The $L_p$-Petty projection inequality for $s$-concave functions

1.   Introduction

Continuous data that strictly falls in the open interval (0, 1) is something we see rather frequently. Practitioners must represent this using the proper distributions, management, etc. This type of analysis includes studying ratios, percentages, etc. The beta distribution, which is utilized in a variety of situations, is one of the most versatile such distributions. However, there is a disadvantage to utilizing the beta distribution, as it is insufficient for some real-world scenarios, such as hydrological data. Considering this, the Topp-Leone distribution ^[1] and Kumaraswamy's distribution ^[2] merit consideration as alternatives to the beta model that have similar structure. The distribution and quantile functions may be represented in closed forms, which is a benefit in this case. To model datasets in the fields of biology, engineering, actuarial science, economics, and financial risk management, among others, several unit distributions have been created. Some of these significant, well-known distributions include the unit logistic distribution ^[3], unit Gompertz and unit Birnbaum-Saunders distributions ^[4,5], extended reduced Kies distribution ^[6], unit-Weibull distributions ^[7], unit generalized half normal distribution ^[8], unit Lindley distribution ^[9], unit Burr XII distribution ^[10], power unit Burr-XII distribution^[11], unit gamma-Gompertz distribution ^[12], unit Teissier distribution^[13], and generalized unit half-logistic geometric distribution^[14], etc.

Recently, Kharazmi et al. ^[15] proposed a new one-parameter unit distribution based on the definition of the arctan function. The new bounded distribution is called the arctan uniform distribution (AUD). The probability density function (PDF) and cumulative distribution function (CDF) of the AUD are given, respectively, by:

and

where $\delta$ is the scale parameter. We depict the PDF (1.1) in Figure 1 for a few different choices of the parameter $\delta$ to examine the effect of $\delta$ on the PDF behavior. It can be concluded that the AUD has asymmetric shapes. Kharazmi et al. ^[15] also provided the moments of this distribution.

Cost-effective sampling is a major issue in some research, especially when measuring the relevant feature is costly, inconvenient, or time-consuming. It is possible to give the sample items that are gathered additional structure using ranked set sampling (RSS) and to leverage this structure to create effective inferential processes. It is possible to rank tiny groups of units exactly, even without true quantification. The ranking might be carried out using eye examination, preliminary data, expert judgment, prior sampling episodes, or other imprecise techniques without the need for real measurement. The RSS method is a great instrument for attaining observational economy since it increases the accuracy attained per unit of the sample. This method of data collection was initially put forth by McIntyre ^[16] as an alternative to the widely used simple random sample (SRS) methodology for improving the effectiveness of the sample mean. It is used extensively in the fields of agriculture, biology, engineering, quality control, and environmental studies (see ^{[17,18,19,20,21,22,23,24,25]}). The procedures below should be followed to implement the RSS of $s$ observations from a population:

(1) Choose ($s$) SRS with size ($s$) each, with $s$ to be a low number.

(2) In each sample, order the units from smallest to greatest. Without actually measuring the units about the variable of interest, ranking is performed.

(3) Only the $a_1$th greatest unit in the $a_1$th sample, $a_1 = 1, .., s$ is used for actual measurements. As a result, the RSS associated with this cycle will be ${Z_{1(1:{m^{ \circ \circ }})}}, {Z_{2(2:{m^{ \circ \circ }})}}, ...., {Z_{{m^{ \circ \circ }}({m^{ \circ \circ }}:{m^{ \circ \circ }})}}.$ Note that ${Z_{{a_1}({a_1}:{m^{ \circ \circ }})}}$ stands for the $a_1$th order statistic from the $a_1$th row.

