Emerging delay-sensitive applications on the Internet increase awareness of the Quality of Service (QoS) parameters of a path for Internet Service Providers (ISPs) as well as users, However, it is costly to frequently monitor delays along individual paths among every edge-router in the ISP. The most widely used way of estimating such statistics is by actively sending probe packets along each path, despite the increased transmission of wasteful traffic introduced by the probe packets itself for frequent and accurate estimations. On the other hand, each router can passively observe local queuing delays experienced at the router. However, while the mean delays can always be concatenated concentrating simply by summing those at tandem routers along a path, the statistics (other than the mean) such as the 90-percentile cannot be estimated accurately by such a simple-sum scheme because of dependence among delays at such routers on the Internet. In this work, a novel scheme to estimate the QoS parameters of a path is proposed, which combines statistics gatherd at each router and data obtained from a small number of sampling along the path. For delays, considering an unknown joint discrete distribution of quantized queuing delays on routers along a path, we find the maximum likelihood estimator for the unknown distribution (under the constraints of the marginal distributions measured at each router) from the samples. Theoretical analysis and numerical simulations indicate that this scheme effectively estimates the delay statistics along a path even with a small number of samples, which allows continual measurements capturing statistics with a broad range of time-scales.