Standard Loss Function Research Articles

Simultaneous One-Sided Tests With Application to Education Evaluation Systems

There is widespread interest in using various statistical inference tools as a part of the evaluations for individual teachers and schools. Evaluation systems typically involve classifying hundreds or even thousands of teachers or schools according to their estimated performance. Many current evaluations are largely based on individual estimates and hypothesis tests, with little or no consideration of controlling simultaneous error rates and their potential effect on the whole educational evaluation system. In this article, we discuss controlling simultaneous errors in classification of teachers or schools by a decision-theoretic approach. We first develop a β-mixture model to estimate the local false discovery rate (local fdr) by the classic p-values from one-sided tests. We discuss a few decision rules based on standard loss functions. We also construct a controversy loss function to further accommodate the incoherence between effect size and local fdr. We apply the proposed approach to evaluate adequate yearly progress (AYP) in math proficiency of Pennsylvania schools.

Optimal operation of cryogenic air separation systems with demand uncertainty and contractual obligations

Cryogenic air separation is an efficient technology for supplying large quantities of nitrogen, argon, and oxygen to chemical, petroleum and manufacturing customers. However, numerous uncertainties make effective operation of these complex processes difficult. This work addresses the problem of determining an optimal operating strategy to maximize the total profit of a cryogenic air separation process while considering demand uncertainty and contractual obligations. A rigorous process model is included as constraints in a nonlinear programming formulation. Uncertain demands are assumed to be normally distributed with known mean and standard deviation, and expected profit in the objective function is evaluated using the standard loss function. A probabilistic fill-rate expression, also based on the loss function, is used to model the contractual obligations by providing a lower bound on the expected product sales. In the single period case with one customer satisfaction constraint, the nonlinear programming formulation can be solved efficiently using the general purpose nonlinear optimization package, Ipopt. This formulation is then extended to include multiple time periods, the potential for product storage, and customer satisfaction constraints on multiple products. To solve the large-scale nonlinear programming formulation that considers a seven-day operating horizon, a tailored parallel nonlinear programming algorithm is used. This approach makes use of a Schur complement decomposition strategy to exploit the block structure of the problem and allow efficient solution in parallel. Using these tools, we solve for a set of optimal operating strategies over the complete space of different fill rates. This produces planning figures that identify key trade-offs between profitability and contractual obligations.

The equivalence of Bayes and robust Bayes estimators for various loss functions

The problem of Bayes and robust Bayes estimation for various bounded and/or symmetric loss functions in a normal model with conjugate and non-informative prior distributions is considered. The prior distribution is not fully specified and covers the conjugate family of priors. It is of interest to know that the Bayes and robust Bayes estimators for symmetric losses are the same as those for the standard square-error loss function.

Forecasting stock market volatility: Further international evidence

This paper evaluates the out-of-sample forecasting accuracy of eleven models for monthly volatility in fifteen stock markets. Volatility is defined as within-month standard deviation of continuously compounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997. The first half of the sample is retained for the estimation of parameters while the second half is for the forecast period. The following models are employed: a random walk model, a historical mean model, moving average models, weighted moving average models, exponentially weighted moving average models, an exponential smoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model, and an EGARCH model. First, standard (symmetric) loss functions are used to evaluate the performance of the competing models: mean absolute error, root mean squared error, and mean absolute percentage error. According to all of these standard loss functions, the exponential smoothing model provides superior forecasts of volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models. Asymmetric loss functions are employed to penalize under-/over-prediction. When under-predictions are penalized more heavily, ARCH-type models provide the best forecasts while the random walk is worst. However, when over-predictions of volatility are penalized more heavily, the exponential smoothing model performs best while the ARCH-type models are now universally found to be inferior forecasters.

Bayesian estimation based on trimmed samples from Pareto populations

Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the inequality and precision parameters of a Pareto distribution based on a trimmed sample and prior information is considered. From an inferential viewpoint, the problem of finding the highest posterior density (HPD) estimates of the Pareto parameters is discussed. The existence and uniqueness of the HPD estimates are established under mild conditions; explicit and accurate lower and upper bounds are also provided. Adopting a decision-theoretic perspective, several Bayesian estimators for standard loss functions are presented. In addition, two-sided and HPD credibility intervals for each Pareto parameter and joint HPD credibility regions for both parameters are derived, which have the corresponding frequentist confidence level in the noninformative case. Finally, an illustrative example concerning annual wage data is included.

