Exponential Inequalities for Self-normalized Processes with Applications
نویسندگان
چکیده
Self-normalized stochastic processes are frequently found in statistical applications. They have the property of (in the standard form) being unit free and frequently eliminate or weaken moment assumptions. The prototypical example of a self-normalized process is Student’s t-statistic, which is used in statistical analysis to test if the mean of a normally distributed sample has a value specified in a null hypothesis when the standard deviation of the underlying distribution is unknown. Let {X i : i ≥ 1} be a sequence of i.i.d. normal random variables with mean 0 and variance σ. The
منابع مشابه
Exponential Inequalities for Self-normalized Martingales with Applications
We propose several exponential inequalities for self-normalized martingales similar to those established by De la Peña. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.
متن کاملSelf - Normalized Processes : Exponential Inequalities , Moment Bounds and Iterated Logarithm Laws
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. ...
متن کاملSelf-normalized Processes: Exponential Inequalities, Moment Bounds and Iterated Logarithm Laws by Victor
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, selfnormalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. T...
متن کاملLimit properties for ratios of order statistics from exponentials
In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for [Formula: see text] with [Formula: see text]. We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, t...
متن کاملFinite sampling inequalities: an application to two-sample Kolmogorov-Smirnov statistics.
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development complements recent results by Wei and Dudley (2012) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving correspondi...
متن کامل