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"Regresi kuantil merupakan salah satu teknik regresi dengan memodelkan kuantil dari variabel dependen bersyarat variabel penjelas. Model yang diperoleh dengan regresi kuantil merupakan suatu gambaran lengkap atas perilaku data baik di bagian tengah maupun ekor (tail) sebaran. Sehingga teknik ini baik digunakan untuk analisa data apabila dicurigai adanya perbedaan pengaruh variabel penjelas terhadap bagian-bagian tertentu variabel dependen. Hal ini dapat dilihat dari hasil taksiran parameter regresi kuantil yang berubah secara monoton. Selain itu regresi kuantil juga bagus digunakan pada data dengan nilai ekstrim yang penting untuk dianalisa. Untuk mendapatkan model regresi kuantil diperlukan proses penaksiran parameter yang dilakukan dengan meminimumkan ekspektasi suatu fungsi loss. Proses optimisasi ini selanjutnya diubah ke dalam program linier dan dapat diselesaikan dengan metode interior point. Metode interior point yang digunakan dalam skripsi ini mengacu pada algoritma Frisch-Newton. Selanjutnya pada skripsi ini, regresi kuantil akan diterapkan pada dua data yang masing-masing memiliki karakteristik yang berbeda.

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Quantile regression is a regression technique by modeling the conditional quantile of the dependent variable. Models obtained with quantile regression is a complete picture of the behavior of the data either in the middle or tail. This technique is well used to analyze data when there is suspected differences in the effect of explanatory variables on the dependent variable. It can be seen from the results of quantile regression parameter estimates which changed monotonically. In addition quantile regression is also good to use on the data with extreme values that are important to be analyzed. To get the required quantile regression model, parameter estimation process is done by minimizing the expectation of a loss function. The optimization process is then converted into a linear program and can be solved by interior point methods. Interior point methods used in this skripsi refers to the Frisch-Newton algorithm. Later in this skripsi, quantile regression will be applied to the two data each has different characteristics."

"Regresi kuantil adalah metode regresi yang menghubungkan kuantil dari variabel respon dengan satu atau beberapa variabel prediktor. Regresi kuantil memiliki kelebihan yang tidak dimiliki oleh regresi linier yaitu robust terhadap outlier dan dapat memodelkan data yang heteroskedastisitas. Regresi kuantil dapat diestimasi parameternya dengan metode Bayesian.Metode Bayesian adalah alat analisis data yang diturunkan berdasarkan prinsip inferensi Bayesian. Inferensi Bayesian adalah proses mempelajari analisis data secara induktif dengan teorema Bayes. Untuk menaksir parameter regresi dengan inferensi Bayesian, perlu dicari distribusi posterior dari parameter regresi dimana distribusi posterior proporsional terhadap perkalian distribusi prior dan fungsi likelihoodnya. Karena perhitungan distribusi posterior secara analitik sulit untuk dilakukan jika semakin banyak parameter yang ditaksir, maka diajukan metode Markov Chain Monte Carlo (MCMC). Penggunaan metode Bayesian dalam regresi kuantil memiliki kelebihan yaitu penggunaan MCMC memiliki kelebihan yaitu mendapatkan sampel nilai parameter dari distribusi posterior yang tidak diketahui, penggunaanyang efisien secara komputasi, dan mudah diimplementasikannya. Yu dan Moyeed (2001) memperkenalkan regresi kuantil Bayesian dengan menggunakan fungsi likelihood dari error yang berdistribusi Asymmetric Laplace Distribution (ALD) dan menemukan bahwameminimumkan taksiran parameter pada regresi kuantil sama dengan memaksimalkan fungsi likelihood dari error yang berdistribusi Asymmetric Laplace Distribution (ALD). Metode yang digunakan untuk menaksir parameter regresi kuantil adalah Gibbs sampling dari distribusi ALD yang merupakan kombinasi dari distribusi eksponensial dan Normal. Penaksiran parameter model regresi dilakukan dengan cara pengambilan sampel pada distribusi posteriordari parameter regresi yang ditemukan dalam skripsi ini. Pengambilan sampel pada distribusi posterior dapat menggunakan metode Gibbs sampling. Hasil yang diperoleh dari Gibbs sampling berupa barisan sampel parameter yang diestimasikan. Setelah mendapatkan barisan sampel, barisan sampel dirata-ratakan untuk mendapatkan taksiran parameter regresinya. Studi kasus dalam skripsi ini adalah membahas pengaruh faktor risiko dari nasabah asuransi kendaraan bermotor terhadap besar klaim yang diajukan oleh nasabah.

