abstract

The moment-based method is a fundamental approach to the extraction of affine invariants. However, only integer-order traditional moments can be used to construct affine invariants. No invariants can be constructed by moments with an order lower than 2. Consequently, the obtained invariants are sensitive to noise. In this paper, the moment order is generalized from integer to non-integer. However, the moment order cannot simply be generalized from integer to non-integer to achieve affine invariance. The difficulty of this generalization lies in the fact that the angular factor owing to shearing in the affine transform can hardly be eliminated for non-integer order moments. In order to address this problem, the Mellin polar coordinate moment (MPCM) is proposed, which is directly defined by a repeated integral. The angular factor can easily be eliminated by appropriately selecting a repeated integral. A method is provided for constructing affine invariants by means of MPCMs. The traditional affine moment invariants (AMIs) can be derived in terms of the proposed MPCM. Furthermore, affine invariants constructed with real-order (lower than 2) MPCMs can be derived using the proposed method. These invariants may be more robust to noise than AMIs. Several experiments were conducted to evaluate the proposed method performance.

 Introduction

Images of an object captured from different viewpoints are often subject to perspective distortions [1–4]. If the object is small compared to the camera-to-scene distance, the perspective effect becomes negligible and the affine model provides a reasonable approximation of the projective model [2,5]. Therefore, the extraction of affine invariant features plays an important role in object recognition and registration [6–10]. This method has been used extensively in numerous fields, such as shape recognition and retrieval [11,12], watermarking [13], aircraft identification [14,15], texture classification [16], and image registration [17,18]. The moment-based method is the most commonly used technique for the extraction of affine invariant features. However, only integer-order moments can be used to construct affine invariants [4]. It has been reported that high-order moments are sensitive to noise [19]. Hence, in practice, a moment of the lowest possible order should be used [20]. For similarity transform (including only translation, scaling, and rotation), Fourier Mellin descriptors [21] can be viewed as the invariants constructed by complex number order moments. However, similarity transform is only a spe cial case of affine transform [4]. Thus far, an order of moments for constructing affine invariants can only be an integer. In this paper, we consider generalizing the moment order from integer to noninteger, and construct affine invariants by means of the proposed moment.

1- Introduction

2- Formulation of calibration with varying window sizes: How to endogenise and make different window sizes comparable

3- Application of the Lagrange regularisation method to a simple linear-regression problem

4- Using the Lagrange regularisation method for detecting the beginning of financial bubbles

5- Conclusion

References

Abstract

Motivated by the question of identifying the start time τ of financial bubbles, we propose an improved calibration approach for time series in which the inception of the latest regime of interest is unknown. By taking into account the tendency of a given model to overfit data, we introduce the Lagrange regularisation of the normalised sum of the squared residuals, χ 2 np(Φ), to endogenously detect the optimal fitting window size := w∗ ∈ [τ : t¯ 2] that should be used for calibration, assuming a fixed pseudo present time t¯ 2. The Lagrange regularisation of χ 2 np(Φ) defines the Lagrange regularised sum of the squared residuals, χ 2 λ (Φ). Its performance is exemplified on a simple Linear Regression problem with a change point and compared against the performances of the Residual Sum of Squares (RSS) := χ 2 (Φ) and RSS/(N-p) := χ 2 np(Φ), where N is the sample size, p is the number of degrees of freedom and Φ is the parameter vector. Applied to synthetic models of financial bubbles with a well-defined transition regime and to a number of financial time series (US S&P500, Brazil IBovespa and China SSEC Indices), χ 2 λ (Φ) is found to provide well-defined reasonable determinations of the starting times for major bubbles such as the bubbles ending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minor speculative bubbles on other Indexes, without any further exogenous information. The application of the method thus allows one to endogenise the determination of the starting time of bubbles, a problem that has yet not received a systematic objective solution. Moreover, the technique appears as a practical solution for comparing goodness-of-fit across unbalanced sample sizes.

