Chebychev's Inequality Research Articles

The Dynamics of Uncertainty or the Uncertainty of Dynamics: Stochastic J-Curves

This paper characterizes the statistical distribution of the response of the U.S. trade account to a dollar depreciation. To accomplish this task, the paper builds and estimates an econometric model of U.S. bilateral trade. Given an exchange-rate shock, this distribution is generated empirically by stochastically simulating this model using random drawings for both innovations and trade elasticities. The paper finds that the distribution of trade-account responses is not stationary, that its variance is directly related to the size of the exchange-rate shock, that the dominant source of uncertainty lies with imports' price elasticities, and that the dispersion of these responses is more pronounced in the short run than in the long run. Based on these properties, the analysis applies Chebychev's inequality to the sample of trade-account responses and finds that hysteresis in price elasticities has a low probability of accounting for the persistence of the U.S. trade deficit. ; These findings have two practical implications. First, forecasts of trade-account responses to exchange-rate shocks should include the associated confidence intervals. Uncertainty in these responses is potentially large and omitting the corresponding confidence intervals is analogous to omitting standard errors of regression estimates. Second, deriving confidence intervals needs to recognize that parameter estimates are random variables and that they contribute, quite significantly in this application, to the width of these intervals.

On the exponential model for survival

SUMMARY A simple sufficient condition for the existence, consistency and asymptotic normality of the regression coefficient in the exponential model for survival is given. The intuition that consistent estimation can only be guaranteed when censoring variables are not too small, depending on the values of the covariates, can be quantified in some examples with the use of this condition. The exponential model for survival time is a fundamental building block in survival analysis, yet there seems to be no easily accessible condition which guarantees existence, consistency and asymptotic normality of the estimators. Well-known heuristics, such as those of Kalbfleisch & Prentice (1980, p. 51), suggest that a sufficient condition will be uniform asymptotic negligibility of the sequence of covariate vectors by comparison with the information matrix. We show that this is indeed the case; see Condition 1 below. The simplicity of this condition encourages applications, and we demonstrate, for example, how Condition 1 reveals the role of censoring in the consistency of the estimator. Since an analogue of Condition 1 is a necessary and sufficient condition for asymptotic normality of the regression coefficients in the ordinary least-squares model (Eicker, 1965), it is probably as general as could be hoped for. Our method is to extend to the survival context the methods of Fahrmeir & Kaufmann (1985), who, in the area of generalized linear models, took advantage of the 'close to linear' structure of those models to strengthen traditional asymptotic arguments for general maximum likelihood estimators. The asymptotics of survival models have been widely studied using the theory of counting processes and martingales, and Borgan (1984) uses this theory to deal with a wide class of models with covariates. To apply his results (Borgan, 1984, p. 14), however, very technical conditions involving uniform convergence of various stochastic processes must be verified. Borgan's methods rely on a traditional approach and cannot yield simple sufficient conditions like Condition 1. We need nothing more sophisticated than Chebychev's inequality and the central limit theorem for sums of independent random variables, although we are restricted to the 'close to linear' exponential model. Another approach by Friedman (1982) requires complex assumptions

The Dynamics of Uncertainty or the Uncertainty of Dynamics: Stochastic J-Curves

This paper characterizes the statistical distribution of the response of the U.S. trade account to a dollar depreciation. To accomplish this task, the paper builds and estimates an econometric model of U.S. bilateral trade. Given an exchange-rate shock, this distribution is generated empirically by stochastically simulating this model using random drawings for both innovations and trade elasticities. The paper finds that the distribution of trade-account responses is not stationary, that its variance is directly related to the size of the exchange-rate shock, that the dominant source of uncertainty lies with imports' price elasticities, and that the dispersion of these responses is more pronounced in the short run than in the long run. Based on these properties, the analysis applies Chebychev's inequality to the sample of trade-account responses and finds that hysteresis in price elasticities has a low probability of accounting for the persistence of the U.S. trade deficit.These findings have two practical implications. First, forecasts of trade-account responses to exchange-rate shocks should include the associated confidence intervals. Uncertainty in these responses is potentially large and omitting the corresponding confidence intervals is analogous to omitting standard errors of regression estimates. Second, deriving confidence intervals needs to recognize that parameter estimates are random variables and that they contribute, quite significantly in this application, to the width of these intervals.

A Combinatoric Approach to the Kaplan-Meier Estimator

The paper considers the Kaplan-Meier estimator $F^{\mathrm{KM}}_n$ from a combinatoric viewpoint. Under the assumption that the estimated distribution $F$ and the censoring distribution $G$ are continuous, the combinatoric results are used to show that $\int |\theta(z)| dF^{\mathrm{KM}}_n(z)$ has expectation not larger than $\int |\theta(z)| dF(z)$ for any sample size $n$. This result is then coupled with Chebychev's inequality to demonstrate the weak convergence of the former integral to the latter if the latter is finite, if $F$ and $G$ are strictly less than 1 on $\mathscr{R}$ and if $\theta$ is continuous.

C66. Sharpened chebychev inequalities

C66. Sharpened chebychev inequalities

Estimation in Sampling Theory with Exchangeable Prior Distributions

Godambe (1955) and Godambe and Joshi (1965) established the joint optimality of the Horvitz-Thompson estimates and the corresponding sampling designs with respect to a class of prior distributions which were product measures. In the present paper the optimality is established replacing the product measures by prior distributions which are exchangeable in appropriately transformed variates. Furthermore, the optimality criterion is justified in terms of Chebychev's inequality.

Consistent Estimation of a Location Parameter in the Presence of an Incidental Scale Parameter

location parameter 0 which is constant for the whole sample sequence (Xv),zN in the presence of an unknown (incidental) scale parameter a- whose value changes from one sample x, to the next. In statistical theory this problem deserves at least the same interest as the central limit problem. The interest in the asymptotic behavior of Jr = n- >jn=1 x, is rooted in the role which JV plays as an estimate for 0. The generalization which was forced upon this problem in probability theory, namely to study the asymptotic behavior of bn7l( .jj=i xv - an), was a mistake from the statistical point of view. The relevant generalization is to consider estimates of 0 other than .n . Despite its importance, this problem is rarely treated in the literature. The only pertinent result known to the author is due to Wolfowitz (1953, p. 17). It assures the existence of a strongly consistent sequence of estimates for the particular case of P being the normal distribution N(0, 1) under the condition (C) sup n-lN EnZ 1 ( v )2 < 00 This result was obtained by the minimum distance method. It does, however, not reveal the full power of this method. It is easy to see that under condition (C), the sequence of means ( n)n,N is consistent for 0, and the sequence of estimates defined by On(x) = x2[10g2 n], n c N, is strongly consistent ([a] denotes the largest integer less than or equal to a). Chebychev's inequality implies