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In financial overlapping return applications, Lo and MacKinlay ( 1988) and Hansen and Hodrick ( 1980) demonstrate how overlapping returns may be used to increase the efficiency of statistics used in variance ratio tests. Several estimation procedures and hypothesis testing frameworks have been improved through the utilization of overlapping returns. Our central aim is to construct unbiased weighted variance and skewness estimators for overlapping return distributions. This motivates the development of unbiased analogues of such estimators which we explore in the cases of the variance and skewness statistics. In addition, for such series, recent data is regularly viewed as more relevant than past information, which has resulted in the creation of weighted generalizations of estimation methodologies. Standard statistical inference and estimation techniques applied to overlapping return financial time series are typically biased.

Applications include variance ratio tests, regression parameter error estimation, and alternative resampling methods. You should perhaps use a Bayesian estimate or Wilson score interval.Overlapping returns are used in many contexts in the finance and econometrics literature. Weighting by the inverse of the SEM is a common and sometimes optimal thing to do. Taking percentages the way you are is going to make analysis tricky even if they're generated by a Bernoulli process, because if you get a score of 20 and 0, you have infinite percentage. You don't have an estimate for the weights, which I'm assuming you want to take to be proportional to reliability. Where $x^* = \sum w_i x_i / \sum w_i$ is the weighted mean. In any case, the formula for variance (from which you calculate standard deviation in the normal way) with "reliability" weights is (Actually, all of this is rubbish-you really need to use a more sophisticated model of the process that is generating these numbers! You apparently do not have something that spits out Normally-distributed numbers, so characterizing the system with the standard deviation is not the right thing to do.) Instead, you need to use the last method. You generate your data from frequencies, but it's not a simple matter of having 45 records of 3 and 15 records of 4 in your data set. In your case, it superficially looks like the weights are frequencies but they're not. you are just trying to avoid adding up your whole sum), if the weights are in fact the variance of each measurement, or if they're just some external values you impose on your data. In particular, you will get different answers if the weights are frequencies (i.e. The key is to notice that it depends on what the weights mean. The formulae are available various places, including Wikipedia.