Introduction

There is an inverse relationship between the tendency of a model to overfit data and the sample size under consideration. In other words, the smaller the sample size, the larger the flexibility for the model with a fixed number of parameters to overfit [1]. Due this characteristic feature, one cannot directly compare goodness-of-fit metrics of statistical models arbitrarily parametrised by the vector Φ of parameters such as the Residual Sum of Squares, RSS := χ 2 (Φ) or its normalised version RSS (N−p) := χ 2 np(Φ), over unequal sized samples. Here, N denotes the sample size while p is the number of degrees of freedom of a model. This has particular relevance when one is interested in selecting the optimal sub-sample of a data set to calibrate a model, a recurrent issue when estimating time series models in a moving window or/and when the model is only valid in a specific time window, which is unknown a priori.

1. Introduction

2. Inverse Gaussian Mixed-Effects Model

3. Estimating Model Parameters

4. Framingham Cholesterol Data

5. Conclusion

Acknowledgements

References

Following [7], we introduce the nested Inverse Gaussian Mixed-Effects model to analyze right-skewed and continuous longitudinal data. The nested random effects don’t follow a specific parameter distribution and rely only on the first two moments assumptions in our model. We apply the truly orthodox best linear unbiased predictor (BLUP) approach to estimate the nested random effects. We derive an optimal estimating equation for the regression parameters under the case of known BLUP of random effects. A real example for Framingham cholesterol data is presented to illustrate our proposed methodology.

Introduction

Skewed continuous longitudinal data frequently appear in many areas of research. Various skew normal models have been proposed to analyze skewed longitudinal data in recent years. For example, [5] for linear mixed models by substituting the skew-normal assumption of random effects for the normal assumption; [9] for Bayesian partial linear model; [1] for Skew-normal antedependence models; [8] for mixed effects model with the skew-normal and skew-t assumption of the distribution of responses and random effects. However, these approaches are generally computationally intensive. The inverse Gaussian regression model is a powerful method for analyzing right-skewed continuous data. Following [6] and [7], we consider a class of nested Inverse Gaussian Mixed-Effects Model for right-skewed and continuous longitudinal data. [7] introduced the nested Tweedie mixed model based on an orthodox BLUP approach. Similarly, this orthodox BLUP approach to our models is still computationally simple and efficient. In addition, our approach consolidates conditional and marginal modeling interpretations.

Mt/G/Infinity queueing systems have been widely used to analyse complex systems, such as telephone call centres, software testing systems, and telecommunication systems. Statistical inferences of performance measures, such as the expected cumulative numbers of arrivals and departures, are indispensable for decision makers in analysing the current scenario, predicting future scenarios, and making cost-effective decisions. In most scenarios, we only obtain interval censored data, namely, counts in fixed time intervals, instead of complete data because we either do not want or are not able to monitor arrivals and departures. We provide a general framework for statistical inference in Mt/G/Infinity queueing systems given interval censored data. A maximum-likelihood estimation (MLE) method is proposed for inferring the arrival rate and service duration. This method is applicable to general forms of the arrival rate functions and general service duration distributions. More importantly, we propose a combination of the bootstrap method and the delta method for inferring the expected cumulative numbers of arrivals and departures. The results of the simulation study demonstrate that the point and interval estimates of the proposed MLE method are satisfactory overall. As the number of intervals increases, the estimates based on the proposed MLE approach the estimates based on MLE with complete data. Our procedure enables estimates to be obtained without the need to keep track of each item, thereby substantially reducing resource consumption for monitoring items and storing data. An application in a software testing system demonstrates that the goodness-of-fit performance of the proposed MLE method is satisfactory.