(4) Carry out the preceding steps $v$ times (cycles) to obtain sample size ${m^{ \circ \circ }} = sv,$ where $s$ is the set size and $v$ is the cycle number. As a result, the observed RSS for this $v$ cycle will be ${Z_{{a_1}({a_1}:{m^{ \circ \circ }}){a_2}}}$, $a_1 = 1, .., s$, $a_2 = 1, .., v$, where $s$ is the set size and $v$ is the cycle count. Hence for simplified form, ${Z_{{a_1}{a_2}}}$ will be used for the rest of the article instead of ${Z_{{a_1}({a_1}:{m^{ \circ \circ }}){a_2}}}.$

Statistical inference relies heavily on the parametric estimate approach employing the sampling design strategy. Numerous studies have examined various estimation techniques for estimating parameters based on RSS designs and their extensions. Reference ^[26] investigated how to estimate the location-scale family distributions' parameters. Some examples included the normal, exponential, and gamma distributions ^[27], the half logistic distribution ^[28], the Gumbel distribution ^[29], the generalized Rayleigh distribution ^[30], the Pareto distribution^[31,32], the x-gamma distribution ^[33], the new Weibull-Pareto distribution^[34], the extended inverted Topp-Leone distribution ^[35], the generalized quasi-Lindley distribution ^[36], and the inverse Kumaraswamy distribution ^[37]. For more, see ^{[38,39,40,41,42]}.

Statistical inference relies heavily on the parametric estimating approach and the sampling design strategy. The statistical literature frequently proposes different estimation approaches, since parameter estimation is important in practice. Generally, maximum likelihood (ML) estimation is the first step in the estimation process. This method's easy-to-understand formulation is the reason for its popularity. The estimators that are produced using this approach, for instance, are normally distributed and asymptotically consistent. Other, more widely used estimating techniques are available in the literature. These techniques include maximum product of spacing (MPS), least squares (LS), weighted LS (WLS), Cramér-von Mises (CM), Anderson-Darling (AD), minimum spacing absolute-log distance (MSALD), right-tail AD (RAD), left-tail AD (LAD), minimum spacing absolute distance (MSAD), percentile (PS), and a few more.

This study's objective is to present an in-depth assessment of several frequentist approaches to the AUD. The wide range of fields in which the RSS approach is used served as the inspiration for this concentration. In addition, the RSS design offers, for a fixed sample size, more efficient estimators than the SRS design. We use certain significant traditional estimating techniques based on the following procedures: RSS and SRS. The following estimation methods are taken into consideration: MPS, LS, ML, WLS, CM, AD, MSALD, RAD, LAD, MSAD, and PS. A simulation task is then used to compare the suggested estimates based on the RSS design to those offered by the SRS approach for the same sample size. Some precision metrics are used in comparison studies. The novelty of this study stems from the lack of prior research evaluating all of these estimating techniques for the AUD based on RSS. For illustration reasons, an insurance data set is investigated as well. Therefore, the study will serve as a guide for selecting the most appropriate estimating technique for the AUD, which we believe applied statisticians would find fascinating.

The following describes how this article is organized: Section 2 addresses the ML estimate (MLE) of the AUD parameter based on the RSS and SRS approaches. A few essential minimum distances of estimation for the proposed AUD are discussed in Section 3. In Section 4, several maximum and minimum product of spacing estimation are covered. The WLS, LS, and PS estimation techniques of the AUD are presented in Section 5. The effectiveness of the supplied estimating techniques is compared and evaluated in Section 6 using a Monte Carlo simulation. In Sections 7 and 8, respectively, an analysis of an insurance dataset is provided, followed by a conclusion.