Monetary Stabilisation with Nominal Asymmetries

Optimal monetary stabilisation in the standard New Keynesian framework usually assumes a policy loss function from outside the model. In this paper, in contrast, the objective arises directly from the model. Credit constraints and sticky nominal debt contracts imply that monetary stabilisation has asymmetric impacts depending on whether consumers are credit‐constrained. The policy problem is to maximise some weighting of the expected utility of the different types of consumer. Features of optimal stabilisation are derived that do not appear to be far out of line with empirical evidence for many countries but that clearly conflict with standard loss function results.

The macroeconomic Loss Function: a Critical Note

The standard loss function counts both positive and negative deviations from the output and inflation targets as losses. But if the sample period is long enough, then output growth in excess of the target, and often also inflation rates that are below target, should be counted as gains instead of losses.

Forecasting Stock Market Volatility: Evidence From Fourteen Countries

This paper evaluates the out-of-sample forecasting accuracy of eleven models for weekly and monthly volatility in fourteen stock markets. Volatility is defined as within-week (within-month) standard deviation of continuously compounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997. The first half of the sample is retained for the estimation of parameters while the second half is for the forecast period. The following models are employed: a random walk model, a historical mean model, moving average models, weighted moving average models, exponentially weighted moving average models, an exponential smoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model, and an EGARCH model. We first use the standard (symmetric) loss functions to evaluate the performance of the competing models: the mean error, the mean absolute error, the root mean squared error, and the mean absolute percentage error. According to all of these standard loss functions, the exponential smoothing model provides superior forecasts of volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models. We also employ the asymmetric loss functions to penalize under/over-prediction. When under-predictions are penalized more heavily ARCH-type models provide the best forecasts while the random walk is worst. However, when over-predictions of volatility are penalized more heavily the exponential smoothing model performs best while the ARCH-type models are now universally found to be inferior forecasters.

The Macroeconomic Loss Function: A Critical Note

The standard loss function counts both positive and negative deviations from the output and inflation targets as losses. But if the sample period is long enough, then output growth in excess of the target, and often also inflation rates that are below target, should be counted as gains instead of losses.

Why Do Countries Float the Way They Float?

Countries that are classified as having floating exchange rate systems (or very wide bands) show strikingly different patterns of behavior. They hold very different levels of international reserves and allow very different volatilities to the movements of the exchange rate relative to the volatility that they tolerate either on the level of reserves or on interest rates. We document these differences and present a model that explains them as the optimal response of a Central Bank that attempts to minimize a standard loss function, in an environment in which firms are credit-constrained and incomplete markets limit their ability to avoid currency mismatches. This model suggests that the difference in the way countries float cold be related to their differing levels of exchange rate pass-through and the differing ability to avoid currency mismatches. We test these implications and find a very strong and robust relationship between the pattern of floating and the ability of a country to borrow internationally in its own currency. We find weaker and less robust evidence on the importance of pass-through to account for differences across countries with respect to their exchange rate/monetary management.

Nonparametric empirical bayes procedures, asymptotic optimality And rates Of convergence For two‐tail tests In exponential family*

This paper provides nonparametric empirical Bayes (EB) solutions to two-tail test in the exponential family , under the standard product loss function which is proportional to (θ-θ1) (θ-θ2) for incorrectly accepting H1. Based on empirical data X1,…X n , and the present data X from nonparametric (in the sense that G is completely unknown and unspecified) EB test procedures are proposed. These procedures are asymptotically optimal (a.o.) whenever Further, for every integer s > 0 a class of non-parametric EB test procedures is proposed. These procedures are shown to be a.o. with rates for 0 < λ ≤ 2 satisfying certain conditions. Examples of exponential families and gamma densities are given where these conditions reduce to some simple moment conditions on G. No assumption on the smoothness of the function u(.), (and hence of the density function of X), is made at all for any of the results of this paper. By an example of a family of distributions, it is demonstrated that the rates arbitrarily close to o(n -1) can be attained by these procedures in some situations. It is noted, however, that the actual rates of convergence really depends on the nature of the unknown prior distribution G.

Why Do Countries Float the Way They Float?

Countries that are classified as having floating exchange rate systems (or very wide bands) show strikingly different patterns of behavior. They hold very different levels of international reserves and allow very different volatilities in the movements of the exchange rate relative to the volatility that they tolerate either on the level of reserves or in interest rates. We document these differences and present a model that explains them as the optimal response of a Central Bank that attempts to minimize a standard loss function, in an environment in which firms are credit-constrained and incomplete markets limit their ability to avoid currency mismatches. This model suggests that the difference in the way countries float could be related to their differing levels of exchange rate pass-through and differences in their ability to avoid currency mismatches. We test these implications and find a very strong and robust relationship between the pattern of floating and the ability of a country to borrow internationally in its own currency. We find weaker and less robust evidence on the importance of pass-through to account for differences across countries with respect to their exchange rate/monetary management.

