In this paper we consider an explicit dynamic risk measure for discrete-time payment processes which have a Markovian structure. The risk measure is essentially a sum of conditional Average Value{at{Risks. Analogous to the static Average Value{at{Risk, this risk measures can be reformulated in terms of the value functions of a dynamic optimization problem, namely a so-called Markov decision problem. This observation gives a nice recursive computation formula. Afterwards, the deflnition of the dynamic risk measure is generalized to a setting with incomplete information about the risk distribution which can be seen as model ambiguity. We choose a parametric approach here. The dynamic risk measure is again deflned as the sum of conditional Average Value{at{Risks or equivalently is the solution of a Bayesian decision problem. Finally, it is possible to discuss the efiect of model ambiguity on the risk measure: Surprisingly, it may be the case that the risk decreases when additional "risk" due to parameter uncertainty shows up. All investigations are illustrated by a simple but useful coin tossing game proposed by Artzner and by the classical Cox{Ross-Rubinstein model.
{"title":"A BAYESIAN APPROACH TO INCORPORATE MODEL AMBIGUITY IN A DYNAMIC RISK MEASURE","authors":"N. Bäuerle, André Mundt","doi":"10.1524/STND.2008.1000","DOIUrl":"https://doi.org/10.1524/STND.2008.1000","url":null,"abstract":"In this paper we consider an explicit dynamic risk measure for discrete-time payment processes which have a Markovian structure. The risk measure is essentially a sum of conditional Average Value{at{Risks. Analogous to the static Average Value{at{Risk, this risk measures can be reformulated in terms of the value functions of a dynamic optimization problem, namely a so-called Markov decision problem. This observation gives a nice recursive computation formula. Afterwards, the deflnition of the dynamic risk measure is generalized to a setting with incomplete information about the risk distribution which can be seen as model ambiguity. We choose a parametric approach here. The dynamic risk measure is again deflned as the sum of conditional Average Value{at{Risks or equivalently is the solution of a Bayesian decision problem. Finally, it is possible to discuss the efiect of model ambiguity on the risk measure: Surprisingly, it may be the case that the risk decreases when additional \"risk\" due to parameter uncertainty shows up. All investigations are illustrated by a simple but useful coin tossing game proposed by Artzner and by the classical Cox{Ross-Rubinstein model.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124587646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-01DOI: 10.1524/stnd.22.4.319.64313
I. Steinke
Summury Let (Xi,j, Yi,j), i = 1,…,n, j = 1,2, be a sample from two populations, where the Xi,j are d-dimensional covariates which have an effect on the response variable Yi,j. It is assumed that the conditional distribution of Yi,j given Xi,j = x is Qg(αj + βjTx) where {Qϑ | ϑ ∊ Θ}, Θ ⊆ R, is a parent family, g is the so-called link function and ϑj = (αj,βj) the parameters of interest. Using the LAN theory, a sequence of locally asymptotically optimal tests φ^n for H0 : ϑ1 = ϑ2 versus HA : ϑ1 ≠ ϑ2 is constructed for an unknown link function g. These tests are asymptotic maximin-tests and adaptive in the sense that the plugging-in of an estimator for the nuisance parameters g does not reduce the local asymptotic power compared to the situation of a known nuisance parameter g. To attain exact α-tests even for finite sample size a permutation test version is given with the same local asymptotic power as φ^n.
{"title":"Locally asymptotically optimal tests in semiparametric generalized linear models in the 2-sample-problem","authors":"I. Steinke","doi":"10.1524/stnd.22.4.319.64313","DOIUrl":"https://doi.org/10.1524/stnd.22.4.319.64313","url":null,"abstract":"Summury Let (Xi,j, Yi,j), i = 1,…,n, j = 1,2, be a sample from two populations, where the Xi,j are d-dimensional covariates which have an effect on the response variable Yi,j. It is assumed that the conditional distribution of Yi,j given Xi,j = x is Qg(αj + βjTx) where {Qϑ | ϑ ∊ Θ}, Θ ⊆ R, is a parent family, g is the so-called link function and ϑj = (αj,βj) the parameters of interest. Using the LAN theory, a sequence of locally asymptotically optimal tests φ^n for H0 : ϑ1 = ϑ2 versus HA : ϑ1 ≠ ϑ2 is constructed for an unknown link function g. These tests are asymptotic maximin-tests and adaptive in the sense that the plugging-in of an estimator for the nuisance parameters g does not reduce the local asymptotic power compared to the situation of a known nuisance parameter g. To attain exact α-tests even for finite sample size a permutation test version is given with the same local asymptotic power as φ^n.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123476583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-01DOI: 10.1524/stnd.22.4.283.64315
M. Giurcanu, V. Spokoiny
Summury In this note we consider the problem of confidence estimation of the covariance function of a stationary or locally stationary zero mean Gaussian process. The constructed confidence intervals are based on the usual empirical covariance estimate and a special estimate of its variance. The results about coverage probability are stated in a nonasymptotic way and apply for small and moderate sample size under mild conditions on the model. The presented numerical results are in agreement with the theoretical issues and demonstrate applicability of the method.