2.   Maximum likelihood estimator

In this section, the MLE of parameter $\delta$ of the AUD is considered based on RSS and SRS. At first, assume that ${Z_{{a_1}{a_2}}} = \left\{ {{Z_{{a_1}{a_2}}}, \, \, {a_1} = 1, ..., s, \, {a_2} = 1, ..., v} \right\}$, is an RSS of size ${m^{ \circ \circ }}$ with PDF (1.1) and CDF (1.2), where $v$ is the cycles count and $s$ is the set size. It can be seen from the structure of RSS that the data are all mutually independent, and, in addition, for each ${a_1} = 1, ..., s$, the data are identically distributed. It should be noted that, if the judgment ranking is perfect, the PDF of $a_1$th order statistics ${Z_{{a_1}{a_2}}}$ is as below:

The likelihood function (LF) of the AUD, based on RSS, is given by:

The log-LF of (2.2), denoted by ${\ell ^ * }$, is as follows:

where $\mathbb{C}(\delta) = \left[{1- \frac{{{{\tan }^{ - 1}}(\delta {z_{{a_1}{a_2}}})}}{{{{\tan }^{ - 1}}(\delta)}}} \right].$ The MLE of $\delta$ says ${\hat \delta _1}$ is obtained by maximizing (2.3), which can be computed as the solution of the following nonlinear equation:

where $\mathbb{C}'(\delta) = \frac{{{{\tan }^{ - 1}}(\delta {z_{{a_1}{a_2}}})}}{{{{\left[{{{\tan }^{ - 1}}(\delta)} \right]}^2}\left({1 + {\delta ^2}} \right)}} - \frac{1}{{\left({1 + {\delta ^2}z_{{a_1}{a_2}}^2} \right){{\tan }^{ - 1}}(\delta)}}.$

Setting (2.4) to zero and solving numerically, we get the MLE ${\hat \delta _1}$ of $\delta$.

Additionally, the MLE of the AUD parameter under SRS is the subject of the following discussion. We assume that ${z_1}, {z_2}, ..., {z_{{m^{ \circ \circ }}}}$ is an observed SRS of size ${m^{ \circ \circ }}$ from the AUD with PDF (1.1). The log-LF, say $\ell _1^ *$, based on SRS, is given by

The MLE ${\ddddot \delta _1},$ of $\delta$, is provided as the solution of the following non-linear equation after setting with zero:

Then, ${\ddddot \delta _1}$ is provided from (2.5) after setting with zero and using the numerical technique.

3.   Minimum distances estimators

This section illustrates four estimation methods that minimize goodness-of-fit statistics, including AD, RAD, LAD, and CM. This series of estimating methods was created based on the discrepancy between the estimated CDF and the actual distribution function. This section presents the estimates of AUD parameters for SRS and RSS using the mentioned methodologies.

3.1.   Anderson-Darling estimators

Reference ^[43] introduced the AD test as an alternative to traditional statistical procedures to identify sample distribution deviations from the presumed distribution. Here, six estimators of parameter $\delta$ are produced based on the RSS and SRS.

Suppose that the ordered items ${Z_{(1:{m^{ \circ \circ }})}}, {Z_{(2:{m^{ \circ \circ }})}}, ..., {Z_{({m^{ \circ \circ }}:{m^{ \circ \circ }})}}$ are an RSS drawn from the AUD with sample size ${m^{ \circ \circ }} = sv$, where $s$ is set size and $v$ is the cycle count. By minimizing the following equation, the AD estimate (ADE) ${\hat \delta _2}$ of $\delta$ for the AUD is generated.

Instead of using (3.1), the ADE ${\hat \delta _2}$ of the AUD may be calculated by solving the nonlinear equation illustrated below:

where,

and

The following function is used to provide the RAD estimate (RADE) ${\hat \delta _3}$ for $\delta$ of the AUD:

Instead of using (3.4), the RADE ${\hat \delta _3}$ of the AUD may be calculated by solving the nonlinear equation illustrated below:

where ${\varphi _1}(.)$ and ${\varphi _2}(.)$ are defined in (3.2) and (3.3).

The following function is used to provide the LAD estimate (LADE) ${\hat \delta _4}$ for $\delta$ of the AUD:

To obtain the LADE ${\hat \delta _4}$ of the AUD, the following nonlinear equation may be solved:

where, ${\varphi _1}(.)$ is defined in (3.2).