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Quantile regression is a regression method that links the quantiles of the response variable with one or more predictor variables. Quantile regression has advantages that linear regression does not have; it is robust against outliers and can model heteroscedasticity data.

The parameters of quantile regression can be estimated using the Bayesian method. The Bayesian method is a data analysis tool derived based on the Bayesian inference principle.

Bayesian inference is the process of studying data analysis inductively with the Bayes theorem. To estimate regression parameters with Bayesian inference, it is necessary to find the posterior distribution of the regression parameters where the posterior distribution is

proportional to the product of the prior distribution and its likelihood function. Since the calculation of the posterior distribution analytically is difficult to do if the more parameters are estimated, the Markov Chain Monte Carlo (MCMC) method is proposed. The use of the Bayesian method in quantile regression has advantages, namely the use of MCMC has the advantages of obtaining sample parameter values from an unknown posterior distribution,

using computationally efficient, and easy to implement. Yu and Moyeed (2001) introduced Bayesian quantile regression using the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) distribution and found that minimizing parameter estimates in quantile regression is the same as maximizing the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) distribution. The method used to estimate quantile regression parameters is Gibbs sampling from the ALD distribution, which is a combination

of the exponential and normal distributions. The estimation of the regression model parameters is done by sampling the posterior distribution of the regression parameters which is found in this thesis. Gibbs sampling method is used to sampling the posterior distribution.

The results obtained from Gibbs sampling are a sample sequence of estimated parameters.

After obtaining the sample sequences, the sample lines are averaged to obtain an estimated regression parameter. The case study in this thesis discusses the effect of risk factors from motor vehicle insurance customers on the size of claims submitted by customers."

"Permasalahan mengenai pencadangan klaim pada perusahaan asuransi merupakan salah satu isu yang harus dihadapi oleh para pelaku bisnis asuransi. Ketersediaan dana klaim oleh perusahaan merupakan hal yang mendasar pada perusahaan asuransi untuk dapat mempertahankan bisnis mereka dan menjaga kelangsungan dari usahanya. Pencadangan klaim ini juga diperlukan perhitungan secara detil mengenai pengalokasian dana yang dimiliki perusahaan berdasarkan penerimaan penjualan produk yang dikeluarkan, untuk menghasilkan profit di dalam bisnis mereka. Berangkat dari keterbatasan model-model terdahulu, tulisan ini ingin memperkenalkan model penghitungan alternatif, yakni model quantile regression. Menurut Chan 2015 model quantile regression ini dianggap memiliki kemampuan untuk melakukan penghitungan pencadangan klaim terhadap data yang memiliki variansi heterogen dan tidak memiliki distribusi yang jelas sebagaimana kebanyakan data asuransi. Penelitian ini akan melakukan penghitungan estimasi cadangan klaim dengan mengadopsi model Quantile Regression. Tujuan utama dari penelitian ini adalah ingin mencoba bagaimana proses penghitungan estimasi dengan model Quantile Regression serta melihat apakah model ini bisa diterapkan pada konteks perusahaan asuransi XYZ di Indonesia. Data yang digunakan dalam penelitian ini adalah data klaim produk asuransi kendaraan bermotor perusahaan XYZ tahun 2013 sampai dengan 2015.

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The issue of claim reserves on insurance companies is one of the issues that insurance businesses have to cope with. The availability of claims within the company is fundamental to insurance companies to maintain their business and keep the business going. This claim reserves is also required in precise calculations regarding the allocation of funds owned by the company based on the sale of products issued, to generate profit in their business. Based on the limitations of the traditional models, this paper wants to introduce an alternative model of estimating claim reserve, it is called quantile regression model. According to Chan 2015 this quantile regression model is considered to have the ability to calculate the reserve of claims against data with heterogeneous variance and have no clear distribution, which is mostly insurance data known for. This research will try to calculate estimation for claim reserve by adopting Quantile Regression model. The main purpose of this research is to try how to calculate the estimation with Quantile Regression model and see if this model can be applied to the context of XYZ insurance company in Indonesia. The data used in this research are the claims data of XYZ company s for motor vehicle insurance products from 2013 to 2015."

