A researcher or a practitioner may frequently confront one among many fundamental sequential estimation problems. It may involve the construction of a fixed width confidence interval (FWCI) or the minimum risk point estimation (MRPE) for a parametric function \(\tau ({\varvec{\theta }})\) of interest. Often they are very difficult to handle requiring some kind of very specialized sequentially designed sampling strategies for gathering appropriate data. The existing literature is vast and the ensuing technicalities are just too many to enumerate. In this paper, a major building block is the immediate availability of an expression of a fixed-sample-size arbitrary and perhaps complicated or awkward looking uniformly minimum variance unbiased estimator (UMVUE) of an appropriate parametric function \(\tau ({\varvec{\theta }})\) of choice from a population distribution \(f(x;{\varvec{\theta }}),\) \(\varvec{ \theta }\in \Theta \) , belonging to an exponential family. The UMVUE may be technically so intractable that a pretty looking closed-form expression of the variance (assumed finite) of the UMVUE may even be out of reach. Under the present culture of big data science, we argue that the required optimal fixed-sample-size ( \(n^{*}\) ) should be assumed very large and hence one should incorporate recording k-tuples of observations (instead of one observation) at-a-time, where k itself is allowed to be very large but staying small compared with \(n^{*}\) . We develop such a unified theory of FWCI and MRPE problems via arbitrary but complicated looking UMVUEs of many kinds of parametric functions \(\tau ({\varvec{\theta }} ) \) which normally stay out of sight from many usual standard textbook examples. We have validated a rich set of first-order (f.o.) unified asymptotic theory so developed here with large k, but held fixed. We have also pushed their immediate practical applications for purposes of making inferences, all within one big tent, by means of simulations. These simulations have allowed us to gather huge volumes of data providing ample evidence in favor of sequential sampling designs by recording k-tuples at-a-time. We do not see any reason to move away from gathering huge datasets that are called for in the context of highly accurate optimization as required. The present era of big data science has led us to suggest with conviction that sequential sampling designs with k-tuples at-a-time are here to stay and claim their due acceptance in the literature. Under the strong and steady grip of big data science in place, we offer no excuses to shy away from designing both k and \(n^{*}\) to be as large as necessary so long as we make sure that we have \(k=o(n^{*})\) .