Next, let us consider the scenario in which the ordered items ${Z_{(1)}}, {Z_{(2)}}, ..., {Z_{({m^{ \circ \circ }})}}$ are SRS seen from AUD with sample size ${m^{ \circ \circ }}.$ The following function is used to provide the ADE ${\ddddot \delta _2}$, for $\delta$ of the AUD:

with respect to $\delta$. The following equation which is equivalent to (3.5) may be solved numerically to provide ${\ddddot \delta _2}$

where

and

The following function is minimized for obtaining the RADE ${\ddddot \delta _3}$ for the AUD.

The RTDE ${\ddddot \delta _3}$ of the AUD is determined by solving the numerically the following nonlinear equation rather than using (3.8):

where ${\varphi '_1}(.)$ and ${\varphi '_2}(.)$ are defined in (3.6) and (3.7).

The following function is used to provide the LADE ${\ddddot \delta _4}$ for $\delta$ of the AUD:

To obtain the LADE ${\ddddot \delta _4}$ of the AUD, the following nonlinear equation may be solved:

where ${\varphi '_1}(.)$ is defined in (3.6).

3.2.   Cramér-von Mises estimators

To support the decision to use minimal distance estimators of the CM type, ^[44] offered empirical evidence showing the estimator's bias is less than that of other minimum distance estimators. Here, the RSS and SRS techniques are used to produce the CM estimate (CME) for the AUD parameter.

Let us assume that the ordered items ${Z_{(1:{m^{ \circ \circ }})}}, {Z_{(2:{m^{ \circ \circ }})}}, ..., {Z_{({m^{ \circ \circ }}:{m^{ \circ \circ }})}},$ with sample size ${m^{ \circ \circ }} = sv$, where $s$ is set size and $v$ is the cycle numbers, are the selected RSS from CDF (1.2). In order to obtain CME ${\hat\delta _5}$ of $\delta$, the following function is minimized with regard to $\delta$ :

Instead of using (3.9), CME can be derived by resolving the following nonlinear equation:

where ${\varphi_1}(.)$ is defined in (3.2).

Currently, suppose that the ordered items ${Z_{(1)}}, {Z_{(2)}}, ..., {Z_{({m^{ \circ \circ }})}}$ are the seen SRS from the AUD with sample size ${m^{ \circ \circ }}$. So, the following function is minimized to determine the CME ${\ddddot \delta _5}$ of $\delta$:

Or equivalent to (3.10), the CME ${\ddddot \delta _5}$ of $\delta$ is produced by minimizing the following function

where ${\varphi '_1}(.)$ is defined in (3.6).

4.   Maximum and minimum product of spacings estimators

The concept of differences in the values of the CDF at successive data points, according to Cheng and Amin ^[45], may be used to get the MPS estimate (MPSE) of the unknown parameter of the AUD. This approach is just as effective as ML estimators and consistent under a wider range of conditions.

Let ${Z_{(1:{m^{ \circ \circ }})}}, {Z_{(2:{m^{ \circ \circ }})}}, ..., {Z_{({m^{ \circ \circ }}:{m^{ \circ \circ }})}}$ be ordered items of the RSS drawn from the AUD with sample size ${m^{ \circ \circ }} = sv$, where $s$ is set size and $v$ is the cycle numbers. The uniform spacings may be defined as follows based on a random sample taken from the AUD.

where $F\left({{z_{(0:{m^{ \circ \circ }})}}\left| \delta \right.} \right) = 0, \, \, \, \, F\left({{z_{({m^{ \circ \circ }} + 1:{m^{ \circ \circ }})}}\left| \delta \right.} \right) = 1$, such that $\sum\limits_{k = 1}^{{m^{ \circ \circ }} + 1} {{\hbar _k}(\delta)} = 1.$

To get the MPSE ${\hat \delta _6}$ of $\delta$, the geometric mean of the spacing should be maximized.

