@tezu.ernet.in
Professor, Department of Mathematical Sciences
Tezpur University
Non-Parametric functional estimation (Density, distribution function and quantile estimation by kernel and Berstein polynomial methods and their application in finance. Bootstrap methods for kernel estimators)\b
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Santanu Dutta and Tushar Kanti Powdel
Springer Science and Business Media LLC
Arup Bose and Santanu Dutta
Springer Science and Business Media LLC
Santanu Dutta and Suparna Biswas
Informa UK Limited
ABSTRACT Expected shortfall (ES) is a well-known measure of extreme loss associated with a risky asset or portfolio. For any 0 < p < 1, the 100(1 − p) percent ES is defined as the mean of the conditional loss distribution, given the event that the loss exceeds (1 − p)th quantile of the marginal loss distribution. Estimation of ES based on asset return data is an important problem in finance. Several nonparametric estimators of the expected shortfall are available in the literature. Using Monte Carlo simulations, we compare the accuracy of these estimators under the condition that p → 0 as n → ∞ for several asset return time series models, where n is the sample size. Not much seems to be known regarding the properties of the ES estimators under this condition. For p close to zero, the ES measures an extreme loss in the right tail of the loss distribution of the asset or portfolio. Our simulations and real-data analysis provide insight into the effect of varying p with n on the performance of nonparametric ES estimators.
Santanu Dutta and Suparna Biswas
Informa UK Limited
ABSTRACT Estimation of market risk is an important problem in finance. Two well-known risk measures, viz., value at risk and median shortfall, turn out to be extreme quantiles of the marginal distribution of asset return. Time series on asset returns are known to exhibit certain stylized facts, such as heavy tails, skewness, volatility clustering, etc. Therefore, estimation of extreme quantiles in the presence of such features in the data seems to be of natural interest. It is difficult to capture most of these stylized facts using one specific time series model. This motivates nonparametric and extreme value theory-based estimation of extreme quantiles that do not require exact specification of the asset return model. We review these quantile estimators and compare their known properties. Their finite sample performance are compared using Monte Carlo simulation. We propose a new estimator that exhibits encouraging finite sample performance while estimating extreme quantile in the right tail region.
Santanu Dutta and Koushik Saha
Informa UK Limited
ABSTRACT We obtain the rates of pointwise and uniform convergence of multivariate kernel density estimators using a random bandwidth vector obtained by some data-based algorithm. We are able to obtain faster rate for pointwise convergence. The uniform convergence rate is obtained under some moment condition on the marginal distribution. The rates are obtained under i.i.d. and strongly mixing type dependence assumptions.
Santanu Dutta
Springer Science and Business Media LLC
The problem of distribution function (df) estimation arises naturally in many contexts. The empirical and the kernel df estimators are well known. There is another df estimator based on a Bernstein polynomial of degree m. For a Bernstein df estimator, plays the same role as the bandwidth in a kernel estimator. The asymptotic properties of the Bernstein estimator has been studied so far assuming m is non random, chosen subjectively. We propose algorithms for data driven choice of m. Such an m is a function of the data, i.e. random. We obtain the convergence rates of a Bernstein df estimator, using a random m, for i.i.d., strongly mixing and a broad class of linear processes. The estimator is shown to be consistent for any stationary, ergodic process satisfying some conditions. Using simulations and analysis of real data the finite sample performance of the different df estimators are compared.
Santanu Dutta
Informa UK Limited
Data-based choice of the bandwidth is an important problem in kernel density estimation. The pseudo-likelihood and the least-squares cross-validation bandwidth selectors are well known, but widely criticized in the literature. For heavy-tailed distributions, the L1 distance between the pseudo-likelihood-based estimator and the density does not seem to converge in probability to zero with increasing sample size. Even for normal-tailed densities, the rate of L1 convergence is disappointingly slow. In this article, we report an interesting finding that with minor modifications both the cross-validation methods can be implemented effectively, even for heavy-tailed densities. For both these estimators, the L1 distance (from the density) are shown to converge completely to zero irrespective of the tail of the density. The expected L1 distance also goes to zero. These results hold even in the presence of a strongly mixing-type dependence. Monte Carlo simulations and analysis of the Old Faithful geyser data suggest that if implemented appropriately, contrary to the traditional belief, the cross-validation estimators compare well with the sophisticated plug-in and bootstrap-based estimators.
Santanu Dutta and Koushik Saha
Informa UK Limited
Asymptotic properties of a kernel density estimator using a random bandwidth are difficult to establish. Under some assumptions we prove the L1 consistency of a class of multivariate kernel density estimators using different bandwidth vector selectors. The expected L1 distance between such an estimator and the density is also shown to converge to zero. Our results hold even when the marginal densities are heavy-tailed. As a special case, we propose a simple estimator that depends on only one parameter, irrespective of the dimension. Its L1 distance from the density goes to zero, exponentially. Simulations suggest that this estimator performs well in terms of the integrated squared error as well.
Suparna Biswas and Santanu Dutta
SAGE Publications
An index fund is a mutual fund that aims to imitate a benchmark index. In India, there has been significant growth in the number of such funds since 2002. These funds are exposed mainly to market risk. The assessment and comparison of the market risk and the risk-adjusted returns of these funds are topics of interest to both researchers and investors. Value-at-risk and expected shortfall are well-known measures of the market risk. The Sharpe ratio and the Treynor ratio measure risk-adjusted return earned in excess of average market return. For each fund, we estimate these measures. Most of the index funds exhibit similar market risk as the NIFTY or the SENSEX index, which they mimic. Moreover, the market risk of these funds seems to be unaffected by multiple-fund management by the respective fund managers. However, there exist significant differences among the risk-adjusted daily returns, earned in excess of the average daily index return, of the various Indian index funds. Ideally, an index fund is expected to exhibit similar risk and risk-adjusted return as the benchmark index. We identify some such Indian index funds.