{"title":"Confidence estimation of the covariance function of stationary and locally stationary processes","authors":"M. Giurcanu, V. Spokoiny","doi":"10.1524/stnd.22.4.283.64315","DOIUrl":"https://doi.org/10.1524/stnd.22.4.283.64315","url":null,"abstract":"Summury In this note we consider the problem of confidence estimation of the covariance function of a stationary or locally stationary zero mean Gaussian process. The constructed confidence intervals are based on the usual empirical covariance estimate and a special estimate of its variance. The results about coverage probability are stated in a nonasymptotic way and apply for small and moderate sample size under mild conditions on the model. The presented numerical results are in agreement with the theoretical issues and demonstrate applicability of the method.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126998407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-01DOI: 10.1524/stnd.22.4.261.64314
S. Delattre, S. Graf, H. Luschgy, G. Pagès
Summury For a probability measure P on Rd and n ∊ N consider en = inf ∫ mina∊αV(||x − a||)dP(x) where the infimum is taken over all subsets α of Rd with card(α) ≤ n and V is a nondecreasing function. Under certain conditions on V, we derive the precise n-asymptotics of en for nonsingular distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.
{"title":"Quantization of probability distributions under norm-based distortion measures","authors":"S. Delattre, S. Graf, H. Luschgy, G. Pagès","doi":"10.1524/stnd.22.4.261.64314","DOIUrl":"https://doi.org/10.1524/stnd.22.4.261.64314","url":null,"abstract":"Summury For a probability measure P on Rd and n ∊ N consider en = inf ∫ mina∊αV(||x − a||)dP(x) where the infimum is taken over all subsets α of Rd with card(α) ≤ n and V is a nondecreasing function. Under certain conditions on V, we derive the precise n-asymptotics of en for nonsingular distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127998788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-01DOI: 10.1524/stnd.22.4.335.64312
Gabriela Ciuperca
Summury We consider a two-phase random design nonlinear regression model, the regression function is discontinuous at the change-point. The errors ∊ are arbitrary, with E(∊) = 0 and E(∊2) < ∞. We prove that Koul and Qian’s results [12] for linear regression still hold true for the nonlinear case. Thus the maximum likelihood estimator r^n of the change-point r is n-consistent and the estimator θ^1n of the regression parameters θ1 is n1/2-consistent. The asymptotic distribution of n1/2(θ^1n − θ01) is Gaussian and n(r^n − r) converges to the left end point of the maximizing interval with respect to the change point. The likelihood process is asymptotically equivalent to a compound Poisson process.
{"title":"Maximum likelihood estimator in a two-phase nonlinear random regression model","authors":"Gabriela Ciuperca","doi":"10.1524/stnd.22.4.335.64312","DOIUrl":"https://doi.org/10.1524/stnd.22.4.335.64312","url":null,"abstract":"Summury We consider a two-phase random design nonlinear regression model, the regression function is discontinuous at the change-point. The errors ∊ are arbitrary, with E(∊) = 0 and E(∊2) < ∞. We prove that Koul and Qian’s results [12] for linear regression still hold true for the nonlinear case. Thus the maximum likelihood estimator r^n of the change-point r is n-consistent and the estimator θ^1n of the regression parameters θ1 is n1/2-consistent. The asymptotic distribution of n1/2(θ^1n − θ01) is Gaussian and n(r^n − r) converges to the left end point of the maximizing interval with respect to the change point. The likelihood process is asymptotically equivalent to a compound Poisson process.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125692905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2004-04-01DOI: 10.1524/stnd.22.4.301.64311
Hanxiang Peng, A. Schick
Summury In this paper we construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. The proposed estimator generalizes the construction of efficient estimators given by Bickel, Ritov and Wellner (1991) for the case of known, but not necessarily equal, marginals.
