# How to test the differences between two correlated samples of different sizes

#### Drive

##### New Member
I have to ask help for a practical question. I have two samples of the monthly returns of mutual funds, in other words each sample is a matrix with n funds and for each fund I have 60 monthly returns (i.e. percent change in prices). The number of funds is always different from one sample to another, but more or less not smaller than 30 and as large as 400. The returns cover the same time window (2015-2019) and the funds invest in similar asset classes, in other words their monthly returns have a high correlation.

I have to make tests on one couple of samples each time. My aim is to test whether one measure of sample A is statistically larger (or smaller) than the same measure for sample B. This "measure" should be larger for sample A than for B when is the mean of the returns, the skewness index of returns, or the Sharpe ratio (calculated on the returns, again), etc. The "measure" should be smaller for sample A than for sample B when it is the standard deviation of returns o the kurtosis of returns.
Of course, for each couple of samples I have to test each measure separatedly, not all the measures at the same time (i.e. I have to perform several tests for each couple of samples).

I have already used the one tailed t-test on the means of each measure (the mean of the mean returns, the mean of the standard deviations of returns, etc.) of two samples at a time, but given that the t-test requires the independence of the two samples I fear it is not correct.

By the way, I cannot even measure the "correlation" between two samples of different sizes. I can measure the correlation between each pair of 60 monthly returns, but not between two samples of mean returns (or of standard deviations, Sharpe ratios, etc.) of different sizes and casual ranking (of course, you can list funds in any way - alphabetic, by investment style, random - there isn't a criterion for their place in the sample).

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