Nonparametric estimation by convex programming

The problem we concentrate on is as follows: given (1) a convex compact set $X$ in ${\mathbb{R}}^n$, an affine mapping $x\mapsto A(x)$, a parametric family $\{p_{\mu}(\cdot)\}$ of probability densities and (2) $N$ i.i.d. observations of the random variable $\omega$, distributed with the density $p_{A(x)}(\cdot)$ for some (unknown) $x\in X$, estimate the value $g^Tx$ of a given linear form at $x$. For several families $\{p_{\mu}(\cdot)\}$ with no additional assumptions on $X$ and $A$, we develop computationally efficient estimation routines which are minimax optimal, within an absolute constant factor. We then apply these routines to recovering $x$ itself in the Euclidean norm.Comment: Published in at http://dx.doi.org/10.1214/08-AOS654 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gee-Smoothing Spline for Semiparametric Estimation of Longitudinal Categorical Data

In this thesis we propose estimation methods of semiparametric marginal models for longitudinal (correlated) categorical data, where the systematic component of the model consists of parametric and nonparametric forms. We develop GEE-Smoothing spline as a method to analyze semiparametric model for longitudinal data. The proposed methods are an extension of parametric generalized estimating equation (GEE) to semiparametric GEE by introducing smoothing spline into parametric GEE. We derive estimation method of GEE-Smoothing spline in the case of longitudinal binary, ordinal, and nominal data. Derivation of the estimating equation of GEE-Smoothing spline for these three types ofcategorical data is the same. However their estimating equations have different forms of the covariance and correlation matrices. In the estimation of the association (correlation) parameter for binary data, we use moment method of Liang & Zeger’s and method of Prentice’s. For ordinal and nominal data, we use different models of the covariance matrices than of binary data. These models need smaller number of the association parameter to be estimated which is different from the existing models of parametric GEE for ordinal data. We also derive and propose the methods to estimate the association parameter for these two types of data. The properties of the estimate for both parametric and nonparametric components of GEE-Smoothing spline are evaluated using simulation studies. We obtained that the estimates of parametric component for binary and ordinal data are unbiased. Whilst for nominal data, the estimates of parametric componentare almost unbiased. Meanwhile the estimates of the nonparametric component for all types of data are biased, with the bias decreases when the samplesize increases. The estimators of both parametric and nonparametric components are also consistent, and the consistency is not affected by the correct or incorrect working correlation used in model. This consistency property holds for correlated and independent data. The efficiency of the estimates of using independent or correlated working correlation in the estimation depends on the type of covariate, such as time varying, subject specific, or mean-balanced covariates. The estimates of both parametric and nonparametric components also follow the central limit theorem (CLT), for both independent and correlated data, and using correct or incorrect working correlation. Both components estimate have normal distribution

MODELLING JAKARTA COMPOSITE INDEKS USING SPLINE TRUNCATED

Regression analysis can be done by parametric and nonparametric approach. The nonparametric approach does not assume an assumption compared to parametric. One nonparametric approach is the spline truncated. Spline is a polynomial piece that provides high flexibility. Spline modeling requires spline and knots. To determine the knots using General Cross Validation (GCV). In this study modeled the value of Jakarta Composite  Index (JCI). JCI provides benefits to know the overall stock price in the stock exchange Indonesia. In this study the best spline model is linear with three knots with R square is 94.34%. Keywords: Jakarta Composite’s Index, Spline truncated, GCV

Analisis Indeks Harga Saham Gabungan (Ihsg) Dengan Menggunakan Model Regresi Kernel