{"title":"Efficient estimation of a linear functional of a bivariate distribution with equal, but unknown, marginals: The minimum chi-square approach","authors":"Hanxiang Peng, A. Schick","doi":"10.1524/stnd.22.4.301.64311","DOIUrl":"https://doi.org/10.1524/stnd.22.4.301.64311","url":null,"abstract":"Summury In this paper we construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. The proposed estimator generalizes the construction of efficient estimators given by Bickel, Ritov and Wellner (1991) for the case of known, but not necessarily equal, marginals.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116821553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-02-01DOI: 10.1524/STND.21.2.149.19004
J. Rahnenführer
Power functions of tests for Gaussian shift experiments on infinite dimensional Hilbert spaces usually can not be calculated explicitly. Therefore one analyzes the behavior of such tests in the neighborhood of the null hypothesis. Useful measures to compare the quality of different testing procedures are the gradient of a one-sided and the curvature of a two-sided test in the null hypothesis. Janssen (1995) showed that a principal component decomposition of the curvature exists based on a Hilbert–Schmidt operator. It follows that these tests have only acceptable power for a finite number of directions. In this paper we prove an even stronger general result for Gauss shifts under just mild additional assumptions. A certain optimality property of a one-sided test implicates that for a small level α the corresponding two-sided test acts only in a single direction. The results are applied to Kolmogorov–Smirnov type tests and the signal detection problem.
{"title":"On preferences of general two-sided tests with applications to Kolmogorov–Smirnov-type tests","authors":"J. Rahnenführer","doi":"10.1524/STND.21.2.149.19004","DOIUrl":"https://doi.org/10.1524/STND.21.2.149.19004","url":null,"abstract":"Power functions of tests for Gaussian shift experiments on infinite dimensional Hilbert spaces usually can not be calculated explicitly. Therefore one analyzes the behavior of such tests in the neighborhood of the null hypothesis. Useful measures to compare the quality of different testing procedures are the gradient of a one-sided and the curvature of a two-sided test in the null hypothesis. Janssen (1995) showed that a principal component decomposition of the curvature exists based on a Hilbert–Schmidt operator. It follows that these tests have only acceptable power for a finite number of directions. In this paper we prove an even stronger general result for Gauss shifts under just mild additional assumptions. A certain optimality property of a one-sided test implicates that for a small level α the corresponding two-sided test acts only in a single direction. The results are applied to Kolmogorov–Smirnov type tests and the signal detection problem.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114882313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1524/STND.21.1.79.20320
W. Schmid, Yarema Okhrin
In this paper we consider a general control scheme. The control statistic Zt is equal to an arbitrary weighted sum of the past observations Xt,...,X1. This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process {Xt} is assumed to be a stationary Gaussian process. The aim of the work is to analyze the behaviour of the tail probability of the run length N=inf{t∈ℕ:Zt−E(Zt)>c√{Var(Zt)}} with respect to the autocorrelation of {Xt}. It is shown under which conditions on the weights and on the autocorrelations of {Xt} the correlation between Zt and Zt−i is a nondecreasing function in the autocorrelations of the observed process. Using this result it can be proved that the probability of a false alarm is a nondecreasing function of the autocorrelations of {Xt}, too. The weight conditions are verified for several well-known charts.
{"title":"Tail behaviour of a general family of control charts","authors":"W. Schmid, Yarema Okhrin","doi":"10.1524/STND.21.1.79.20320","DOIUrl":"https://doi.org/10.1524/STND.21.1.79.20320","url":null,"abstract":"In this paper we consider a general control scheme. The control statistic Zt is equal to an arbitrary weighted sum of the past observations Xt,...,X1. This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process {Xt} is assumed to be a stationary Gaussian process. The aim of the work is to analyze the behaviour of the tail probability of the run length N=inf{t∈ℕ:Zt−E(Zt)>c√{Var(Zt)}} with respect to the autocorrelation of {Xt}. It is shown under which conditions on the weights and on the autocorrelations of {Xt} the correlation between Zt and Zt−i is a nondecreasing function in the autocorrelations of the observed process. Using this result it can be proved that the probability of a false alarm is a nondecreasing function of the autocorrelations of {Xt}, too. The weight conditions are verified for several well-known charts.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"262 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116034438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1524/STND.2006.24.4.415
J. Kallsen, C. Kühn
The neutral valuation approach for contingent claims in incomplete markets is based on the assumption that investors are identical utility maximizers and that derivative supply and demand are balanced. It is closely related to (marginal) utility-based pricing in the sense of Hugonnier et al. (2005), where however only buy-and-hold investments in the derivative are possible.