Santanu Dutta
Informa UK Limited
The problem of bandwidth selection for kernel-based estimation of the distribution function (cdf) at a given point is considered. With appropriate bandwidth, a kernel-based estimator (kdf) is known to outperform the empirical distribution function. However, such a bandwidth is unknown in practice. In pointwise estimation, the appropriate bandwidth depends on the point where the function is estimated. The existing smoothing methods use one common bandwidth to estimate the cdf. The accuracy of the resulting estimates varies substantially depending on the cdf and the point where it is estimated. We propose to select bandwidth by minimizing a bootstrap estimator of the MSE of the kdf. The resulting estimator performs reliably, irrespective of where the cdf is estimated. It is shown to be consistent under i.i.d. as well as strongly mixing dependence assumption. Two applications of the proposed estimator are shown in finance and seismology. We report a dataset on the S & P Nifty index values.
Santanu Dutta
Informa UK Limited
We consider the problem of data-based choice of the bandwidth of a kernel density estimator, with an aim to estimate the density optimally at a given design point. The existing local bandwidth selectors seem to be quite sensitive to the underlying density and location of the design point. For instance, some bandwidth selectors perform poorly while estimating a density, with bounded support, at the median. Others struggle to estimate a density in the tail region or at the trough between the two modes of a multimodal density. We propose a scale invariant bandwidth selection method such that the resulting density estimator performs reliably irrespective of the density or the design point. We choose bandwidth by minimizing a bootstrap estimate of the mean squared error (MSE) of a density estimator. Our bootstrap MSE estimator is different in the sense that we estimate the variance and squared bias components separately. We provide insight into the asymptotic accuracy of the proposed density estimator.
Santanu Dutta and Alok Goswami
Elsevier BV
We obtain the rates of pointwise and uniform convergence of kernel density estimators using random bandwidths under i.i.d. as well as strongly mixing dependence assumptions. Pointwise rates are faster and not affected by the tail of the density.
Subhrangshu Sekhar Sarkar, Santanu Dutta, and Pinky Dutta
SAGE Publications
An index fund is a mutual fund that aims to imitate some benchmark index. There are several advantages of investing in an index fund, namely, exposure to a diversified portfolio, minimization of company-specific risks, high liquidity, etc. In India, there has been a significant growth in the number of index funds from 2002 onwards. Today there are more than 20 index funds imitating the NIFTY or SENSEX. In this article, we review and compare a number of Indian index funds. The CRISIL composite ranking of an index fund reflects the quarterly performance of that fund and is subject to fluctuations. We are interested in those index funds that do not deviate significantly from the underlying benchmark index in the long run. This ensures that an investor gets the benefit of any strong rally in the benchmark index, which the index fund imitates. We identify some index funds that satisfy our criteria, and are also ranked above three (indicating above average performance) in the CRISIL rankings of March 2011 and December 2010.
Arup Bose and Santanu Dutta
Elsevier BV
Smoothing methods for density estimators struggle when the shape of the reference density differs markedly from the actual density. We propose a bootstrap bandwidth selector where no reference distribution is used. It performs reliably in difficult cases and asymptotically outperforms well known automatic bandwidths.
Santanu Dutta
Elsevier BV
Abstract Given i.i.d. d -dimensional ( d > 1 ) data, a bootstrap estimate of the mean integrated squared error (MISE) of a product kernel density estimate is proposed. We propose to select the 1 × d bandwidth vector by minimizing the bootstrap estimate of the MISE. Some asymptotic properties of the proposed estimators are obtained. For fixed sample size, the bootstrap MISE estimator by Sain et al. (1994) can fail to estimate the MISE, especially for large smoothing along any one of the d directions. Our MISE estimator overcomes this demerit. For a Gaussian kernel the exact formula of the bootstrap MISE estimator is obtained. The smoothed cross-validation method (SCV) is similar to the proposed bootstrap scheme. The proposed bootstrap estimate can be asymptotically more accurate than the SCV estimate of the MISE. We provide insight into the accuracy of the SCV and the bootstrap bandwidth vectors in terms of minimizing the MISE. The product kernel density estimate, using the bootstrap bandwidths, for Old Faithful geyser data seem to compare well with the kernel density estimate using full bandwidth matrix chosen by plug-in method.
S. Dutta and A. Goswami
Allerton Press
The problem of estimating the mode of a discrete distribution is considered. New characterizations of discrete unimodal and multi-modal distributions are obtained. The proposed mode estimator is essentially the sample mode, modulo appropriate modifications when the sample mode is not well defined. In the case of i.i.d. observations coming from a unimodal discrete distribution, our proposed mode estimator is shown to possess a number of strong asymptotic properties. Many of these results extend to the case of multi-modal discrete distributions as well. Our method also applies — and we have similar asymptotic results — to the problem of mode estimation based on finitely many observations on a Markov chain whose equilibrium distribution is the underlying unimodal distribution. For unimodal discrete distributions, we also propose a consistent large sample test of mode based on the proposed statistic. Applications of mode estimation problem in Monte-Carlo optimization problem using the Hastings Metropolis chain and in prediction problem using binary response variable, specially in the context of dose-response experiments, are also illustrated.