Introduction

An Mt/G/∞ queueing system is a relatively simple queueing system that has a nonhomogeneous Poisson arrival process with a time-dependent deterministic arrival rate function λa ≡ λa(t), independent and identically distributed (i.i.d.) service durations that are independent of the arrival process, and infinitely many servers. Time-varying queueing models, including Mt/G/∞ models, are standard models for describing the dynamics of large-scale service systems, such as telecommunication systems, call centres, and healthcare systems, e.g., hospitals (Pender, 2016). Researchers have applied Mt/G/∞ models to service systems, such as telemarketing, police patrol, fire fighting, hospitals, copy machine repairs, and automatic teller machine operations. In these applications, an operating policy was to keep customer delays close to zero—a scenario that is consistent with the use of an infinite-server model (Green & Kolesar, 1998). Mt/G/∞ models have been applied to storage systems to assess the day-by-day adequacy of stock (Crawford, 1977), to analyse stock requirements (Hillestad & Carrillo, 1980), and to evaluate war-readiness spare requirements for aircraft (Crawford, 1981). They also have been applied to software testing programs (Vizarreta et al., 2018; Yang, 1996) and internet traffic systems (Fay, Roueff & Soulier, 2007). The Mt/G/∞ model is the offered load model for wireless and packet network systems, which describes the total packet carrying capacity of the channels or links in a packet network (Malhotra, Dey, van Doorn & Koonen, 2001; Palm, 1943; Singhai, Joshi & Bhatt, 2009).

Exponentially weighted moving average (EWMA) control charts are typically used for faster detection of shifts in the process mean, relative to a Shewhart control chart, when the degree of shift is small. Normal guidelines suggest using a small (large) value of the weighting constant, λ, for detecting smaller (larger) shifts in the process mean. Prior research has suggested that the choice of λ should depend on the observed data and have considered the use of a weighting constant, that varies and adapts as monitoring continues and new data are collected. One such adaptive control chart, called the AEWMA chart, utilizes a rather computationally complex scheme to determine the weighting constant λ and it requires knowledge of the size of the shift, to specify whether it is “small” or “large”. A complex twophase optimization scheme is then solved to yield “good solutions”. As an alternative, we propose an adaptive EWMA-type control chart that does not require knowledge of the degree of the shift. Further, the computational scheme is easier and completed in one stage. The performance of the proposed chart is studied using simulations, where the degree of the shift in the process mean is varied over a wide range of values. Based on the average run length (ARL), as a performance measure, the proposed chart is demonstrated to perform uniformly better than the traditional EWMA chart with a constant weight. The proposed chart also performs better than the AEWMA chart for moderate to large shifts.

Introduction

The traditional Shewhart control chart technique, introduced in 1924 by Walter Shewhart, has been used in a variety of manufacturing and service industries. The commonly used control charts are those based on the sample mean (X¯), the range (R), the standard deviation (s), individual data (X) and the moving range (MR). In the first two situations (X¯ and R, X¯ and s), it is assumed that independent and identically distributed observations (X) are selected from a process, in subgroups, where the subgroup size is denoted by n. In the third situation, the subgroup size is 1 and the process is monitored based on individual observations (Mitra, 2016). The popular Shewhart control charts, while easy to implement, have the disadvantage that for small shifts in the process parameters, for example, the mean, it takes a rather long time to detect the shift. Consequently, control charts have been developed with the purpose of faster detection of small changes in the process parameters. These include the cumulative sum (CUSUM) chart and the exponentially-weighted moving average (EWMA) control chart (Albin, Kang, & Shea, 1997; Chen & Elsayed, 2002; Domangue & Patch, 1991; Klein, 1996; Lucas & Saccucci, 1990; Roberts, 1959). The traditional EWMA control chart is based on the monitoring statistic.

We propose a novel approach based on the Marshall–Olkin (MO) copula to estimate the impact of systematic and idiosyncratic components on cross-border systemic risk. To use the data on non-failed banks in the suggested method, we consider the time to bank failure as a censored variable. Therefore, we propose a pseudo-maximum likelihood estimation procedure for the MO copula for a Type I censored sample. We derive the log-likelihood function, the copula parameter estimator and the bootstrap confidence intervals. Empirical data on the banking system of three European countries (Germany, Italy and the UK) shows that the proposed censored model can accurately estimate the systematic component of cross-border systemic risk.