Monetary Policy with a Nonlinear Phillips Curve and Asymmetric Loss

Recent theoretical and empirical work has cast doubt on the hypotheses of a linear Phillips curve and a symmetric quadratic loss function underlying traditional thinking on monetary policy. This paper studies the one-period optimal monetary policy problem under an asymmetric loss function corresponding to the "opportunistic approach" to disinflation and a convex Phillips curve. The policy-inaction range and its properties are derived analytically. Numerical simulations are then used to assess the implications of asymmetric loss for the distributional properties of the equilibrium levels of inflation and unemployment. For parameter values relevant to the U.S., it is found that the asymmetric loss function yields an average inflation rate in excess of the target, and that bias is larger than the standard symmetric loss function. For moderate policy-maker preferences, the asymmetric loss function also yields a smaller gap between average unemployment and the natural rate, and higher (lower) variance of inflation (unemployment) compared to the symmetric benchmark. Calibrating the model to match the observed average unemployment rate requires a high degree of inflation aversion and small asymmetry.

The finite sample properties of simultaneous equations' estimates and estimators Bayesian and non-Bayesian approaches

After discussing the need for good finite sample estimation procedures for simultaneous equations models and showing the inadequacies of asymptotically justified estimators, it is shown how the Bayesian method of moments (BMOM) provides an exact, finite sample analysis of unrestricted reduced form systems. Then optimal, finite sample estimates of structural coefficients are derived using three standard loss functions and they are compared to traditional Bayesian optimal estimates. Monte Carlo experimental evidence from four studies on the relative performance of Bayesian and non-Bayesian estimators is reviewed with the finding that the performance of Bayesian estimators is better.

Mitigating the Effects of Multicollinearity Using Exact and Stochastic Restrictions: The Case of an Aggregate Agricultural Production Function in Thailand: Comment

Mittelhammer, Young, Tasanasanta, and Donnelly in estimating an aggregate agricultural production function use a variety of techniques to deal with multicollinearity present in their sample, including exact and stochastically restricted least squares and principal components regression.' They claim that consideration of these techniques allowed mitigation of a serious multicollinearity problem present in their data and permitted more precise parameter estimates. The authors also hold that two of the techniques they considered generally outperformed OLS in terms of risk and overall reasonableness ... (p. 199). While the empirical evidence Mittelhammer et al. present is convincing and the product of sound econometrics, we feel that some important limitations of the estimation techniques they employed as alternatives to ordinary least squares (OLS) were not sufficiently stressed. Furthermore, we will provide estimators that are unambiguously superior to least squares under a variety of standard loss functions.

Minimax estimators for a multinormal precision matrix

Let S p×p ∼ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ −1 are given for L 1, an Empirical Bayes loss function; and L 2, a standard loss function ( R i ≡ E( L i ∣ Σ), i = 1, 2). The estimators are Σ ̂ −1 = aS −1 + br(S)I p×p , a, b ≥ 0, r(·) a functional on R p(p+2) 2 . Stein, Efron, and Morris studied the special cases Σ a −1 = aS −1 (E Σ ̂ k−p−1 −1 = Σ −1) and Σ ̂ 1 −1 = aS −1 + (b/tr S)I , for certain, a, b. From their work R 1(Σ −1, Σ ̂ 1 −1; S) ≤ R 1(Σ −1, Σ ̂ a −1; S) (∀Σ) , a = k − p − 1, b = p 2 + p − 2; whereas, we prove R 2(Σ −1 Σ ̂ a −1; S) ≤ R 2(Σ −1, Σ ̂ 1 −1; S) (∀Σ) . The reversal is surprising because L 1(Σ −1, Σ ̂ 1 −1; S) → L 2(Σ −1, Σ ̂ 1 −1; S) a.e. (for a particular L 2). Assume R (compact) ⊂ S , S the set of p × p p.s.d. matrices. A “divergence theorem” on functions F p× p : R → S implies identities for R i , i = 1, 2. Then, conditions are given for R i (Σ −1, Σ ̂ −1; S) ≤ R i (Σ −1, Σ ̂ 1 −1; S) ≤ R i (Σ −1, Σ ̂ a −1; S) (∀Σ) , i = 1, 2. Most of our results concern estimators with r( S) = t(U)/ tr( S), U = p ∣ S∣ 1/p/ tr( S).