The RS Generalized Lambda Distribution Based Calibration Model

We propose a flexible linear calibration model with errors from RS (Ramberg & Schmeiser, 1974) generalized lambda distribution (GλD). We demonstrate the derivation of the maximum likelihood estimates of RS GλD parameters and examine the estimation performance using a simulation study for sample sizes ranging from 30 to 200. The use of RS GλD calibration model not only provides statistical modeller with a richer range of distributional shapes, but can also provide more precise parameter estimates compared to the standard Normal calibration model or skewed Normal calibration model proposed by Figueiredoa, Bolfarinea, Sandovala and Limab (2010).


Introduction
The statistical calibration model is a reverse regression technique, where we use the response variable to predict the corresponding explanatory variable.There are number of applications of this technique in science.For example, we may use radiometric dating to ascertain the age of a tree and further verify our result using tree rings.Our aim, however, is to use radiometric dating to estimate age of new trees, and the problem is whether we should minimize errors in the observation or minimize errors in age determination.There are many similar problems in substance concentration determination in biology and chemistry, physical quantities determination in physics and blood pressure/cholsterol level measurement in medicine.The literature on calibration problem has a long history, and one of the earliest works can be found in Eisenhart (1939).
The usual calibration experiment is a two stage process involving two random variables X and Y.The first stage is known as the calibration trial, where we observe the n values of the response variable y 1 , • • • , y n from a given set of explanatory values x 1 , • • • , x n and we can estimate the link function between X and Y.The second stage is known as the calibration experiment, where we observe k ≥ 1 value(s) of the response variable Y as y 01 , • • • , y 0k which are mapped from some unknown value x 0 from the explanatory variable X.We can express these two stages by the following equations.
As an extension to Normal distribution, Azzalini (1985) introduced the skewed Normal distribution.The skewed Normal distribution is defined as where φ(•) and Φ(•) are the p.d.f. and c.d.f. of a standard normal distribution respectively.Specially, when ξ = 0 and ω = 1, we obtain the standard skewed Normal distribution.
Based on (1.2), Figueiredoa et al. (2010) defined a skew-normal calibration model by assuming that ε i and ε 0 j are i.i.d. and follow a skewed Normal distribution with ξ = 0 denoted by S N(0, ω, λ).This gives us the following calibration model: In (1.3), the conditional distribution of y i given x i and y 0 j given x 0 are governed by skewed Normal distributions.This skewed Normal calibration model allows the modeller to cope with some degree of skewness in the error distribution.However, this is still limited as the skewed Normal distribution have limited range of shapes.The skewed Normal distribution still cannot handle heavy tailed, U shape, uniform, triangular or exponential upward/downward patterns.These shapes however, can be captured using GλD (generalized lambda distributions), and we propose a further extension to the calibration model by using RS GλD.
Our article is organized as follows.In Section 2, we introduce the GλD family.In Section 3, we outline the RS GλD calibration model and discuss possible ways to estimate parameters of the model using maximum likelihood estimation.In Section 4, we demonstrate the estimation performance of our proposed model across a range of different sample sizes from 30 to 200.As a further test to our proposed model to the literature, we compare the performance of RS GλD calibration model against Normal and skewed Normal calibration model with respect to a real life dataset used by Figueiredoa et al. (2010) in Section 5. A discussion of our proposed method is given in Section 6.
The fundamental motivation for the development of FKML GλD is that the distribution is defined over all λ 3 and λ 4 (Freimer et al., 1988).The only restriction on FKML GλD is λ 2 > 0. This is more convenient to deal with computationally than RS GλD and hence it is sometimes the preferred GλD for some researchers.
We restrict our attention in this article to the more difficult problem of fitting RS GλD calibration model to data.Without loss of generality, the method we outlined below can be easily adapted to build FKML GλD calibration model.

