Dep. of Hotel and Tourism Management

Mean-preserving and covariance preserving matchings are introduced that can be obtained with conditional, randomized matching on sub-populations of a large control group. Under moment conditions it is shown that these matchings are, respectively, equal percent bias reducing (EPBR) and variance proportionate modifying (PM) for linear functions of the covariates and their standardizations. The results provide additional insight into and theory for EPBR and PM properties and confirm empirical and simulation findings that matchings can have the EPBR and PM properties also when the covariates are not exchangeable, or the treatment means are not equal.

The coarsest balancing score for multiple treatments T is the minimal sufficient statistic s for the covariates' distributions, D T , of populations {P t , t ∈ T } receiving each treatment t ∈ T. A unit in P r with covariates x is a good match with respect to T for a unit in P t with covariates y, when x and y provide similar information for D T , i.e. when s(x) ≈ s(y). For finite, countably finite and often continuous treatments, s(x) is shown to be equivalent toẽ, a vector of propensities' ratios. The units in {P t , t ∈ T } can be divided into subpopulations where causal comparisons are simultaneously valid. Satisfactory s-matchings are obtained for simulated covariates in R 3. The use ofẽ's estimate rather than s's estimate allows to avoid the x-curse of dimensionality, but the available data's size in each case is critical for the final choice.

Some units from a population receive the same treatment that is different from treatments available for other reservoir populations. The minimal sufficient statistic s for the pre-treatment x-covariates's distributions in the populations is the coarsest balancing score. s is used to select matching units for simultaneous causal comparisons of multiple treatments.Necessary and sufficient conditions on the posterior distribution of the treatment variable (given x) determine whether a statistic is either sufficient or minimal sufficient for the x-covariates' distributions. Results in the literature are thus extended. Strong ignorability of treatment assignment given s(x) is also established. Consequently, the expected treatments' differences given s(x) are shown to be simultaneously unbiased for the average causal effects of all treatments' differences. The existing statistical theory for s and its estimates support their use in causal inference.

Concerns are expressed for the Monotonic Imbalance Bounding (MIB) property (Iacus et al. 2011) and for MIB matching because i) the definition of the MIB property leads to inconsistencies and the nature of the imbalance measure is not clearly defined, ii) MIB property does not generalize Equal Percent Bias Reducing (EPBR) property, iii) MIB matching does not provide statistical information available with EPBR matching.

The inherent bias pathology of the maximum likelihood (ML) estimation method is confirmed for models with unknown parameters θ and ψ when MLÊ ψ is function of MLEθ. To reduceψ's bias the likelihood equation to be solved for ψ is updated using the model for the data Y in it. Model updated (MU) MLE,ψ M U , often reduces either totally or partiallyψ's bias when estimating shape parameter ψ. For the Pareto modelψ M U reduces alsoψ's variance. The results explain the difference that puzzled R. A. Fisher, between biasedψ and the unbiased estimate he obtained for two models with the "2-stage procedure". MUMLE's implementation is equivalent to the abandoned 2-stage procedure thus justifying its use. MUMLE and Firth's bias correcting likelihood are also obtained with the Minimum Message Length method thus motivating its use in frequentist inference and, more generally, model updating with a prior distribution.

In several linear regression data sets, $Y (\in R)$ on ${\bf X} (\in R^p),$ visual comparisons of $L_1$ and $L_2$-residuals' plots indicate bad leverage cases. The phenomenon is confirmed theoretically by introducing Location Breakdown Point (LBP) of a functional $T$: any point where the derivative of $T$'s Influence Function either takes values at infinities or does not exist. Guidelines for the plots' visual comparisons as diagnostic are provided. The new tools used include E-matrix and suggest influence diagnostic RINFIN which measures the distance in the derivatives of $L_2$-residuals} at $({\bf x},y)$ from model $F$ and from gross-error model $F_{\epsilon, {\bf x},y}.$ The larger RINFIN$({\bf x},y)$ is, the larger $({\bf x},y)$'s influence in $L_2$-regression residual is. RINFIN allows measuring group influence of $k$ ${\bf x}$-neighboring data cases in a size $n$ sample using their average, $(\bar {\bf x}_k,\bar y_k),$ as one case with weight $\epsilon=k/n.$ Fo...

