Nonparametric Estimation Method for the Distribution Function Using Various Types of Ranked Set Sampling

This article is concerned with estimating the cumulative distribution function  based on the local polynomial regression  depending on  and . A new  estimator depends on  is derived. The resulting proposed estimator is used to introduce three ways of estimating  (the method of moments, the maximum likelihood method, and ), based on  and . The method of moments and the maximum likelihood method based on  were suggested by Al-Saleh and Ahmad (2019). In this study we get the same result of  estimate for the method of moments and the maximum likelihood method,  estimator based on  can have some advantages over their competitors for fixed or non-fixed samples; on the empirical side, we use kernel (normal, epanechikov) with bandwidth (0.1, 0.2, …, 0.9) and three levels of degree of kernel, We have concluded that kernel epanechikov is a little better than normal, with bandwidth 0.8 giving the best result. Compared to the others, bandwidth and degree 2 are also like that. We have concluded that  of  based on  is better than  of  based on  because the data in  is stable and is less prone to ranking errors for fixed or non-fixed samples. Depending on the relative efficiency, we get that there is no big difference between both kernels, nonetheless we conclude that relative efficiency depends on whether the  is better than the  and has more efficiency. We recommend using  based on estimators.

1. Introduction

Cumulative Distribution Function CDF is a strong tool for understanding and evaluating random variables, as well as forecasting future occurrences; it is a foundational idea in probability theory and statistics. Also, establish that the occurrence is likely to take place until a specific point. When we utilize it, we encounter several obstacles. One of the challenges is studying nonparametric analysis in-depth and identifying several population features, such as odds, survival analysis, hazards, etc. Estimating the CDF using ranked sets sample RSS is more efficient than basic random sample SRS because RSS frequently yields more precise estimates with the same sample size, as well studying variables and measuring them is not easy; sometimes it is too expensive or time-consuming, but ranking the variables is easy or has a negligible cost. The first to introduce the rank-set sample was McIntyre (1952) for estimating the paster of yields in Australia and expressed expectations about how to develop the estimator that would be more effective for the paster of yields. Halls and Dell (1966) conducted a field trial evaluating its applicability to the estimation of forage yields in a pine-hardwood forest, the terminology ranked set sampling was, coined by Halls and Dell. Takahasi and Wakimoto (1968) proved the first theoretical result is that, when ranking is perfect, the ranked set sample mean is an unbiased estimator of the population mean, and the variance of the ranked set sample mean is always smaller than the variance of the mean of a simple random sample of the same size. Research has continued in ranked sets since (1997). Muttlak (1997) suggested studying median ranked sets sampling to estimate the population mean instead of ranked sets sampling, and it is a strategy to minimize the error in ranking. Gulati (2004) studied the empirical distribution function of Stokes and Sagar with smooth estimators and properties using simulation to compare the smooth and empirical estimators. Frey (2012) derived the constraint to estimate the cumulative distribution function with the mean of the population to create a Woodruff-type confidence interval for the population quantile. Al-Saleh and Ahmad (2019) suggested a new technique of ranked sets sampling, which was called Moving extreme ranked sets sampling, to estimate the cumulative distribution function and then compared the proposed estimator with the corresponding estimator based on both. Zamanzade (2020) established two estimators in moving extreme ranked sets sampling with simulation and also showed that the proposed estimators provide a substantial improvement over their competitors and prove that the estimators are utilized to estimate the stress-strength probability. Abdallah and Al-Omari (2022) considered the problem of estimating the cumulative distribution function and the odds measure under moving extreme ranked set sampling.

The paper is structured as follows: Section 2 describes ranked sets sampling and median ranked sets sampling. Section 3 describes local polynomial regression. Section 4 Estimation of cumulative distribution function using the Maximum Likelihood Method, Method of Moments, and local polynomial regression. Section 5 Simulation study and conclusions.

2. Description RSS & MRSS

2-1. Methodology of ranked sets sample (RSS)

McIntyre (1952) was the first one who suggested the ranked sets sampling as a strategy to estimate the paster of yields. In the  technique, taking samples is much cheaper than measurement of the variable. We will describe how to select ranked sets sampling in the following steps:

1. Draw randomly sample units from the population of interest. Divided the units into  groups each one of the groups has a size of . 
2. Based on the judgment rank the unit sets without actual measurement by eyes or by a bit price method for the variable of interest.
3. From the first set, select the smallest order observation, discard the other units, and then from the second set, select the second smallest order observation, and discard the other units. The process continues until get the maximum order observation.
4. From steps 1-3 we can get the ranked sets sample RSS of one group
5. To obtain with size , we can repeat steps (1-3) times, where .

