In this study, at-site frequency analysis (AFA) of an annual daily maximum rainfall (ADMR) series was carried out using the method of linear moments (L-moments) and their variants such as trimmed linear moments (TL-moments) and higher order linear moments (LH-moments). The ADMR series we investigated was observed at 28 meteorological observatories across Pakistan as retrieved from the Pakistan Meteorological Department (PMD). The basic aim of the study was to find best-fit (i.e., the most suitable) probability distribution among the class of various probability distributions. Initially different goodness-offit (GOF) measures such as the Kolmogorov-Smirnov test (KST), Anderson-Darling test (ADT), root mean square error (RMSE) and L-moments ratio diagram (LRD) were applied to determine not only the best-fit distributions but also the best linear estimation method for AFA. We observed that no single probability distribution could be declared as the best-fit distribution for all the stations. Five distributions were found to be the most appropriate: generalized extreme value (GEV), three parameter lognormal (LN3), Pearson type III (P3), generalized logistic (GLO), and generalized pareto (GPA). The TL-moments method was also applied for parameter estimation to mitigate the effect of outliers on final estimates. LH-moments were used for estimating the upper part of probability distributions and larger events in the data samples. LH moments alleviate the unwanted affects due to small sample values that may be obvious during estimation of events related to larger return periods. Using different GOF tests, we observed that the L-moments method was best for eight stations, TL-moments with trimming (1, 0), and LH-moments with level η =2, 3, 4 were best for six and 14 stations, respectively. A theoretical relationship between TL-moments and LH-moments was also revisited, which revealed that LH-moments are special cases of TL-moments when we are motivated to make trimming only from the lower side.