Introduction

The 2007–2008 financial crisis has shown how a shock that originates in one country or asset class can quickly propagate to other markets and across borders. A key aspect of financial contagion is given by the linkages among banks. In the Euro area, the cross-border exposures arose as a prominent issue with the European sovereign debt crisis in 2011 and 2012, where large exposure of many EU banks to stressed sovereigns were revealed by the European Banking Authority (EBA, 2013). In a broader perspective, correlated exposures have recently been shown to be a major source of systemic risk. Given the importance of this research field, this paper is focused on systemic risk in the European banking sector. By definition, systemic risk involves a collection of interconnected institutions that have mutually beneficial business relationships through which insolvency can quickly propagate during periods of financial distress Billio, Getmansky, Lo, and Pelizzon (2012). Systemic risk is mainly due to idiosyncratic and systematic shocks (see De Bandt, Hartmann, & Peydró, 2009 and European Central Bank, 2010). The former affects only the health of a single financial institution, while the latter affects the whole economy, e.g. all financial institutions together at the same time. The component of systemic risk due to idiosyncratic shocks is also known as contagion risk in the literature (De Bandt et al., 2009). One of the the main aims of this paper is to propose a new methodological approach for the analysis of systemic risk to jointly model idiosyncratic and systematic shocks. We propose to apply the copula approach to measure systemic risk between the banking sectors of two countries. To our knowledge, the only papers that previously applied copulae to assess banking system stability are Baglioni and Cherubini (2013a), Baglioni and Cherubini (2013b), Straetmans (2010) and Weiss (2012). In other words, the approach is quite novel to the area of banking and systemic risk. The contributions of this paper are twofold. The first of these is to apply the Marshall and Olkin (MO) copula for modelling systemic risk between two countries. The second innovative aspect is how time to failure is considered for non-failed banks as rightcensored.

In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal.

Introduction

There are increasing cases in practice where the data are collected in the form of random functions or curves. This type of data is becoming more prevalent throughout science, engineering and financial market, as automated on-line data collection facilities are becoming more ubiquitous. Many classical statistical tools and models for multivariate analysis, such as principal components analysis, canonical correlation analysis and linear model are then extended to the infinite-dimensional functional domain. In this paper, we consider the functional linear model where Y is a scalar response, X : I → R is a square integrable functional predictor defined over compact domain I ⊂ R, α0 is the intercept, β0 : I → R is the slope function. and  is the random noise with mean 0 and finite variance σ 2 . Functional linear model was introduced by J.O. Ramsay and C.J. Dalzell [11] and first written in its commonly encountered form (1) by T. Hastie and C. Mallows [7]. Some recent research on the statistical analysis of (1) includes [2, 3, 6, 15, 16]. In this paper, we focus on the random design where X is a path of a square integrable stochastic process defined over I and is independent of . Without loss of much generality, throughout the paper we assume E(X) = 0 and the intercept α0 = 0, since the intercept can be easily estimated. Let L 2 be the Hilbert space of square integrable functions on I (with respect to Lebesgue measure) with standard inner production < u, v >= R I u(s)v(s)ds and norm kuk = R I u 2 (s)ds
1/.

abstract

Constrained approaches to maximum likelihood estimation in the context of finite mixtures of normals have been presented in the literature. A fully data-dependent soft constrained method for maximum likelihood estimation of clusterwise linear regression is proposed, which extends previous work in equivariant data-driven estimation of finite mixtures of normals. The method imposes soft scale bounds based on the homoscedastic variance and a cross-validated tuning parameter c. In our simulation studies and real data examples we show that the selected c will produce an output model with clusterwise linear regressions and clustering as a most-suited-to-the-data solution in between the homoscedastic and the heteroscedastic models.