"Written for specialists working in optimization, mathematical programming, or control theory. The general theory of path-following and potential reduction interior point polynomial time methods, interior point methods, interior point methods for linear and quadratic programming, polynomial time methods for nonlinear convex programming, efficient computation methods for control problems and variational inequalities, and acceleration of path-following methods are covered. In this book, the authors describe the first unified theory of polynomial-time interior-point methods. Their approach provides a simple and elegant framework in which all known polynomial-time interior-point methods can be explained and analyzed; this approach yields polynomial-time interior-point methods for a wide variety of problems beyond the traditional linear and quadratic programs. The book contains new and important results in the general theory of convex programming, e.g., their "conic" problem formulation in which duality theory is completely symmetric. For each algorithm described, the authors carefully derive precise bounds on the computational effort required to solve a given family of problems to a given precision. In several cases they obtain better problem complexity estimates than were previously known. Several of the new algorithms described in this book, e.g., the projective method, have been implemented, tested on "real world" problems, and found to be extremely efficient in practice."

"In the past decade, primal-dual algorithms have emerged as the most important and useful algorithms from the interior-point class. This book presents the major primal-dual algorithms for linear programming in straightforward terms. A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software. This is an excellent, timely, and well-written work. The major primal-dual algorithms covered in this book are path-following algorithms (short- and long-step, predictor-corrector), potential-reduction algorithms, and infeasible-interior-point algorithms. A unified treatment of superlinear convergence, finite termination, and detection of infeasible problems is presented. Issues relevant to practical implementation are also discussed, including sparse linear algebra and a complete specification of Mehrotra's predictor-corrector algorithm. Also treated are extensions of primal-dual algorithms to more general problems such as monotone complementarity, semidefinite programming, and general convex programming problems."

"This compact book, through the simplifying perspective it presents, will take a reader who knows little of interior-point methods to within sight of the research frontier, developing key ideas that were over a decade in the making by numerous interior-point method researchers. It aims at developing a thorough understanding of the most general theory for interior-point methods, a class of algorithms for convex optimization problems. The study of these algorithms has dominated the continuous optimization literature for nearly 15 years. In that time, the theory has matured tremendously, but much of the literature is difficult to understand, even for specialists. By focusing only on essential elements of the theory and emphasizing the underlying geometry, A Mathematical View of Interior-Point Methods in Convex Optimization makes the theory accessible to a wide audience, allowing them to quickly develop a fundamental understanding of the material.The author begins with a general presentation of material pertinent to continuous optimization theory, phrased so as to be readily applicable in developing interior-point method theory. This presentation is written in such a way that even motivated Ph.D. students who have never had a course on continuous optimization can gain sufficient intuition to fully understand the deeper theory that follows. Renegar continues by developing the basic interior-point method theory, with emphasis on motivation and intuition. In the final chapter, he focuses on the relations between interior-point methods and duality theory, including a self-contained introduction to classical duality theory for conic programming; an exploration of symmetric cones; and the development of the general theory of primal-dual algorithms for solving conic programming optimization problems.Rather than attempting to be encyclopedic, A Mathematical View of Interior-Point Methods in Convex Optimization gives the reader a solid understanding of the core concepts and relations, the kind of understanding that stays with a reader long after the book is finished."

"Analisis regresi digunakan untuk mengetahui hubungan antara satu variabel respon dan satu atau lebih variabel penjelas. Ketika variabel respon berupa data count yaitu data yang berupa bilangan bulat non-negatif, analisis regresi yang sering digunakan adalah analisis regresi Poisson. Pada regresi Poisson terdapat asumsi kesamaan nilai mean dengan nilai variansinya. Dalam data count sering didapati kondisi dimana nilai variansi lebih besar dari nilai meannya atau disebut overdispersi. Pada data yang overdispersi, regresi Poisson kurang tepat jika digunakan karena nilai standard error dari taksiran parameter yang dihasilkan akanunderestimate sehingga beresiko memberikan kesimpulan yang tidak tepat. Model regresi Poisson-Inverse Gaussian dapat digunakan pada data count yang overdispersi dan memiliki tail panjang. Penaksiran parameter model regresi Poisson-Inverse Gaussian menggunakan metode maksimum likelihood dan solusi dari fungsi log -likelihood-nya menggunakan pendekatan numerik yaitu Newton-Raphson. Uji kesesuaian model yang digunakan mencakup statistik pseudo R-Squared, uji rasio likelihood, dan Uji Wald.