or, alternatively, there is maximizing the function that follows:

The MPSE ${\hat \delta _6}$ of $\delta$ can also be obtained by numerically resolving the following nonlinear equations:

where ${\varphi _1}({z_{(k:{m^{ \circ \circ }})}}\left| \delta \right)$ is defined in (3.2) and ${\varphi _1}\left({{z_{(k - 1:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$ has the same expression with ${z_{(k - 1:{m^{ \circ \circ }})}}.$

Similarly, the minimum spacing distance estimator of $\delta$ is created by minimizing the following function.

where $\Delta ({u_1}, {u_2})$ is the suitable distance. According to Ref. ^[46], for $\Delta ({u_1}, {u_2}) = \left| {{u_1} - {u_2}} \right|$, $\Delta ({u_1}, {u_2}) = \left| {\log {u_1} - \log {u_2}} \right|$ are referred to the MSAD and MSALD, respectively. As a result, the MSAD estimate (MSADE) and MSALD estimate (MSALDE) of $\delta$ are provided by minimizing the following functions:

and

with respect to $\delta$. Equivalently to (4.1) and (4.2), the MSADE ${\hat \delta _7}$ and MSALDE ${\hat \delta _8}$ are provided by solving the nonlinear equations

and

where ${\varphi _1}\left({{z_{(k:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$ and ${\varphi _1}\left({{z_{(k - 1:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$ are defined above.

In addition to the above, the MPSE ${\ddddot \delta _6}$ of $\delta$ for the AUD under SRS is obtained. Let ${Z_{(1)}}, {Z_{(2)}}, ..., {Z_{({m^{ \circ \circ }})}}$ be SRS of size ${m^{ \circ \circ }}$ from CDF (1.2), and the uniform spacings in this situation are

where $F\left({{z_{(0)}}\left| \delta \right.} \right) = 0, \, \, \, \, F\left({{z_{({m^{ \circ \circ }} + 1)}}\left| \delta \right.} \right) = 1$, such as $\sum\limits_{l = 1}^{{m^{ \circ \circ }} + 1} {\hbar _l^ \bullet (\delta)} = 1$.

The MPSE ${\ddddot \delta _6}$ of $\delta$ is provided by maximizing the following function:

Equivalent to (4.3), the MPSE ${\ddddot \delta _6}$ of $\delta$ is produced by solving the following nonlinear equation numerically:

where ${\varphi '_1}({z_{(l)}}\left| \delta \right)$ is defined in (3.6), and ${\varphi '_1}({z_{(l - 1)}}\left| \delta \right)$ has the same expression with ${z_{(l - 1)}}.$

Furthermore, MSADE ${\ddddot \delta _7}$ and MSALDE ${\ddddot \delta _8}$ are obtained by solving numerically the following equations:

and

with respect to $\delta$. Equivalently to (4.4) and (4.5), the MSADE ${\ddddot \delta _7}$ and MSALDE ${\ddddot \delta _8}$ are provided by solving the nonlinear equations

and

where, ${\varphi _1}\left({{z_{(l)}}\left| \delta \right.} \right)$ and ${\varphi _1}\left({{z_{(l - 1)}}\left| \delta \right.} \right)$ are defined above.

5.   Other estimation methods

This section offers the LS estimate (LSE), WLS estimate (WLSE), and PS estimate (PSE) for the AUD parameter based on RSS and SRS methods.

5.1.   Least and weighted least squares estimators

Let ${Z_{(1:{m^{ \circ \circ }})}}, {Z_{(2:{m^{ \circ \circ }})}}, ..., {Z_{({m^{ \circ \circ }}:{m^{ \circ \circ }})}}$ be an observed ordered RSS with size ${m^{ \circ \circ }} = sv$, from the AUD. The LSE ${\hat \delta _9}$ and WLSE ${\hat \delta _{10}}$ are derived by minimizing the following functions with regard to $\delta$:

and

These estimators ${\hat \delta _9}$ are ${\hat \delta _{10}}$, which are equivalent to (5.1) and (5.2), and can be obtained by solving the following equations numerically:

and

where ${\varphi _1}\left({{z_{(k:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$ is defined before.