Introduction

Let {(yi, xi)}n = {(y1, x1),…,(yn, xn)} be a sample of n independent units, where yi is the outcome variable and xi are the J covariates. A clusterwise linear regression model assumes that the density of yi|xi is given by f (yi|xi;ψ) = G g=1 pg f g (yi|xi;σ 2 g ,βg ) = G g=1 pg 1 2πσg 2 exp  − (yi − x i βg )2 2σ 2 g  , (1) where G is the number of clusters, ψ = {(p1,…, pG;β1,…,βG;σ 2 1 ,…,σ 2 G ) ∈ RG( J+2) : p1 +···+ pG = 1, pg ≥ 0,σ 2 g > 0, g = 1,…, G} is the set of model parameters, and pg , βg , and σ 2 g are respectively the mixing proportions, the vector of J regression coefficients, and the variance term for the g-th cluster. The model in Equation (1) is also known under the name of finite mixture of linear regression models, or switching regression model [21,22,15]. The parameters of finite mixtures of linear regression models are identified if some mild regularity conditions are met [10]. The clusterwise linear regression model of Equation (1) can naturally serve as a classification model. Based on the model, one computes the posterior membership probabilities for each observation as follows: and then classify each observation according, for instance, to fuzzy or crisp classification rules. The problem of clustering sample points grouped around linear structures has been receiving a lot of attention in the statistical literature because of its important applications (see, for instance, [16], and references therein. For the robust literature, among the others, see [6,7]). In order to estimate ψ, one has to maximize the following sample likelihood function L(ψ; y) =n i=1  G g=1 pg 1 2πσg 2 exp  − (yi − x i βg )2 2σ 2 g  , (3) which can be done using iterative procedures like the EM algorithm [5], whose clustering can be interpreted as a fuzzy partition [9]. Unfortunately, maximum likelihood (ML) estimation of univariate unconditional or conditional normals suffers from the well-known issue of unboundedness of the likelihood function: whenever a sample point coincides with the group’s centroid and the relative variance approaches zero, the likelihood function increases without bound ([14]; also the multivariate case suffers from the issue of unboundedness. See [4]). Hence a global maximum cannot be found. Yet, ML estimation does not fail: Kiefer [15] showed that there is a sequence of consistent, asymptotically efficient and normally distributed estimators for switching regressions with different group-specific variances (heteroscedastic switching regressions). These estimators correspond, with probability approaching one, to local maxima in the interior of the parameter space. Nonetheless, although there is a local maximum which is also a consistent root, there is no tool for choosing it among the local maxima. Day [4] showed, for multivariate mixtures of normals, that potentially each sample point – or any pair of sample points being sufficiently close together, or co-planar [24] – can generate a singularity in the likelihood function of a mixture with heteroscedastic components. This gives rise, both in univariate and multivariate contexts, to a number of spurious maximizers [18].

Geotechnical engineers face a number of uncertainties [1, 2]. Soil materials formed from geological weathering processes, and by physical means to deliver the soil to the current position [3]. In the forming process, the soil is influenced by various stress, pore fluid, and physical and chemical changes. Therefore, it is not surprising that there are always some outliers in geotechnical data. When dealing with geotechnical problems, empirical correlations between in situ or laboratory test results and geotechnical parameters are often used in geotechnical design. When establishing such empirical correlations, mostly adopted method is regression analysis, including linear or nonlinear regression [4]. Linear least squares regression (LLR) is a modeling approach by far the most widely used. When people say they use “regression”, “linear regression” or “least squares” to adapt their data, they usually mean doing LLR. LLR is not only the most widely used method of modeling, but also have adapted to a variety of circumstances, beyond its immediate scope [5]. A mathematical method that finds the best-fit curve for a given set of points is to minimizing the sum of the squares of the distances of regression data deviating from the curve. The sum of squares of the offset distances is used instead of the absolute values of the offset distances because this allows the residuals to be treated as a continuously differentiable quantity. Whereas, because of the use of the square of the offset, peripheral points may have a disproportionate effect on fit. Whether the results are desirable or not, it depends on the issue of question [6]. The statistical observations of outliers are significantly different from the other sample values. Least-squares regression is obviously the best option if errors are normally distributed. Then, other means is eagerly required if these errors are not normally distributed. One particular distribution is the long tail error distribution of great concern. One solution is still to use the LLR method after removing the largest remaining value as outliers. However, this solution may be infeasible if several larger residual values exist by reason that the poor nature of the outlier tests. In addition, outlier testing is an acceptance/rejection process. The testing processes are neither smooth or statistically efficient. Robust regression (RR) is another option for least-squares regression in the case of the data contaminated with outliers. It can also be used to detect influential observations when the data is exposed to outliers [7]. It is difficult to define absolute outliers from geotechnical testing data, but it is possible to indicate the least predictable or relatively outlying data points using statistical tools. The objective of this paper is to demonstrate the advantages of RR analysis used in geotechnical data comparing with least square regression analysis. The procedure of RR is discussed shortly, based on LLR method. And then, Regression analysis is operated on undrained shear strength (su) data derived from CPTU test with both RR and LLR. Comparing regression result, the outlier of su data can be detected based on RR method.

