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Yannis G. Yatracos

Publications and source records attributed to Yannis G. Yatracos.

12 recordsLinked to original sources
arXiv2025

Plug-in error bounds for a mixing density estimate in $R^d,$ and for its derivatives

A mixture density, $f_p,$ is estimable in $R^d, \ d \ge 1,$ but an estimate for the mixing density, $p,$ is usually obtained only when $d$ is unity; $h$ is the mixture's kernel. When $f_p$'s estimate has form $f_{\hat p_n}$ and $p$ is $\tilde q$-smooth, vanishing outside a compact in $R^d,$ plug-in upper bounds are obtained herein for the $L_u$-error (and risk)of $\hat p_n$ and its derivatives; $d \ge 1, 1 \le u \le \infty.$ The bounds depend on $f_{\hat p_n}$'s $L_u$-error (or risk), $h$'s Fourier transform, $\tilde h,$ and the bandwidth of kernel $K$ used in approximations. The choice of $\hat p_n,$ via $f_{\hat p_n},$ suggests that $\hat p_n$'s error rate could be only nearly optimal when $f_{\hat p_n}$ is optimal, but competing estimates and their error rates may not be available for $d>1.$ In examples with $d$ unity, the upper bound is optimal when $h$ is super smooth, misses the optimal rate by the factor $(\log n)^ξ, \ ξ>0,$ when $h$ is smooth, and is satisfactory when $\tilde h$ has periodic zeros.

math.ST↗
arXiv2020

Fiducial Matching for the Approximate Posterior: F-ABC

F-ABC is introduced, using universal sufficient statistics, unlike previous ABC papers, e.g. Bernton et al. (2019), and avoiding in the approximate posterior artifacts due to a Kernel. The nature of matching tolerance is examined and indications for determining its values are presented. F-ABC does not face concerns associated with ABC. Asymptotics and simulation results are also presented.

stat.ME↗
arXiv2017

The Derivative of Influence Function, Location Breakdown Point, Group Leverage and Regression Residuals' Plots

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 $F_{ε, {\bf x},y}.$ 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 $ε=k/n.$ For high dimensional, simulated data, the misclassification proportion of bad leverage cases in data's RINFIN-ordering decreases to zero as $p$ increases, thus reconfirming the blessing of high dimensionality in the detection of remote clusters. The visual diagnostic and RINFIN are successful in applications and complement each other.

math.ST↗
arXiv2015

Statistical Inference with Data Augmentation and Parameter Expansion

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.

math.ST↗
arXiv2015

Information in stock prices and some consequences: A model-free approach

The price of a stock will rarely follow the assumed model and a curious investor or a Regulatory Authority may wish to obtain a probability model the prices support. A risk neutral probability ${\cal P}^*$ for the stock's price at time $T$ is determined in closed form from the prices before $T$ without assuming a price model. The findings indicate that ${\cal P}^*$ may be a mixture. Under mild conditions on the prices the necessary and sufficient condition to obtain ${\cal P}^*$ is the coincidence at $T$ of the stock price ranges assumed by the stock's trader and buyer. This result clarifies the relation between market's informational efficiency and the arbitrage-free option pricing methodology. It also shows that in an incomplete market there are risk neutral probabilities not supported by each stock and their use can be limited. ${\cal P}^*$-price $C$ for the stock's European call option expiring at $T$ is obtained. Among other results it is shown for "calm" prices, like the log-normal, that i) $C$ is the Black-Scholes-Merton price thus confirming its validity for various stock prices, ii) the buyer's price carries an exponentially increasing volatility premium and its difference with $C$ provides a measure of the market risk premium.

q-fin.GN↗
arXiv2014

A new method to obtain risk neutral probability, without stochastic calculus and price modeling, confirms the universal validity of Black-Scholes-Merton formula and volatility's role

A new method is proposed to obtain the risk neutral probability of share prices without stochastic calculus and price modeling, via an embedding of the price return modeling problem in Le Cam's statistical experiments framework. Strategies-probabilities $P_{t_0,n}$ and $P_{T,n}$ are thus determined and used, respectively,for the trader selling the share's European call option at time $t_0$ and for the buyer who may exercise it in the future, at $T; \ n$ increases with the number of share's transactions in $[t_0,T].$ When the transaction times are dense in $[t_0,T]$ it is shown, with mild conditions, that under each of these probabilities $\log \frac{S_T}{S_{t_0}}$ has infinitely divisible distribution and in particular normal distribution for "calm" share; $S_t$ is the share's price at time $t.$ The price of the share's call is the limit of the expected values of the call's payoff under the translated $P_{t_0,n}.$ It coincides for "calm" share prices with the Black-Scholes-Merton formula with variance not necessarily proportional to $(T-t_0),$ thus confirming formula's universal validity without model assumptions. Additional results clarify volatility's role in the transaction and the behaviors of the trader and the buyer. Traders may use the pricing formulae after estimation of the unknown parameters.

q-fin.PR↗
arXiv2014

MLE's bias pathology, Model Updated Maximum Likelihood Estimates and Wallace's Minimum Message Length method

The inherent bias pathology of the maximum likelihood (ML) estimation method is confirmed for models with unknown parameters $θ$ and $ψ$ when MLE $\hat ψ$ is function of MLE $\hat θ.$ To reduce $\hat ψ$'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, $\hat ψ_{MU},$ often reduces either totally or partially $\hat ψ$'s bias when estimating shape parameter $ψ.$ For the Pareto model $\hat ψ_{MU}$ reduces also $\hat ψ$'s variance. The results explain the difference that puzzled R. A. Fisher, between biased $\hat ψ$ 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.

math.ST↗
arXiv2014

Simultaneous causal inference for multiple treatments via sufficiency

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.

math.ST↗
arXiv2013

MLE's bias pathology motivates MCMLE

Maximum likelihood estimates are often biased. It is shown that this pathology is inherent to the traditional ML estimation method for two or more parameters, thus motivating from a different angle the use of MCMLE.

math.ST↗
arXiv2013

Option pricing, Bayes risks and Applications

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 Levy 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.

q-fin.PR↗
arXiv2013

Concerns on Monotonic Imbalance Bounding Matching Methods

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.

math.ST↗