The aim of this paper is to prove the phenotypic convergence of cryptocurrencies, in the sense that individual cryptocurrencies respond to similar selection pressures by developing similar characteristics. In order to retrieve the cryptocurrencies phenotype, we treat cryptocurrencies as financial instruments (genus proximum) and find their specific difference (differentia specifica) by using the daily time series of log-returns. In this sense, a daily time series of asset returns (either cryptocurrencies or classical assets) can be characterized by a multidimensional vector with statistical components like volatility, skewness, kurtosis, tail probability, quantiles, conditional tail expectation or fractal dimension. By using dimension reduction techniques (Factor Analysis) and classification models (Binary Logistic Regression, Discriminant Analysis, Support Vector Machines, K-means clustering, Variance Components Split methods) for a representative sample of cryptocurrencies, stocks, exchange rates and commodities, we are able to classify cryptocurrencies as a new asset class with unique features in the tails of the log-returns distribution. The main result of our paper is the complete separation of the cryptocurrencies from the other type of assets, by using the Maximum Variance Components Split method. More, we observe a divergent evolution of the cryptocurrencies species, compared to the classical assets, mainly due to the tails behaviour of the log-returns distribution. The codes used here are available via www.quantlet.de.

A statistical decision problem is hidden in the core of option pricing. A simple form for the price C of a European call option is obtained via the minimum Bayes risk, R B , of a 2-parameter estimation problem, thus justifying calling C Bayes (B-)price. The result provides new insight in option pricing, among others obtaining C for some stock-price models using the underlying probability instead of the risk neutral probability and giving R B an economic interpretation. When logarithmic stock prices follow Brownian motion, discrete normal mixture and hyperbolic Lévy motion the obtained B-prices are "fair" prices. A new expression for the price of American call option is also obtained and statistical modeling of R B can be used when pricing European and American call options.

The inherent bias pathology of the maximum likelihood (ML) estimation method is confirmed for models with unknown parameters θ and ψ when MLÊ ψ is function of MLEθ. To reduceψ's bias the likelihood equation to be solved for ψ is updated using the model for the data Y in it. Model updated (MU) MLE,ψ M U , often reduces either totally or partiallyψ's bias when estimating shape parameter ψ. For the Pareto modelψ M U reduces alsoψ's variance. The results explain the difference that puzzled R. A. Fisher, between biasedψ and the unbiased estimate he obtained for two models with the "2-stage procedure". MUMLE's implementation is equivalent to the abandoned 2-stage procedure thus justifying its use. MUMLE and Firth's bias correcting likelihood are also obtained with the Minimum Message Length method thus motivating its use in frequentist inference and, more generally, model updating with a prior distribution.

For a large class of distributions and large samples, it is shown that estimates of the variance σ 2 and of the standard deviation σ are more often Pitman closer to their target than the corresponding shrinkage estimates which improve the mean squared error. Our results indicate that Pitman closeness criterion, despite its controversial nature, should be regarded as a useful and complementary tool for the evaluation of estimates of σ 2 and of σ. Résumé : Nous montrons dans cet article que certains estimateurs de la variance σ 2 et de l'écart type σ sont plus souvent proches de leur cible au sens de Pitman que les estimateurs correspondants obtenus par "shrinkage", pourtant connus pour améliorer l'erreur quadratique moyenne. Nos résultats sont valables asymptotiquement et pour une grande famille de lois de probabilité. Ils indiquent en particulier que le critère de proximité de Pitman, malgré sa nature controversée, devrait être envisagé comme un outil utile à l'évaluation de la qualité des estimateurs de σ 2 et de σ .

For a large class of distributions and large samples, it is shown that estimates of the variance σ2 and of the standard deviation σ are more often Pitman closer to their target than the corresponding shrinkage estimates which improve the mean squared error. Our results indicate that Pitman closeness criterion, despite its controversial nature, should be regarded as a useful and complementary tool for the evaluation of estimates of σ2 and of σ. Index Terms — Variance estimation, standard deviation, shrinkage, Pitman closeness, mean squared error. 2010 Mathematics Subject Classification: 62F10, 62C15.

Statistical pragmatism embraces all efficient methods in statistical inference. Augmentation of the collected data is used herein to obtain representative population information from a large class of non-representative population's units. Parameter expansion of a probability model is shown to reduce the upper bound on the sum of error probabilities for a test of simple hypotheses, and a measure, R, is proposed for the effect of activating additional component(s) in the sufficient statistic.

