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Information about the physical processes that generate the primordialfluctuations in the early universe can be gained by testing the Gaussian natureof the fluctuations through cosmic microwave background radiation (CBR)temperature anisotropy experiments. One of the crucial aspects of densityperturbations that are produced by the standard inflation scenario is that theyare Gaussian, whereas seeds produced by topological defects left over from anearly cosmic phase transition tend to be non-Gaussian. To carry out this test,sophisticated statistical tools are required. In this paper, we will discussseveral such statistical tools, including multivariant skewness and kurtosis,  Euler-Poincare characteristics, the three point temperature correlationfunction, and the Hotelling's $T^{2}$ statistic defined through bispectralestimates of a one dimensional dataset. The effect of noise present in thecurrent data is discussed in detail and the COBE 53 GHz dataset is analyzed.Our analysis shows that, on the large angular scale to which COBE is sensitive,the statistics are probably Gaussian. On the small angular scales, theimportance of Hotelling's $T^{2}$ statistic is stressed, and the minimum samplesize required to test Gaussianity is estimated. Although the current datasetavailable from various experiments at half-degree scales is still too small,improvement of the dataset by roughly a factor of two will be enough to testthe Gaussianity statistically. On the arcminute scales, we analyze the recentRING data through bispectral analysis, and the resultComment: 44 pages, Latex (15 Figures not included

Statistical tests for the Gaussian nature of primordial fluctuations through CBR experiments