GλD Based Calibration Model
We consider the following usual calibration model: We assume that i and j are i.i.d.GλD(0, λ 2 , λ 3 , λ 4 ).In general, we consider x 1 , • • • , x n to be known and fixed and α, β, λ 2 , λ 3 and λ 4 are parameters we need to estimate.Our GλD calibration model takes the following form: Consequently, the likelihood function for RS GλD is: where and 0
Theoretically, the MLE of θ is the solution of (3.7) when it is set to be equal to 0. The derivatives ∂ f 1 ∂θ and ∂ f 2 ∂θ are given below.
It is difficult to obtain the exact solutions of setting (3.7) to zero using the above formulations, owing to the fact that RS GλD is defined by its inverse quantile function and there is a high degree of complexity involved in solving the above equations.As an alternative, we carry out the maximum likelihood estimation by maximising (3.6) directly using Nelder-Mead optimisation algorithm as is customary done for maximum likelihood estimation problems involving GλD (see Su, 2010Su, , 2007aSu, , 2007b)).This is a preferred and more reliable method of estimation as opposed to trying to satisfy the exact conditions to which all of the above equations equal to zero.The GLDEX package in R (Su, 2010(Su, , 2007a) ) facilitates the Nelder-Mead optimisation algorithm for GλD.
Our algorithm is as follows: 1) Generate a set of initial values for α, β, x 0 , λ 2 , λ 3 , λ 4 .There are a number of strategies that can be used to determine the best set of initial values.One strategy is to generate initial values α, β, x 0 using Normal or skewed Normal calibration model and then generate some low discrepancy quasi random numbers for λ 2 , λ 3 , λ 4 over a range of values and select the set of initial values that maximises (3.6).Alternatively all initial values can be randomly generated using low discrepancy quasi random numbers.
4) Check the minimal support of GλD(λ 1 , λ 2 , λ 3 , λ 4 ) is lower or equal to the lowest value of y 0 .Similarly, check that the maximum support of GλD(λ 1 , λ 2 , λ 3 , λ 4 ) is greater or equal to the largest value of y 0 .This is to ensure that the fitted GλD will span the entire dataset.If these conditons are not met, choose another set of initial values and repeat from 2).
5) Conduct Nelder Mead optimisation by maximising (3.6) directly using the above initial values to obtain the required estimates.
We repeat this process 1000 times, which give us 1000 x0m estimates of x 0 .The mean x0 , Bias(x 0 ) and MSE(x 0 ) are calculated as follows: The results of above simulations are shown in Tables 1 and 2. As expected, the MSE decreases as we increase the sample size or increase the value of inverse scale parameter λ 2 .In terms of bias, we observe that the performance appear to be fairly consistent across sample sizes, this gives confidence in the use of RS GλD calibration model for smaller samples, even though there are are more parameters that need to be estimated from this model.There also appears to be a tendency for RS GλD calibration model to slightly overestimate as nearly all the bias results are positive.Increasing the shape parameter λ 3 does not always result in increase in MSE, this is because the shape parameter spaces of λ 3 and λ 4 for RS GλD are fairly complex.
Table 1.Simulations results with x 0 = 15, α = 3, β = 1.5, λ 4 = 1  We further considered using RS GλD to approximate generalized extreme value distribution (GEV) with location, scale and shape parameters being 0.1860, 0.4016, 0.1511 respectively.We choose RS GλD with λ 1 = 0, λ 2 ≈ −0.0374, λ 3 ≈ −0.0027, λ 4 ≈ −0.0212 for this demonstration (Figure 1).We then generate simulated data based on GEV and use our approximated RS GλD to estimate x 0 with α = 3, β = 1.5 and repeat this over 1000 simulation runs.The result of this simulation is given in Table 3.We observe that the RS GλD calibration model tends to overestimate the true x 0 by a small margin, but the bias appears to decrease as sample size increases.

Application
We apply the RS GλD calibration model to a dataset which measures teenager testicular volume (ml 3 ).This dataset is from Chipkevitch, Nishimura, Tu and Galea-Rajas (1996) and consists of 42 observations.Figueiredoa et al. (2010) considered two measurement methods from Chipkevitch et al. (1996): dimensional measurement with a caliper (DM) and measurement by ultrasonography (US) and the data is given in Table 4.In their paper, Figueiredoa et al. (2010) consider the x 0 value of 16.4, which is observed twice by ultrasonography.They subsequently treated this value as unknown, with corresponding y 0 j values of y 01 = 10.3 and y 02 = 17.3.Then, they estimate x 0 using their skewed Normal calibration model and compared this with the standard Normal calibration model.We did the same using the RS GλD calibration model and our results are shown in Table 5.The theoretical derivation of the variability of our estimates under RS GλD is not readily tractable as in the cases of skewed Normal and Normal distributions.As we need to numerically derive our calculations, small errors in numerical procedures could accumulate into large errors even if we could evaluate the exact theoretical solution.
As a workaround, we adopt the following procedure.Once we obtained the parameters of our model, α, β, x 0 , λ 2 , λ 3 , λ 4 , we conduct simulations to estimate the variability of our estimate.We use our estimated parameters from the RS GλD calibration model and x i (excluding x i = 16.4) from the original data to randomly generate y 0 j and y i according to (3.1) and (3.2).We then maximise the likelihood in (3.6) using Nelder Mead Simplex algorithm with initial values being our original estimated parameters.We repeat the process 1000 times and calculate the sample standard deviations of our estimated parameters.
Table 5 lists the estimated parameters and their standard deviations from RS GλD, skewed Normal and Normal calibration models.We compute the Akaike, Bayesian and Hannan-Quinn information criterion (AIC, BIC, and HQ) to allow model selection between three models.All three criterion favors the RS GλD calibration model.In addition, the RS GλD model is much more efficient compared to the other models, with the smallest variability in its parameter estimates.

Concluding Remarks
We propose a new calibration model with RS GλD errors, which is an extremely flexible model that can cope with a wide range of different error distributions.Our method also lends to the development of FKML GλD calibration model, which may have better properties with regard to numerical convergence.Our simulations studies suggest our proposed model perform well for small sample sizes across a range of inverse scale and shape parameters of RS GλD.We further demonstrate that the RS GλD calibration model can outperform skewed Normal or Normal calibration model, with lower AIC, BIC and HQ information criterion and lower variability in our parameter estimates in the context of a real life data.These simulation results are promising and future statistical models should aim to develop statistical technique that are tailored to data, rather than requiring empirical data to satisfy a particular statistical model.One possible extension of our model is the development of a mixture RS GλD calibration model, which would extend the flexibility of our model even further but also present a very challenging problem for data with small samples.

Table 2 .
Simulations results with x 0