Let  be the  element set, be the judgment order statistics of the  sample of size , and the  cycle of the  repeated. You should notice that we utilize square brackets [.], when the ranked sets sample is imperfect ranking it means there is an error in ranking, if there is no error in ranking it means that the ranked sets sample is perfect ranking we use the round brackets (.), it’s very important to note that for each    are independent and identically distributed . And for each    are just independent.

2-2. Methodology of median ranked sets sample ( )

Muttlak (1997) suggested a new strategy of ranked sets sampling, which is called median ranked sets sampling , to minimize errors in the process of ranking units within groups and to increase the efficiency of the estimator in the presence of errors in ranking, also to increase the efficiency over  with perfect ranking. The following summarizes the  procedure for drawing a sample of size . We will describe how to select median ranked sets sampling  in the following steps:

• Drow randomly sample units from the population of interest.
• Divided the units into  groups each one of the groups has a size of .
• If the sample size of group is odd, the odd will be measured by rule  it is equal to the units in the medial of the groups, if the sample size of the  group is even, the even will be measured the first half group units with rule  and the second half with rule .
• Steps 1-3 can be replayed times, if necessary to get  of size .

If the  groups are odd, the median ranked sets sample odd, symbolized , where the units of the  for variable , is described as follows:

   Be the judgment order statistics of the  sample of size  and the cycle of the  repeated.

If the  groups are even, the median ranked sets sample even, symbolized , Also  is the units of the  for variable , is described as follows:

Where  be the judgment order statistics of the  sample of size  and the cycle of the  repeated.

3. Local polynomial regression (LPR)

Local polynomial regression is a nonparametric technique used to generalize kernel regression, also used to model functions and smoothing one of the statistics plots, which is called the scatter plot. One of the most important uses is to find the relationship between the dependent variable and the independent variable,   is better than other types of regression for having a good performance near the boundary. For each point of  a  which is low order weighted least square regression is fit at each point of . By the Fan and Gijbels (1996), (  are defined according to the fixed model in equation (1).

                                                                                                                                  (1)

Where ,  is the variance of  at point ,  is a residual error with normal distribution with mean zero and variance . For estimating  we use a Tylor series.

                                                                                                     (2)

We need the point in the area of z because it gives us a higher weight than the other point remaining, we can estimate the unknown parameters in equation (2) by using  weighted least square, depending on the following formula:

 and   

 represents kernel function,  represents bandwidth.  represents the diagonal elements matrix of weight, where  is the  with  in the first entry and  elsewhere.

4. Estimation of cumulative distribution function (CDF)

4.1. Estimation of (CDF) using The Maximum likelihood Method (MLE):

4.1.1. Based on ranked sets sampling (RSS).

Al-Saleh and Ahmad (2019) suggested and proved  using  based on

   the represent RSS that we selected from the population with  and , and then we will use the maximum likelihood estimation  for estimating    depending on the . They assumed that the variable  is distributed according to a binomial distribution with mass parameter  and success probability . Therefore, the estimator of the probability distribution function is defined according to the following relationship:

 is the mean obtained by estimating the  based on  by using .

4.1.2. Based on odd median ranked sets sampling ( ).

Let , be median ranked sets sample odd  of size , that we selected from the population with  and , and then we will use the maximum likelihood estimation  for estimating    depending on the , note that for each ,    are independent and identically distributed  each unit distributed Bernoulli distribution with probability of success , and  represent indicator.

Let , then variable  distributed binomial with mass parameter  and success probability .

The likelihood function is determined according to equation (4):

Therefore, the estimator of the probability distribution function is defined according to the following relationship:

is the mean obtained by estimating the  based on  by using .

4.1.3. Based on even median ranked sets sampling ( ).

The  element set is  of size , that we selected from the population with  and , and then we will use the maximum likelihood estimation  for estimating    depending on the , depending on the withdrawal method of median ranked sets, sample even note that for each

 are independent and identically distributed  each unit distributed Bernoulli distribution with a probability of success , and  represent indicator.

Let , then variable  distributed binomial with mass parameter  and success probability .

The likelihood function is determined according to equation (6):

Therefore, the estimator of the probability distribution function is defined according to the following relationship:

is the mean obtained by estimating the  based on  by using .