In this paper, we develop a novel algorithm for constructing time-uniform, asymptotic confidence sequences for quantiles under local differential privacy (LDP). The procedure combines dynamically chained parallel stochastic gradient descent (P-SGD) with a randomized response mechanism, thereby guaranteeing privacy protection while simultaneously estimating the target quantile and its variance. A strong Gaussian approximation for the proposed estimator yields asymptotically anytime-valid confidence sequences whose widths obey the law of the iterated logarithm (LIL). Moreover, the method is fully online, offering high computational efficiency and requiring only $\mathcal{O}(\kappa)$ memory, where $\kappa$ denotes the number of chains and is much smaller than the sample size. Rigorous mathematical proofs and extensive numerical experiments demonstrate the theoretical soundness and practical effectiveness of the algorithm.

BACKGROUND: Expression microarrays are increasingly used to characterize environmental responses and host-parasite interactions for many different organisms. Probe selection for cDNA microarrays using expressed sequence tags (ESTs) is challenging due to high sequence redundancy and potential cross-hybridization between paralogous genes. In organisms with limited genomic information, like marine organisms, this challenge is even greater due to annotation uncertainty. No general tool is available for cDNA microarray probe selection for these organisms. Therefore, the goal of the design procedure described here is to select a subset of ESTs that will minimize sequence redundancy and characterize potential cross-hybridization while providing functionally representative probes. RESULTS: Sequence similarity between ESTs, quantified by the E-value of pair-wise alignment, was used as a surrogate for expected hybridization between corresponding sequences. Using this value as a measure of dissimilarity, sequence redundancy reduction was performed by hierarchical cluster analyses. The choice of how many microarray probes to retain was made based on an index developed for this research: a sequence diversity index (SDI) within a sequence diversity plot (SDP). This index tracked the decreasing within-cluster sequence diversity as the number of clusters increased. For a given stage in the agglomeration procedure, the EST having the highest similarity to all the other sequences within each cluster, the centroid EST, was selected as a microarray probe. A small dataset of ESTs from Atlantic white shrimp (Litopenaeus setiferus) was used to test this algorithm so that the detailed results could be examined. The functional representative level of the selected probes was quantified using Gene Ontology (GO) annotations. CONCLUSIONS: For organisms with limited genomic information, combining hierarchical clustering methods to analyze ESTs can yield an optimal cDNA microarray design. If biomarker discovery is the goal of the microarray experiments, the average linkage method is more effective, while single linkage is more suitable if identification of physiological mechanisms is more of interest. This general design procedure is not limited to designing single-species cDNA microarrays for marine organisms, and it can equally be applied to multiple-species microarrays of any organisms with limited genomic information.