Linear Penalized Spline Model Estimation Using Ranked Set Sampling Technique

Abstract

Benefits of using Ranked Set Sampling (RSS) rather than Simple Random Sampling (SRS) are indeed significant when estimating population mean or estimating linear models. Significance of this sampling method clearly appears since it can increase efficiency of the estimated parameters and decrease sampling costs. This paper investigates and introduces RSS method to fit spline and penalized spline models parametrically. It shows that the estimated parameters using RSS are more efficient than the estimated parameters using SRS for both spline and penalized spline models. The superiority of RSS approach is demonstrated using a simulation study as well as the "Air Pollution"environmental real data study. The approach in this paper can be illustrated for general smoothing spline models; for example B-spline,Radial spline etc, straightforwardly.

Another look at regression analysis using ranked set samples with application to an osteoporosis study

“…The efficiencies of regression analysis with RSS data in Muttlak (1995) are slightly more than the corresponding ones with SRS data. Similar performances were observed in Al Kadiri (2016), Algarni (2016), andAl Kadiri (2017), where they also used the OLS approach to estimate the nonlinear relationships between X[𝑟]𝑗 and 𝑦 [𝑟]𝑗 using spline regression models. In all these studies, RSS-based models have shown slight improvements in the relative efficiencies.…”

“…In this paper, we proposed two approaches for regression analysis in the prediction problems context using RSS data. We show that our proposed methods provide efficient approaches to imbed the extra rank information of RSS data into the construction of more efficient predictive models and our fitted models are more efficient than the current ones based on the OLS methodology (Muttlak, 1995;Algarni, 2016;Al Kadiri, 2017). We showed that under the RSS design, the error terms associated with order statistics in their corresponding regression models have different distributions with different variances and one can use a weighted least squares method or a multilevel modeling approach to incorporate the rank information of such data and develop better predictive models than those with SRS data.…”

“…We have calculated the average MSE ($$\widehat{\text{MSE}}$$) and MSPE ($$\widehat{\text{MSPE}}$$) values based on $$B=5000$$ replications for each model (see the supplementary material). Table 2 represents the relative efficiencies of each method compared with corresponding ones under SRS and shows that using our proposed method's efficiencies of our estimated/predicted values is significantly increased (sometimes up to a factor of 5), while in Al Kadiri (2017), RSS‐based models show only a slight efficiency improvement.…”

“…We generated ranking variables Z such that $$\rho _{Y, Z} \in \lbrace 0.5, 0.75, 0.85\rbrace$$ to represent different ranking abilities. Training data sets were used to estimate the underlying relationship using spline regression models under the usual SRS setting, the method of Al Kadiri (2017) which we call it OLS and our proposed approaches. For the adjusted response method, imperfect ranking probabilities are estimated using a pilot sample and the response variable Y in the sample (i.e., $$p_{rj}= \mathbb {P}[Y_{[r]}= Y_{(j)}]$$).…”

“…See Irbtd (2001), Ozturk (2002), and Murff and Sager (2006) for other research along these lines. Regression methodology with RSS data has also been extended to more flexible models such as spline and polynomial regressions by Al Kadiri (2016), Algarni (2016), and Al Kadiri (2017). In all these studies, the usual least squares method is used by simply treating RSS data as simple random samples.…”