Saham merupakaninvestasi yang banyak dipilih para investor, salah satu indikator yang menunjukkan pergerakan harga saham adalah Indeks Harga Saham Gabungan (IHSG). IHSG merupakan data runtun waktu sehingga untuk menganalisisnya dapat menggunakan metode runtun waktu klasik. Namun dengan metode tersebut banyak asumsi yang harus dipenuhi, sehingga diperlukan metode alternatif salah satunya metode regresi nonparametrik karena dalam model regresi nonparametrik tidak ada asumsi khusus sehingga model ini merupakan metode alternatif yang dapat digunakan dalam analisis IHSG. Dalam makalah ini dibandingkan nilai MSE yang dihasilkan dari analisis runtun waktu klasik, regresi parametrik linier sederhana dan regresi nonparametrik kernel. Data IHSG yang digunakan adalah periode minggu pertama Januari 2011 sampai dengan minggu ke empat Februari 2012. Data tersebut merupakan data closing price saham mingguan pada periode perdagangan terakhir. Hasil perbandingan nilai MSE dari dataIHSG yang sering fluktuatif pada tiga analisis didapatkan nilai MSE terkecil adalah pada analisis menggunakan regresi nonparametrik kernel dengan fungsi triangle dan badwidth h sebesar 58.2 dengan nilai MSE = 6987.787. Model terbaik tersebut dapat digunakan untuk memprediksikan nilai IHSG selanjutnya

Pemodelan Data Inflasi Indonesia pada Sektor Transportasi, Komunikasi, dan Jasa Keuangan Menggunakan Metode Kernel dan Spline

In this research, we study data modeling of Indonesian inflation in the transportation, communication and financial services sector using the kernel and spline models. Determination of the optimal models based on the smallest of GCV value and determination of the best model based on the smallest out sampels of Mean Square Error (MSE) value. By modeling the yoy (year on year) inflation data in Indonesia in the transportation, communication and financial services sector In January 2007 to January 2015, shows that the kernel model using Gaussian kernel function obtained optimal model with a bandwidth 0.24 and the optimal spline model with order 5 and 4 points knots. Based on out sampels data in February to August 2015, obtained out sampels MSE value of the spline model is smaller than the kernel model. So that the spline model is better than the kernel model to analyze the inflation data of transportation, communication and financial services sector

Pemodelan Regresi Nonparametrik Menggunakan Pendekatan Polinomial Lokal pada Beban Listrik di Kota Semarang

Semarang is the provincial capital of Central Java, with infrastructure and economic's growth was high. The phenomenon of power outages that occurred in Semarang, certainly disrupted economic development in Semarang. Large electrical energy consumed by industrial-scale consumers and households in the San Francisco area, monitored or recorded automatically and presented into a historical data load power consumption. Therefore, this study modeling the load power consumption at a time when not influenced by the use of electrical load (t-1)-th. Modeling using nonparametric regression approach with Local polynomial. In this study, the kernel used is a Gaussian kernel. In local polynomial modeling, determined optimum bandwidth. One of the optimum bandwidth determination using the Generalized Cross Validation (GCV). GCV values obtained amounted to 1425.726 with a minimum bandwidth of 394. Modelling generate local polynomial of order 2 with MSE value of 1408.672

Pemodelan Indeks Harga Saham Gabungan (Ihsg) Menggunakan Multivariate Adaptive Regression Splines (Mars)

Composite Stock Price Index (CSPI) is a historical information about the movement of joint-stock until a certain date. CSPI is often used by inventors to see a representation of the overall stock price, it can analyze the possibility of increase or decrease in stock price. Following old examination, some economy macro variables affecting CSPI is inflation, interest rate,and exchange rate the Rupiah againts the u.s.dollar. MARS method is particularly suitable to analyze a CSPI because many variables that affected. Furthermore, in the real world is very difficult to find a spesific data pattern. The analysis is MARS analysis. The purpose is an obtained a MARS model to be used to analyze the CSPI movement\u27s. Selection MARS model can be used CV method. The MARS model is an obtained from combination of BF, MI, dan MO. In this case, happens the best models with BF=9, MI=2, dan MO=1. Accuracy for MARS model can see MAPE values is 14,32588% it means the model can be used