Convergence rates for uniform B-spline density estimators on bounded and semi-infinite domains

Simulations that generate very large data sets in one and many dimensions are increasingly common. Nonparametric density estimates based on these data sets are often required and estimators that can be generated and manipulated efficiently are needed. In computer graphics, for example, nonparametric density estimates over surfaces can be used to represent lighting functions [1]. These simulations may generate as many as 50–100,000 data points for each light in the scene. Another application area is that of statistical genetics where simulations are needed for hypothesis tests when the exact distribution of the test statistics is not known [2]. For example, the likelihood ratio test statistics is, under reasonable conditions, asymptotically chi-squared. But in the case that the parameter for the null hypothesis is on the boundary of the parameter domain, the distribution of the test statistics may be unknown. The fact that spline functions can be efficiently evaluated on a digital computer has led to the use of splines for many statistical applications [3–9] and for density estimation [10]. We note that the paper by Lii [7] uses spline interpolation of the cumulative distribution function and shows that the bias is O(h3). But this result depends on knowing certain endpoint information. If this information is incorrect, then the estimate would have much larger bias at the endpoints. In addition to these papers, a B-spline nonparametric density estimator with uniformly spaced knots convenient for large data sets was discussed by Gehringer and Redner [11]. These