with the adapted calibration estimators, we conducted a simulation study using asymmetric real-life datasets. We calculate the bias, MSE, and PREs of calibration estimators. The results demonstrate that the proposed calibration estimators are more efficient than the adapted calibration estimators for asymmetric datasets. In future studies, the work can be expanded to incorporate different sampling schemes, and new proposals can be compared to existing approaches.

Author Contributions

Conceptualization, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; methodology, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; software, H.A. and U.S.; validation, H.A., M.H., U.S.; formal analysis, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; investigation, H.A. and U.S.; resources, H.A. and U.S.; data curation, H.A. and U.S.; writing—original draft preparation, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; writing—review and editing, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; visualization, H.A. and U.S.; supervision, M.H.; project administration, H.A., M.H., U.S., W.E., Y.T., S.I. and S.S.; funding acquisition, W.E. and Y.T. All authors have read and agreed to the published version of the manuscript.

Funding

The study was funded by Researchers Supporting Project number (RSP2023R488), King Saud University, Riyadh, Saudi Arabia.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

All the dataset information is already available in References [16,20,21].

Acknowledgments

The study was funded by Researchers Supporting Project number (RSP2023R488), King Saud University, Riyadh, Saudi Arabia.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Population 1 for h = 1.

Figure 2. Population 1 for h = 2.

Figure 3. Population 1 for h = 3.

Figure 4. Population 1 for h = 4.

Figure 5. Population 2 for h = 1.

Figure 6. Population 2 for h = 2.

Figure 7. Population 2 for h = 3.

Figure 8. Population 2 for h = 4.

Figure 9. Population 3 for h = 1.

Figure 10. Population 3 for h = 2.

Figure 11. Population 3 for h = 3.

Figure 12. Population 3 for h = 4.

Figure 13. Population 4 for h = 1.

Figure 14. Population 4 for h = 2.

Figure 15. Population 4 for h = 3.

Figure 16. Population 4 for h = 4.

Table 1. Bias of proposed and adapted estimators.

Estimator
	$G_{{R M}_{(A_{1})}}$	$G_{{R M}_{(A_{2})}}$	$G_{{R M}_{(P_{1})}}$	$G_{{R M}_{(P_{2})}}$
Population 1	2.020741	1.240396	1.51145	0.9898991
Population 2	2.525566	1.871007	2.026478	1.421637
Population 3	0.8161305	0.8058786	0.7483107	0.7522832
Population 4	8.117104	9.373521	6.609163	5.893258

Table 2. MSE of proposed and adapted estimators.

Estimator
	$G_{{R M}_{(A_{1})}}$	$G_{{R M}_{(A_{2})}}$	$G_{{R M}_{(P_{1})}}$	$G_{{R M}_{(P_{2})}}$
Population 1	4.083395	1.538583	2.284482	0.9799003
Population 2	6.378484	3.500668	4.106615	2.021053
Population 3	0.666069	0.6494403	0.5599689	0.56593
Population 4	65.88738	87.86289	43.68103	34.73049

Table 3. PRE.

Population 1	Population 2	Population 3	Population 4
$\frac{G_{{R M}_{(A_{1})}}}{G_{{R M}_{(P_{1})}}} \times 100 = 178.7449$	$\frac{G_{{R M}_{(A_{1})}}}{G_{{R M}_{(P_{1})}}} \times 100 = 155.3222$	$\frac{G_{{R M}_{(A_{1})}}}{G_{{R M}_{(P_{1})}}} \times 100 = 118.9475$	$\frac{G_{{R M}_{(A_{1})}}}{G_{{R M}_{(P_{1})}}} \times 100 = 150.8375$
$\frac{G_{{R M}_{(A_{2})}}}{G_{{R M}_{(P_{2})}}} \times 100 = 154.0143$	$\frac{G_{{R M}_{(A_{2})}}}{G_{{R M}_{(P_{2})}}} \times 100 = 173.2101$	$\frac{G_{{R M}_{(A_{2})}}}{G_{{R M}_{(P_{2})}}} \times 100 = 114.7563$	$\frac{G_{{R M}_{(A_{2})}}}{G_{{R M}_{(P_{2})}}} \times 100 = 252.9849$

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Abbasi, H.; Hanif, M.; Shahzad, U.; Emam, W.; Tashkandy, Y.; Iftikhar, S.; Shahzadi, S. Calibration Estimation of Cumulative Distribution Function Using Robust Measures. Symmetry 2023, 15, 1157. https://doi.org/10.3390/sym15061157

AMA Style

Abbasi H, Hanif M, Shahzad U, Emam W, Tashkandy Y, Iftikhar S, Shahzadi S. Calibration Estimation of Cumulative Distribution Function Using Robust Measures. Symmetry. 2023; 15(6):1157. https://doi.org/10.3390/sym15061157

Chicago/Turabian Style

Abbasi, Hareem, Muhammad Hanif, Usman Shahzad, Walid Emam, Yusra Tashkandy, Soofia Iftikhar, and Shabnam Shahzadi. 2023. "Calibration Estimation of Cumulative Distribution Function Using Robust Measures" Symmetry 15, no. 6: 1157. https://doi.org/10.3390/sym15061157

APA Style

Abbasi, H., Hanif, M., Shahzad, U., Emam, W., Tashkandy, Y., Iftikhar, S., & Shahzadi, S. (2023). Calibration Estimation of Cumulative Distribution Function Using Robust Measures. Symmetry, 15(6), 1157. https://doi.org/10.3390/sym15061157

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Calibration Estimation of Cumulative Distribution Function Using Robust Measures

Abstract

1. Introduction

2. First Adapted Calibration Estimator of CDF Using Robust Measure

2.1. First Adapted Calibration Estimator of CDF

2.2. Second Adapted Calibration Estimator of CDF

3. Proposed Estimator

3.1. First Proposed Calibration Estimator of CDF

3.2. Second Proposed Calibration Estimator of CDF

4. Numerical Study

4.1. Apple Data (Population 1 and 2)

4.2. COVID-19 Data (Populations 3 and 4)

4.3. Interpretation

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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