不完全市场中或有债权的中性估值方法是基于这样的假设:投资者是相同的效用最大化者，衍生品的供求是平衡的。它与Hugonnier et al.(2005)意义上的(边际)基于效用的定价密切相关，然而，在衍生品中只有买入并持有的投资是可能的。
{"title":"On utility-based derivative pricing with and without intermediate trades","authors":"J. Kallsen, C. Kühn","doi":"10.1524/STND.2006.24.4.415","DOIUrl":"https://doi.org/10.1524/STND.2006.24.4.415","url":null,"abstract":"The neutral valuation approach for contingent claims in incomplete markets is based on the assumption that investors are identical utility maximizers and that derivative supply and demand are balanced. It is closely related to (marginal) utility-based pricing in the sense of Hugonnier et al. (2005), where however only buy-and-hold investments in the derivative are possible.","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124361207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1524/STND.2006.24.3.373
M. Laan, S. Dudoit, A. Vaart
Suppose that we observe a sample of independent and identically distributed realizations of a random variable, and a parameter of interest can be defined as the minimizer, over a suitably defined parameter set, of the expectation of a (loss) function of a candidate parameter value and the random variable. For example, squared error loss in regression or the negative log-density loss in density estimation. Minimizing the empirical risk (i.e., the empirical mean of the loss function) over the entire parameter set may result in ill-defined or too variable estimators of the parameter of interest. In this article, we propose a cross-validated e-net estimation method, which uses a collection of submodels and a collection of e-nets over each submodel. For each submodel s and each resolution level e, the minimizer of the empirical risk over the corresponding e-net is a candidate estimator. Next we select from these estimators (i.e. select the pair (s,e)) by multi-fold cross-validation. We derive a finite sample inequality that shows that the resulting estimator is as good as an oracle estimator that uses the best submodel and resolution level for the unknown true parameter. We also address the implementation of the estimation procedure, and in the context of a linear regression model we present results of a preliminary simulation study comparing the cross-validated e-net estimator to the cross-validated L1-penalized least squares estimator (LASSO) and the least angle regression estimator (LARS).
{"title":"The Cross-Validated Adaptive Epsilon-Net Estimator","authors":"M. Laan, S. Dudoit, A. Vaart","doi":"10.1524/STND.2006.24.3.373","DOIUrl":"https://doi.org/10.1524/STND.2006.24.3.373","url":null,"abstract":"Suppose that we observe a sample of independent and identically distributed realizations of a random variable, and a parameter of interest can be defined as the minimizer, over a suitably defined parameter set, of the expectation of a (loss) function of a candidate parameter value and the random variable. For example, squared error loss in regression or the negative log-density loss in density estimation. Minimizing the empirical risk (i.e., the empirical mean of the loss function) over the entire parameter set may result in ill-defined or too variable estimators of the parameter of interest. In this article, we propose a cross-validated e-net estimation method, which uses a collection of submodels and a collection of e-nets over each submodel. For each submodel s and each resolution level e, the minimizer of the empirical risk over the corresponding e-net is a candidate estimator. Next we select from these estimators (i.e. select the pair (s,e)) by multi-fold cross-validation. We derive a finite sample inequality that shows that the resulting estimator is as good as an oracle estimator that uses the best submodel and resolution level for the unknown true parameter. We also address the implementation of the estimation procedure, and in the context of a linear regression model we present results of a preliminary simulation study comparing the cross-validated e-net estimator to the cross-validated L1-penalized least squares estimator (LASSO) and the least angle regression estimator (LARS).","PeriodicalId":380446,"journal":{"name":"Statistics & Decisions","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125191972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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