The Garman-Klass volatility estimator revisited

The Garman-Klass unbiased estimator of the variance per unit time of a zero-drift Brownian Motion B, based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1=(CLOSE-OPEN, OPEN-MIN, MAX-OPEN). This estimator, with efficiency 7.4 with respect to the classical estimator (CLOSE-OPEN)^2, is widely believed to be of minimal variance. The current report disproves this belief by exhibiting an unbiased estimator with slightly but strictly higher efficiency 7.7322. The essence of the improvement lies in the observation that the data should be compressed to the statistic S2 defined on W(t)= B(0)+[B(t)-B(0)]sign[(B(1)-B(0)] as S1 was defined on the Brownian path B(t). The best S2-based quadratic unbiased estimator is presented explicitly. The Cramer-Rao upper bound for the efficiency of unbiased estimators, corresponding to the efficiency of large-sample Maximum Likelihood estimators, is 8.471. This bound cannot be attained because the distribution is not of exponential type. Regression-fitted quadratic functions of S2 (with mean 1) markedly out-perform those of S1 when applied to random walks with heavy-tail-distributed increments. Performance is empirically studied in terms of the tail parameter

In search of characterization of the preference for safety under the Choquet model

Victor prefers safety more than Ursula if whenever Ursula prefers some constant to some uncertain act, so does Victor. This paradigm, whose Expected Utility version takes the form of Arrow & Pratt's more risk averse concept, will be studied in the Choquet Uncertainty model, letting u and μ (v and ν) be Ursula's (Victor's) utility and capacity. A necessary and sufficient condition (A) on the pairs (u, μ) and (v, ν) will be presented for dichotomous weak increased uncertainty aversion, the preference by Victor of a constant over a dichotomous act whenever such is the preference of Ursula. This condition, pointwise inequality between a function defined in terms of v (u-1(⋅)) and another defined purely in terms of the capacities, preserves the flavor of the "more pessimism than greediness" characterization of monotone risk aversion by Chateauneuf, Cohen & Meilijson in the Rank-dependent Utility Model and its extension by Grant & Quiggin to the Choquet Utility Model. A sufficient condition (B) in terms of the capacities only, satisfied in particular if ν (⋅) = f (μ (⋅)) for some convex f, will be presented for more simplicity seeking, the preference by Victor over any act for some dichotomous act, that leaves Ursula indifferent. Condition A is thus a characterization of weak increased uncertainty aversion for convex f. An example will be exhibited disproving the more far reaching conjecture under which the dichotomous case implies the general case.Choquet Utility, greediness, pessimism, Rank-dependent Utility, Risk aversion, uncertainty.

The observed Fisher information attached to the EM algorithm, illustrated on Shepp and Vardi estimation procedure for positron emission tomography

The Shepp & Vardi (1982) implementation of the EM algorithm for PET scan tumor estimation provides a point estimate of the tumor. The current study presents a closed-form formula of the observed Fisher information for Shepp & Vardi PET scan tumor estimation. Keywords: PET scan, EM algorithm, Fisher information matrix, standard errors

Four notions of mean preserving increase in risk, risk attitudes and applications to the Rank-Dependent Expected Utility model

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant tothe Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk - MPIR, monotoneMPIR and two versions of location-independent risk (renamed here left and right monotone MPIR) - and shows which choice questions are consistently modeled by each of these four orders.Location-independent risk, monotone increase in risk, rank-dependent expected utility.

More pessimism than greediness: a characterization of monotone risk aversion in the Rank-Dependent Expected Utility model

This paper studies monotone risk aversion, the aversion to monotone, meanpreserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by twofunctions, a utility function u in conjunction with a probability-perception function f.Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel & Lehmann [3, 4] in Non-parametric Statistics. We present a characterization of the pairs (u; f) of monotone risk averse decision makers, based on an index of greediness Gu of the utility function u and an index of pessimism Pf of the probability perception function f: the decision maker is monotone risk averse if and onlyif Pf exceeds Gu. A novel element is that concavity of u is not necessary. In fact, u must be concave only if Pf = 1.Risk aversion, pessimism, greediness, Rank-dependent Expected Utility

In search of characterization of the preference for safety under the Choquet model

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/Documents de travail du Centre d'Economie de la Sorbonne 2011.31 - ISSN : 1955-611XVictor prefers safety more than Ursula if whenever Ursula prefers some constant to some uncertain act, so does Victor. This paradigm, whose Expected Utility version takes the form of Arrow & Pratt's more risk averse concept, will be studied in the Choquet Uncertainty model, letting u and μ (v and ν) be Ursula's (Victor's) utility and capacity. A necessary and sufficient condition (A) on the pairs (u, μ) and (v, ν) will be presented for dichotomous weak increased uncertainty aversion, the preference by Victor of a constant over a dichotomous act whenever such is the preference of Ursula. This condition, pointwise inequality between a function defined in terms of v (u-1(⋅)) and another defined purely in terms of the capacities, preserves the flavor of the "more pessimism than greediness" characterization of monotone risk aversion by Chateauneuf, Cohen & Meilijson in the Rank-dependent Utility Model and its extension by Grant & Quiggin to the Choquet Utility Model. A sufficient condition (B) in terms of the capacities only, satisfied in particular if ν (⋅) = f (μ (⋅)) for some convex f, will be presented for more simplicity seeking, the preference by Victor over any act for some dichotomous act, that leaves Ursula indifferent. Condition A is thus a characterization of weak increased uncertainty aversion for convex f. An example will be exhibited disproving the more far reaching conjecture under which the dichotomous case implies the general case.Victor aime plus la sécurité que Ursula si, dès que Ursula préfère une constante à un acte incertain, il en est de même pour Victor. Ce paradigme, qui, dans le modèle EU, n'est autre que le concept d'Arrow-Pratt : "plus d'aversion pour le risque que", sera étudié dans le modèle CEU, modèle de Choquet de décision dans l'incertain, où on appelle u et μ (v et ν) l'utilité et la capacité d'Ursula (de Victor). Nous présentons une condition nécessaire et suffisante (A) sur les paires (u, μ) et (v, ν) pour l'accroissement faible d'aversion pour l'incertain dichotomique, la préférence de Victor pour une constante à un acte dichotomique dès que Ursula a cette préférence. Cette condition, inégalité ponctuelle entre une fonction en termes de v (u-1(⋅)) et une autre uniquement en termes de capacités, garde la forme de la caractérisation de l'aversion pour le risque monotone de Chateauneuf, Cohen et Meilijson et de son extension à l'incertain monotone de Grant et Quiggin dans le modèle de Choquet. Nous présentons une condition suffisante (B) de "plus de goût pour la simplicité" (préférence de Victor pour un acte dichotomique sur tout autre acte, qui laisse Ursula indifférente), uniquement en termes de capacités, satisfaite en particulier si ν (⋅) = f (μ (⋅)) pour une f convexe. Nous exposons un contre-exemple à la conjecture suivant laquelle le cas dichotomique impliquerait le cas général