## Single index model statistics

8 Semiparametric Single Index Models 8.1 Index Models A object of interest such as the conditional density f (y j x) or conditional mean E(y j x) is a single index model when it only depends on the vector x through a single linear combination x0 : Most parametric models are single index, including Normal regression, Logit, Probit, Tobit, Abstract. A single index model (SIM) summarizes the effects of the explanatory variables X 1, , X d within a single variable called the index. As stated at the beginning of Part II, the SIM is one possibility for generalizing the GLM or for restricting the multidimensional regression E(Y|X) to overcome the curse of dimensionality and the lack of interpretability. Single index models contain smooth terms with arguments that are linear combinations of other covariates. e.g. \(s(X\alpha)\) where \(\alpha\) has to be estimated. For identifiability, assume \(\|\alpha\|=1\) with positive first element. One simple way to fit such models is to use -31" data-mini-rdoc="mgcv::gam">gam to profile out the smooth model coefficients and smoothing An extended single‐index model is considered when responses are missing at random. A three‐step estimation procedure is developed to define an estimator for the single‐index parameter vector by a joint estimating equation. The proposed estimator is shown to be asymptotically normal. An algorithm for computing this estimator is proposed. In this paper, the varying-coefficient single-index model (VCSIM) is proposed. It can be seen as a generalization of the semivarying-coefficient model by changing its constant coefficient part to a nonparametric component, or a generalization of the partially linear single-index model by replacing the constant coefficients of its linear part with varying coefficients.

## In Section 2, we construct a. Cramér-von Mises test statistic and give its asymptotic distribution. In Section 3, we propose some bootstrap test statistics and show

SPLINE ESTIMATION OF SINGLE-INDEX MODELS Li Wang and Lijian Yang University of Georgia and Michigan State University Abstract: For the past two decades, the single-index model, a special case of pro-jection pursuit regression, has proven to be an eﬃcient way of coping with the high-dimensional problem in nonparametric regression. 8 Semiparametric Single Index Models 8.1 Index Models A object of interest such as the conditional density f (y j x) or conditional mean E(y j x) is a single index model when it only depends on the vector x through a single linear combination x0 : Most parametric models are single index, including Normal regression, Logit, Probit, Tobit, Abstract. A single index model (SIM) summarizes the effects of the explanatory variables X 1, , X d within a single variable called the index. As stated at the beginning of Part II, the SIM is one possibility for generalizing the GLM or for restricting the multidimensional regression E(Y|X) to overcome the curse of dimensionality and the lack of interpretability. Single index models contain smooth terms with arguments that are linear combinations of other covariates. e.g. \(s(X\alpha)\) where \(\alpha\) has to be estimated. For identifiability, assume \(\|\alpha\|=1\) with positive first element. One simple way to fit such models is to use -31" data-mini-rdoc="mgcv::gam">gam to profile out the smooth model coefficients and smoothing

### Abstract. A single index model (SIM) summarizes the effects of the explanatory variables X 1, , X d within a single variable called the index. As stated at the beginning of Part II, the SIM is one possibility for generalizing the GLM or for restricting the multidimensional regression E(Y|X) to overcome the curse of dimensionality and the lack of interpretability.

The single-index model (SIM) is a simple asset pricing model to measure both the risk and the return of a stock. The model has been developed by William In a semiparametric single index model, the object of interest depends on x through nals of Statistics, 1993) suggest picking Я and the bandwidth h jointly by 15 Aug 2013 Plug-in Principle Estimators. Plug-in principle: Estimate model parameters using sample statistics. ˆ.. = ˆ . ˆ . 2.. ˆ . = 1. − 1.

### Estimating and testing the model index coefficients β is one of the most important objectives in the statistical analysis. However, the commonly used assumption on

FUNCTIONAL SINGLE INDEX MODELS FOR LONGITUDINAL DATA By Ci-Ren JiangandJane-LingWang1 University of California, Berkeley and University of California, Davis A new single-index model that reﬂects the time-dynamic eﬀects of the single index is proposed for longitudinal and functional response data, possibly measured with errors, for both { Single Index Model (Review) { Multi Index Models { Capital Asset Pricing Model 1 The Single Index Model (Review) One possible model for the returns is R i = i + iR m + i where i,and i are constants, R m is the return of a market index and i is a random variable with mean 0 and variance ˝2 i. If the 2 i, i and ˝ Single-Index Model 2 / 25. Correlation Structure of Security Returns is a Key Input Recall from Mean-Variance Portfolio: Z = 1 (R RF1) (1) This implies that correlation structure of security returns, 1, is a We can use historical or past time-series data on security Rit and summary features of the data contribute to the actual knowledge discovery. The DSI model has connections to several familiar multivariate models. For example, in the special case when q D 1, the DSI model in (1) reduces to the familiar single index model (Ichimura, 1993; Härdle et al., 1993). In addition, DSI is an extension and gener- single index model when it only depends on the vector x through a single linear combination x0 : Most parametric models are single index, including Normal regression, Logit, Probit, Tobit, and Poisson regression. In a semiparametric single index model, the object of interest depends on x through the function g(x0 ) where 2 Rk and g : R ! R are

## model (the Single Index Model) and obtain result using the empirical data. In this case the sample size is 248 data points spanning over one year period from 1st January 2015 to 31 December 2015. Technique In this paper Sharpe‟s Single Index Model is used, which basically selects assets on the basis of

Single-Index Model 2 / 25. Correlation Structure of Security Returns is a Key Input Recall from Mean-Variance Portfolio: Z = 1 (R RF1) (1) This implies that correlation structure of security returns, 1, is a We can use historical or past time-series data on security Rit and summary features of the data contribute to the actual knowledge discovery. The DSI model has connections to several familiar multivariate models. For example, in the special case when q D 1, the DSI model in (1) reduces to the familiar single index model (Ichimura, 1993; Härdle et al., 1993). In addition, DSI is an extension and gener- single index model when it only depends on the vector x through a single linear combination x0 : Most parametric models are single index, including Normal regression, Logit, Probit, Tobit, and Poisson regression. In a semiparametric single index model, the object of interest depends on x through the function g(x0 ) where 2 Rk and g : R ! R are Single-index model A model of stock returns that decomposes influences on returns into a systematic factor, as measured by the return on the broad market index, and firm specific factors. Related: Market Model Single-Index Model The relationship between a security's performance and the performance of a portfolio containing it. The market model states Estimating the Single Index Model Eric Zivot August 15, 2013. Estimating the Single Index Model Sharpe’s Single (SI) model: Assuming the SI model generates the observed data, the estimators Foundations of Finance: Index Models 2 I. Readings and Suggested Practice Problems BKM, Chapter 10, Section 1 (Skim Section 4) Suggested Problems, Chapter 10: 5-13 II. A Single Index Model An Index Model is a Statistical model of security returns (as opposed to an economic, equilibrium-based model).

To circumvent this difficulty, we consider the single-index model estimation problem from a sparsity Mathematical Methods of Statistics, 17:279-304, 2008. Learning Single-Index Models in Gaussian Space. Rishabh Dudeja. RD2714@ COLUMBIA.EDU. Department of Statistics, Columbia University. Daniel Hsu. 31 Mar 2010 Ann Inst Stat Math (2011) 63:1277–1293. DOI 10.1007/s10463-010-0287-4. Variable selection in a class of single-index models. Li-Ping Zhu 24 May 2006 In the statistical literature the relations as in(1. 1)and. (1.2)are referred to as thesingle-index regressionmodels. These models are often used in Project Euclid - mathematics and statistics online. Abstract. A new single-index model that reflects the time-dynamic effects of the single index is proposed for