A statistical decision problem is hidden in the core of option pricing. A simple form for the price C of a European call option is obtained via the minimum Bayes risk, R B , of a 2-parameter estimation problem, thus justifying calling C Bayes (B-)price. The result provides new insight in option pricing, among others obtaining C for some stock-price models using the underlying probability instead of the risk neutral probability and giving R B an economic interpretation. When logarithmic stock prices follow Brownian motion, discrete normal mixture and hyperbolic Lévy motion the obtained B-prices are "fair" prices. A new expression for the price of American call option is also obtained and statistical modeling of R B can be used when pricing European and American call options.

Approximate Bayesian Computation (ABC) provides posterior models for a stochastic parameter Θ when the observed, size n sample x(∈ R nxd), has intractable likelihood; d ≥ 1. For x from c.d.f. F θ , with unknown θ ∈ Θ, the ABC-steps are: a single sample, x * , is drawn from F θ * , with known θ * ∈ Θ; a nearly sufficient summary, S(x), is determined for matching x and x * within some-tolerance for a distance-measure ρ; if x and x * match, θ * is included in the approximate posterior with weight, K(x, x * ;); K is arbitrary kernel. We introduce Fiducial (F)-ABC, with M x * drawn from F θ *. The goal is a "one-for-all F-models" approach, with θ *-weights not K-artifacts and with universal sufficient S the empirical measure, µ X , when d > 1, the empirical cumulative distribution,F X , when d=1, and, respectively for ρ, the Total-Variation and the Kolmogorov distance, d K. Light is thrown to 's nature via d K , guidelines are given to determine its value and the "0-1" restrictive influence on θ * is reduced. θ *-weight is the proportion of x * matching x which, for many models, increases to 1 as θ * converges to θ, unlike K(x, x * ;). The number of simulations for implementation is moderate. Under few, mild assumptions, F-ABC posterior converges to its target and rates of concentration to T (F θ) are obtained; T functional. When M = 1, F-ABC is reduced to ABC. When M > 1, F-ABC posterior includes either selected θ * or all θ * used, reducing 's influence. In simulations, nonparametric F-ABC posterior improves the concentration of parametric ABC posterior at θ and "F-ABC posterior for all θ * " is satisfactory.

Some units from a population receive the same treatment that is different from treatments available for other reservoir populations. The minimal sufficient statistic s for the pre-treatment x-covariates's distributions in the populations is the coarsest balancing score. s is used to select matching units for simultaneous causal comparisons of multiple treatments.Necessary and sufficient conditions on the posterior distribution of the treatment variable (given x) determine whether a statistic is either sufficient or minimal sufficient for the x-covariates' distributions. Results in the literature are thus extended. Strong ignorability of treatment assignment given s(x) is also established. Consequently, the expected treatments' differences given s(x) are shown to be simultaneously unbiased for the average causal effects of all treatments' differences. The existing statistical theory for s and its estimates support their use in causal inference.

Minimum Kolmogorov and Wasserstein distance estimates, $$\hat{\theta }_{MKD}$$ θ ^ MKD and $$\hat{\theta }_{MWD},$$ θ ^ MWD , respectively, of model parameter, $$\theta (\in \Theta ),$$ θ ( ∈ Θ ) , are empirically compared, obtained assuming the model is intractable. For the Cauchy and Lognormal models, simulations indicate both estimates have expected values nearly $$\theta ,$$ θ , but $$\hat{\theta }_{MKD}$$ θ ^ MKD has in all repetitions of the experiments smaller SD than $$\hat{\theta }_{MWD},$$ θ ^ MWD , and $$\hat{\theta }_{MKD}$$ θ ^ MKD ’s relative efficiency with respect to $$\hat{\theta }_{MWD}$$ θ ^ MWD improves as the sample size, n,  increases. The minimum expected Kolmogorov distance estimate, $$\hat{\theta }_{MEKD},$$ θ ^ MEKD , has eventually bias and SD both smaller than the corresponding Wasserstein estimate, $$\hat{\theta }_{MEWD},$$ θ ^ MEWD , and $$\hat{\theta }_{MEKD}$$ θ ^ MEKD ’s relative efficiency improves as n increases. These results hold also for stable ...