Additionally, suppose that ${Z_{(1)}}, {Z_{(2)}}, ..., {Z_{({m^{ \circ \circ }})}}$ is an ordered SRS of size ${m^{ \circ \circ }}$ taken from the AUD. The LSE and WLSE ${\ddddot \delta _9}, {\ddddot \delta _{10}}$ of $\delta$ are produced by solving numerically the following equations:

and

where ${\varphi _1}\left({{z_{(k:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$ is defined before.

5.2.   Percentiles estimators

One of the often employed methods for estimating the Weibull distribution's parameters is the percentile approach, which differs from other estimation techniques in terms of its ease of computation and effectiveness in parameter estimation ^[47]. Here, PSE ${\hat \delta _{11}}$ of $\delta$ of the AUD is provided using RSS and SRS methods.

Consider ${Z_{(1:{m^{ \circ \circ }})}}, {Z_{(2:{m^{ \circ \circ }})}}, ..., {Z_{({m^{ \circ \circ }}:{m^{ \circ \circ }})}}$ as an observed ordered RSS, with size ${m^{ \circ \circ }} = sv$, available from the AUD. From the PSE of the AUD's parameter one may get ${\hat \delta _{11}}$ by minimizing the following function and assuming that ${p_k} = \frac{k}{{{m^{ \circ \circ }} + 1}}$ is the estimate of $F\left({{z_{(k:{m^{ \circ \circ }})}}\left| \delta \right.} \right)$

with respect to $\delta$.

In the case of the SRS method, let ${Z_{(1)}}, {Z_{(2)}}, ..., {Z_{({m^{ \circ \circ }})}}$ be an ordered SRS of size ${m^{ \circ \circ }}$ drawn from AUD. The PSE ${\ddddot \delta _{11}}$ of $\delta$ is obtained, by minimizing the following equation:

with respect to $\delta$.

6.   Simulation

This section focuses on the examination of various estimation methods presented in this paper. The goal is to assess the efficacy of these methods in estimating model parameters through the generation of random datasets derived from the proposed model. Subsequently, these datasets will be ranked, and the estimation methods will be employed to identify the most recommended one. The simulation will be conducted with the assumption of a flawless ranking, outlined as follows:

● To generate an RSS from the AUD with a fixed set size $s = 5$ and different cycle numbers $v = 3, 10, 24, 40, 60,$ and $90$, the corresponding sample sizes $m^ {\circ \circ} = sv = 15, 50, 120, 200, 300,$ and $450$ are employed.

● Generate an SRS from the AUD with the specified sample sizes, $m^ {\circ \circ}$ = 15, 50, 120, 200, 300, and 450.

● We have a set of estimates corresponding to each sample size, using the true parameter values ($\delta$) of 0.15, 0.6, 1.0, 1.5, 2.0, and 2.5.

● To evaluate the effectiveness of the estimation methods, three measures are employed, which include the following:

Average of absolute bias (bias), $|bias(\widehat{\pmb \delta})| =$ $\frac{1}{M}\sum_{i = 1}^{M}|\widehat{\pmb \delta}_i-\pmb \delta|$, mean squared errors (MSE), $MSE = \frac{1}{M}\sum_{i = 1}^{M}(\widehat{\pmb \delta}_i-\pmb \delta)^2$, mean absolute relative errors (MRE) $MRE = \frac{1}{M}\sum_{i = 1}^{M}|\widehat{\pmb \delta}_i-\pmb \delta|/\pmb \delta$.

● The measures outlined in the previous step serve as objective benchmarks for evaluating the accuracy and reliability of the estimated parameters. Utilizing these evaluation metrics enables a comprehensive assessment of the performance of the estimation techniques. This evaluation process provides valuable insights into the effectiveness and appropriateness of these techniques for the particular model under consideration.