Groundwater and surface water has been exploited due to several noxious impurities and  among all chemical dyes is one of the most detrimental impurity that possess serious  environmental side effect on human being and marine life (Garg et al., 2003;Tanzifi et al., 2018; Yao et al., 2018 Gupta et al., 2011b; Gupta et al., 2013a). Synthetic dyes and pigments are widely utilized as colorants in different industrial processes including the pharmaceutical, textile,  leather, paper, gasoline, food industries etc. and a huge amount of unused synthetic dyes is being 8 released in water streams by various industries worldwide annually. Dyes being colored in nature  is visible even at very low concentration and it resistive nature also decreases the sunlight penetration in water. Therefore, their release into the environment presents a major source of contamination (Gupta et al., 2013b; Gupta et al., 2014; Tian et al., 2018). Dyes molecules are normally produced from the aromatic structure and their degradation products are toxic or even carcinogenic and mutagenic. In addition to this, it lead to increased biological oxygen demand (BOD) and chemical oxygen demand (COD) levels of aquatic resources (Crini, 2006; Nekouei et al., 2017). Thus, it is the need of today to eliminate the toxic dyes from effluents of industries, before it is released into the environment. Several technologies for instance flocculation–coagulation (Szygula et al., 2009; Wang et al., 2012), photodegradation (Gupta et al., 2011c; Smirnova et al., 2015; Tadjarodi et al., 2013), biodegradation (Li et al., 2004), membrane separation(Bouazizi et al., 2017; Chen et al., 2017; 20 Ruan et al., 2016; Saravan et al., 2013; 2015; Tahri et al., 2016), aerobic or anaerobic treatment (Fernando et al., 2013), electrochemical (Aquino et al., 2010; Riera-Torres and Gutiérrez, 2010), and oxidizing agents (Ghoneim et al., 2011) have been used worldwide for the removal of noxious dyes from wastewater. In spite of the growth of different techniques for treatment of dyeeffluent, but still there are some specific challenges, for example effective, rapid and economic 2 for treatment of water and wastewater at a commercial level are going on. Among these methods  for water treatments, the adsorption technique by solid adsorbents is one of the most efficient  approaches for the uptake of dyes from industrial wastewater (Gupta et al., 2011a). The key  advantage of this technology is that it is scalable and large volume of effluents can be passed in a single run, which lead to high quality of treated wastewaters without the formation of harmful materials (Gupta et al., 2010b).

The aero-material spare parts are essential material basis for aviation equipment maintenance engineering. With the development of aviation equipment, more and more complex equipment, maintenance of spare parts required for the variety and quantity are more and more spare parts financing, supply and storage process is also more complex. In this paper, we use linear regression method to forecast the consumption of spare parts [1]. The regression analysis and forecasting method is based on the analysis of the correlation between independent variables and dependent variables, the regression model between variables is built, and the regression model is used as the forecasting method. There are a lot of ways of regression analysis. Depending on the number of independent variables in the relationship the regression models can be divided into simple regression analysis and multivariate regression analysis. Depending on the correlation between independent variables and dependent variables, the regression models can be divided into linear regression forecasting and nonlinear regression forecasting. This paper focuses on simple linear regression prediction.

