The examination of baseline noise and the impact on the interpretation of low-template DNA samples

It is common practice for DNA STR profiles to be analyzed using an analytical threshold (AT), but as more low template DNA (LT-DNA) samples are tested it has become evident that these thresholds do not adequately separate signal from noise. In order to confidently examine LT-DNA samples, the behavior and characteristics of the background noise of STR profiles must be better understood. Thus, the background noise of single source LT-DNA STR profiles were examined to characterize the noise distribution and determine how it changes with DNA template mass and injection time. Current noise models typically assume the noise is independent of fragment size but, given the tendency of the baseline noise to increase with template amount, it is important to establish whether the baseline noise is randomly found throughout the capillary electrophoresis (CE) run or whether it is situated in specific regions of the electropherogram. While it has been shown that the baseline noise of negative samples does not behave similarly to the baseline noise of profiles generated using optimal levels of DNA, the ATs determined using negative samples have shown to be similar to those developed with near-zero, low template mass samples. The distinction between low-template samples, where the noise is consistent regardless of target mass, and standard samples could be made at approximately 0.063 ng for samples amplified using the Identifiler^TM Plus amplification kit (29 cycle protocol), and injected for 5 and 10 seconds. At amplification target masses greater than 0.063 ng, the average noise peak height increased and began to plateau between 0.5 and 1.0 ng for samples injected for 5 and 10 seconds. To examine the time dependent nature of the baseline noise, the baselines of over 400 profiles were combined onto one axis for each target mass and each injection time. Areas of reproducibly higher noise peak heights were identified as areas of potential non-specific amplified product. When the samples were injected for five seconds, the baseline noise did not appear to be time dependent. However, when the samples were injected for either 10 or 20 seconds, there were three areas that exhibited an increase in noise; these areas were identified at 118 bases in green, 231 bases in yellow, and 106 bases in red. If a probabilistic analysis or AT is to be employed for DNA interpretation, consideration must be given as to how the validation or calibration samples are prepared. Ideally the validation data should include all the variation seen within typical samples. To this end, a study was performed to examine possible sources of variation in the baseline noise within the electropherogram. Specifically, three samples were prepared at seven target masses using four different kit lots, four capillary lots, in four amplification batches or four injection batches. The distribution of the noise peak heights in the blue and green channels for samples with variable capillary lots, amplifications, and injections were similar, but the distribution of the noise heights for samples with variable kit lots was shifted. This shift in the distribution of the samples with variable kit lots was due to the average peak height of the individual kit lots varying by approximately two. The yellow and red channels showed a general agreement between the distributions of the samples run with variable kit lots, amplifications, and injections, but the samples run with various capillary lots had a distribution shifted to the left. When the distribution of the noise height for each capillary was examined, the average peak height variation was less than two RFU between capillary lots. Use of a probabilistic method requires an accurate description of the distribution of the baseline noise. Three distributions were tested: Gaussian, log-normal, and Poisson. The Poisson distribution did not approximate the noise distributions well. The log-normal distribution was a better approximation than the Gaussian resulting in a smaller sum of the residuals squared. It was also shown that the distributions impacted the probability that a peak was noise; though how significant of an impact this difference makes on the final probability of an entire STR profile was not determined and may be of interest for future studies

Nonparametric estimation of multivariate convex-transformed densities

We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$ for $x\in \mathbb {R}^d$ and for a fixed monotone function $h$ and an unknown convex function $g$. The canonical example is $h(y)=e^{-y}$ for $y\in \mathbb {R}$; in this case, the resulting class of densities [\mathcal {P}(e^{-y})={p=\exp(-g):g is convex}] is well known as the class of log-concave densities. Other functions $h$ allow for classes of densities with heavier tails than the log-concave class. We first investigate when the maximum likelihood estimator $\hat{p}$ exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations $h$, including decreasing and increasing functions $h$. The resulting models for increasing transformations $h$ extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to $h(y)=\exp(y)$. We then establish consistency of the maximum likelihood estimator for fairly general functions $h$, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of $p$ and its vector of derivatives at a fixed point $x_0$ under natural smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from convex analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Kiefer--Wolfowitz theorem for convex densities

Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\hat{F}_n$, which is, in fact, the least concave majorant of the empirical distribution function $\mathbb {F}_n$, differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1}\log n)^{2/3}$ almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions $F$ with convex decreasing densities $f$, but with the maximum likelihood estimator $\hat{F}_n$ of $F$ replaced by the least squares estimator $\widetilde{F}_n$: if $X_1,..., X_n$ are sampled from a distribution function $F$ with strictly convex density $f$, then the least squares estimator $\widetilde{F}_n$ of $F$ and the empirical distribution function $\mathbb {F}_n$ differ in the uniform norm by no more than a constant times $(n^{-1}\log n)^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall [J. Approximation Theory 1 (1968) 209--218], Hall and Meyer [J. Approximation Theory 16 (1976) 105--122], building on earlier work by Birkhoff and de Boor [J. Math. Mech. 13 (1964) 827--835]. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor [A Practical Guide to Splines (2001) Springer, New York].Comment: Published at http://dx.doi.org/10.1214/074921707000000256 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org