● By repeating this process multiple times through numerous iterations, we can obtain a reliable and robust assessment of the estimation techniques. This repeated evaluation helps ensure that the performance results are consistent and representative, contributing to a more thorough understanding of the effectiveness of these techniques in estimating the model parameters.

● The results of the evaluation measures are presented in Tables 1–12, encompassing both SRS and RSS. These tables offer a comprehensive summary of the outcomes obtained. In these tables, the magnitude of each value signifies its relative effectiveness when compared to all the estimation approaches examined in the study. Lower-ranked values indicate stronger and more significant performance relative to the investigated estimation methods. These tables serve as a valuable reference for assessing the relative power and significance of the different estimation techniques.

● The ratio of the MSE for SRS to the MSE for RSS is provided in Table 13. This ratio helps to gauge the comparative performance of SRS and RSS in terms of MSE, offering insights into the efficiency of these sampling methods.

● For a thorough and detailed analysis of the estimates, we present both their partial and total ranks in Tables 14 and 15 for the SRS and RSS, respectively. These rank tables offer a more nuanced and comprehensive perspective on the performance and comparative effectiveness of each estimation approach, facilitating a deeper understanding of their relative strengths and weaknesses.

Upon careful analysis of the simulation results and the rankings presented in the tables, several conclusions can be drawn:

● It is noteworthy that for both SRS and RSS datasets, our model estimates exhibit the consistency property. This property implies that as the sample size increases, the estimates converge to the true parameter values.

● All three measures used exhibit a consistent trend: They decrease as the sample size increases. This pattern suggests that larger sample sizes lead to more accurate and precise parameter estimates.

● Based on our simulation results for both SRS and RSS datasets, it appears that the MPSE has the advantage in determining the quality of our estimates.

● From Table 13, it can be observed that the estimates obtained from the RSS datasets are more efficient compared to those obtained from the SRS datasets. This suggests that RSS is a more efficient sampling method in terms of producing estimates with lower MSE.

7.   Application

To highlight the practical utility of the proposed estimation methods, a real dataset was meticulously selected and is comprehensively elucidated in this section. The objective was to illustrate the practical applications of these proposed estimates by conducting an in-depth analysis of the real-world dataset. This analysis serves as a demonstration of how these estimation techniques can be applied to real-world data, showcasing their effectiveness and relevance in practical research and decision-making contexts. The used real dataset presents the firm's risk management cost-effectiveness and it was studied by Abd El-Bar et al. ^[48]. Its values are: 0.0432, 0.1271, 0.793, 0.0407, 0.0076, 0.037, 0.18, 0.1129, 0.09, 0.0535, 0.0783, 0.0093, 0.0851, 0.1753, 0.0036, 0.1597, 0.002, 0.1357, 0.0215, 0.0065, 0.079, 0.0329, 0.0458, 0.1192, 0.0431, 0.1245, 0.0255, 0.1396, 0.0122, 0.15, 0.14, 0.0529, 0.2222, 0.0315, 0.0389, 0.0297, 0.0608, 0.1833, 0.0279, 0.0694, 0.15, 0.0818, 0.2912, 0.1261, 0.0931, 0.0216, 0.0525, 0.1938, 0.0433, 0.0351, 0.0629, 0.0125, 0.0571, 0.0094, 0.0885, 0.0411, 0.004, 0.0582, 0.2172, 0.0434, 0.0509, 0.65, 0.0913, 0.1, 0.0375, 0.2886, 0.0206, 0.0028, 0.0407, 0.0849, 0.0612, 0.1333, 0.9755.