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Regression analysis is used to investigate the relationship between one response variable and one or more regressor variables. If the response variable is count data, that has non negative integer value, the regression analysis that usually used is Poisson Regression. Poisson regression has an assumption that mean of response variable equal to its variance. On count data frequently found that the variance is greater than mean, or called overdispersion. On overdispersion case, poisson regression is inconvenient to used because it may underestimate the standard error of regression parameters and consequently it risk to give misleading inference. Poisson Inverse Gaussian regression model can be used on overdispersion and long tail count data. Parameter estimation of Poisson Inverse Gaussian Regression Model can be obtained through the maximum likelihood method and the solution of log likelihood function may be solved by using numerical method called Newton Raphson. Goodness of fit testing of this model includes pseudo R Squared, rasio likelihood test, and Wald test."

"Generalized Estimating Equation (GEE) adalah metode penaksiran parameter model regresi yang pengamatan-pengamatannya saling berkorelasi, yang dapat disebabkan oleh lokasi atau kelompok, yang biasa disebut sebagai data terklaster. Penaksiran parameter dalam metode GEE menggunakan suatu fungsi yang dibangun dari bentuk umum distribusi keluarga eksponensial yang erat kaitannya dengan Generalized Linear Model (GLM). Ada dua model pendekatan dalam menganalisis data terklaster menggunakan metode GEE yaitu Population Averaged (PA) dan Clustered Specific (CS). Tugas akhir ini membahas mengenai bagaimana menaksir parameter model regresi linier Clustered Specific Generalized Estimating Equation (CS-GEE) pada data terklaster, serta aplikasinya pada data kemiskinan di Provinsi Jawa Timur. Uji kesesuaian model regresi linier CS-GEE yang digunakan adalah Uji Wald dengan menggunakan Naïve standard error.

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Generalized Estimating Equation (GEE) is a regression model parameter estimation method where correlation exist in the observations due to the locations or groups, used known as clustered data. Parameter estimation of GEE method using a function constructed from a general form of exponential family distributions that closely related to the Generalized Linear Model (GLM). There are two models approach in analyzing the clustered data that is Population Averaged (PA) and Clustered Specific (CS).

This skripsi discusses about how to estimate linear regression model parameters of Clustered Specific Generalized Estimating Equation (CS-GEE) on clustered data, as well as its application on the poverty data in East Java. Goodness of fit testing of this model is Wald test by using the Naïve standard error."

"Model regresi data panel dinamis merupakan model regresi data panel yang melibatkan lag dari variabel dependen sebagai variabel eksplanatori yang berkorelasi dengan error. Lag dari variabel dependen tersebut dinamakan variabel endogen eksplanatori. Adanya variabel endogen eksplanatori menyebabkan estimasi parameter menggunakan metode OLS menghasilkan taksiran yang bias dan tidak konsisten. Oleh karena itu dibutuhkan metode lain untuk menaksir parameter, salah satunya adalah metode yang dikembangkan oleh Arellano dan Bond. Arellano dan Bond mengembangkan metode penaksiran parameter melalui proses first differencing dan metode instrumental variabel sehingga taksiran yang dihasilkan oleh metode ini memiliki sifat tak bias, konsisten dan efisien. Metode Arellano dan Bond tersebut kemudian dikembangkan oleh Blundell dan Bond dengan cara mengkombinasikan momen kondisi dan matriks instrumen antara model first difference dan model level untuk menghasilkan taksiran yang sama-sama tak bias dan konsisten tetapi lebih efisien yang dinamakan GMM-System Estimator.

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Regression model of dynamic panel data is a regression model of panel data involving lag of dependent variable as explanatory variables which are correlated with the error. Lag of dependent variable is called endogenous explanatory variables. The presence of this lag cause the estimates of the parameters produce the estimator that are biased and inconsistent using OLS method. Therefore, other methods are needed to estimate the parameters, one of is the method developed by Arellano and Bond.

Arellano and Bond developed a method of parameter estimation through a process of first-differencing and instrumental variable method so that the estimator are unbiased, consistent and efficient. This method is then developed by Blundell and Bond with combine the moment conditions and matrix of instruments between first-difference model and level model to produce the estimator that are both unbiased and consistent but more efficient thus it called GMM-System estimator."