2. The principle of simple linear regression model

The linear regression is a linear method used to simulate the relationship between one dependent variable and many explanatory variables. The case of one explanatory variable is called simple linear regression model, while the case of multiple explanatory variables is called multivariate linear regression model [2-4]. As a commonly used statistical methods and for its principles is clear, model is simple and easy to use, classical linear regression model has been a very wide range of applications in the aviation equipment maintenance and support.

Suppose that we are interested in estimating a linear regression model Y = β0 + X 0 1β1 + X 0 2β2 + Z 0 γ + u := W0 θ + u, E (u| W) = 0, (1) using a random sample, where X1 ∈ R d1 , X2 ∈ R d2 and Z ∈ R d3 . The reason for distinguishing between the regressors X1, X2 and Z will become clear shortly. In addition, while d1 = 0 is allowed, d2, d3 > 0 must be the case in our setup. When W = (1, X0 1 , X0 2 , Z0 ) 0 ∈ R d+1, where d := d1 + d2 + d3, is exogenous and a single random sample of (Y, X1, X2, Z) can be obtained, the ordinary least squares (OLS) estimator of θ = (β0, β0 1 , β0 2 , γ0 ) 0 is consistent. In reality, however, we often face the problem that (Y, X1, X2, Z) cannot be taken from a single data source. It is not uncommon that economists who use survey data for empirical analysis must collect all necessary variables from more than one source. Examples include Lusardi (1996), Bj¨orklund and J¨antti (1997), Currie and Yelowitz (2000), Dee and Evans (2003), Borjas (2004), Bover (2005), Fujii (2008), Bostic et al. (2009), and Murtazashvili et al. (2015), to name a few. Ridder and Moffitt (2007) provide an excellent survey. This is the setting in which we are interested. Specifically, suppose that instead of observing a complete data set (Y, X1, X2, Z), we have the following two overlapping subsets of data, (Y, X1, Z) and (X2, Z), i.e., some of the regressors are not available in the initial data set, where the initial data set is the one containing observations on the dependent variable along with a few other regressors. In such a setting, it is natural to construct a matched data set via exploiting the proximity of the common regressor(s) Z across the two samples. This is often called “probabilistic record linkage”. Here are two examples of the setting.

statement of the problem and motivation The problem under study: variable selection in functional regression The study of regression models is clearly among the leading topics in statistics. In particular, these models play a central role in the theory of statistics with functional data, often called Functional Data Analysis (FDA); see [7, 15, 16] for an overview on FDA. Throughout this paper, we will consider “functional data” consisting of independent X1 = X1(t), . . . , Xn = Xn(t) observations (trajectories) drawn from a second-order (L 2 ) stochastic process X = X(t), t ∈ [0, 1], with continuous trajectories and continuous mean and covariance functions, denoted by m = m(t) and K(s, t), respectively. All the involved random variables are supposed to be defined in a common probability space (Ω, A, Pr). We are interested on functional regression models with scalar response, of type Yi = g(Xi) + εi , where g is a real function defined on a suitable space X where the trajectories of our process are supposed to live. The random variables εi are independent errors (and also independent from the Xi) with mean zero and common variance σ 2 . More specifically, we are concerned with variable selection issues; see, [4, Sec. 1], [11] for additional information and references. Basically, a variable selection functional method is an automatic procedure that takes a function {x(t), t ∈ [0, 1]} to a finite-dimensional vector (x(t1), . . . , x(tp)). The overall idea for variable selection is to choose the variables x(ti) (or, equivalently, the “impact points” t1, . . . , tp ∈ [0, 1]; see [22]), in an “optimal way” so that the original functional problem (regression, classification, clustering,…) is replaced with the corresponding multivariate version, based on the selected variables. In the regression setting, this would amount to replace the functional model Yi = g(Xi) + εi with a finite dimensional version of type Yi = φ{X(t1), . . . , X(tp)} + ei . Nevertheless, note that still the problem is of a functional nature, since the methods to select the ti are generally based upon the full data trajectories.