Table 16 provides a comprehensive summary of the descriptive analyses performed on the dataset under investigation. Figure 2 displays various graphical representations, including histograms, kernel density plots, violin plots, box plots, total time on test (TTT) plots, and quantile-quantile (Q-Q) plots. These visualizations and descriptive statistics collectively offer insights into the characteristics and distributions of the data, enhancing our understanding of the dataset's key features and patterns. The dataset was subjected to a Kolmogorov-Smirnov (KS) test to assess its compatibility with a specific model. The MLE was utilized to obtain the parameter estimates. The K-S distance (KSD) was computed to be 0.0812933, and the p-value (KSP) was found to be 0.720254. Based on these results, it is apparent that the AUD is a suitable candidate for fitting the firm's real dataset. To visually demonstrate this suitability, Figure 3 presents various graphical representations, including the probability-probability (P-P) plot, estimated CDF, estimated survival function (SF), and a histogram with the estimated PDF. These visualizations collectively suggest that the AUD is a suitable choice for modeling and fitting the firm's real dataset, as they align well with the distributional characteristics of the model.

Based on the theoretical findings discussed earlier, the dataset underwent an examination using two sampling techniques, SRS and RSS. Tables 17 and 18 present the SRS and RSS estimates, respectively, derived from the AUD. These estimates are provided for different sample sizes over five cycles, employing various estimation techniques. The process of generating the RSS and SRS observations was facilitated using the R-package. These tables collectively display the results of the estimation techniques applied to the dataset, allowing for a comprehensive comparison of the sampling methods and estimation procedures used. To demonstrate the superiority of RSS over SRS to various estimation methods, we conducted an evaluation using several goodness-of-fit statistics for the model. These statistics encompassed the Anderson-Darling test statistics (ADTS), Cramér-von Mises test statistics (CMTS), and KS test statistics (KSTS), along with their KSP. These tests and their p-values were utilized to assess how well the data conforms to the model, and their results can provide insights into the effectiveness of RSS compared to SRS in capturing the underlying distribution of the dataset. Estimates that outperformed their counterparts typically displayed larger p-values (greater than 5%) and lower goodness-of-fit values. Table 19 provides a comparative analysis between the SRS and RSS designs in terms of their goodness-of-fit values and KSPs. This comparison helps in assessing the relative effectiveness of SRS and RSS in fitting the dataset to the model, with a focus on identifying which design and estimation techniques yield better goodness-of-fit results. The fitting of the model to the dataset can be observed in Figures 4 and 5. Notably, the RSS design demonstrates superior performance compared to the SRS design in terms of efficiency. This is evident from the smaller goodness-of-fit values and the correspondingly larger KSPs. This superiority is consistently observed across all estimates, even when the same number of measurement units is considered. These findings underscore the advantages of RSS over SRS in terms of fitting the dataset to the model and obtaining more efficient estimates.

8.   Conclusions

A brand-new bounded distribution called the arctan uniform distribution may be used to simulate several bounded real-world datasets that already exist. When accurately measuring the observation is difficult or expensive, RSS is a valuable strategy. In the present work, the parameter estimator of the arctan uniform distribution is regarded using RSS and SRS approaches. The PS, WLS, AD, ML, MSALD, CM, LS, MPS, RAD, LAD, and MSAD are a few of the well-known conventional estimating techniques that are used. A Monte Carlo simulation based on some accuracy measures is used to assess the effectiveness of the generated estimates. Based on the results of our simulations for both the SRS and RSS datasets, it appears that the MPS approach is preferred in evaluating the quality of suggested estimates compared to the others. A similar pattern of decline with larger sample sizes is seen in all criteria measures. This trend indicates that parameter estimates are more accurate and trustworthy with higher sample numbers. Estimates derived from the RSS datasets are more trustworthy than those derived from the SRS datasets. This suggests that RSS is a sampling strategy that produces estimates with a lower mean squared error than other sampling methods. Real data findings provide more evidence that the RSS design is superior to the SRS approach.

Use of AI tools declaration

The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RG23048).

Conflict of interest

The authors declare no conflict of interest.