This thesis consists of a summary and five papers, dealing with self-normalized sums of independent, identically distributed random variables, and three-decision procedures for directional conclusions. In Paper I, we investigate a general set-up for Student's t-statistic. Finiteness of absolute moments is related to the corresponding degree of freedom, and relevant properties of the underlying distribution, assuming independent, identically distributed random variables. In Paper II, we investigate a certain kind of self-normalized sums. We show that the corresponding quadratic moments are greater than or equal to one, with equality if and only if the underlying distribution is symmetrically distributed around the origin. In Paper III, we study linear combinations of independent Rademacher random variables. A family of universal bounds on the corresponding tail probabilities is derived through the technique known as exponential tilting. Connections to self-normalized sums of symmetrically distributed random variables are given. In Paper IV, we consider a general formulation of three-decision procedures for directional conclusions. We introduce three kinds of optimality characterizations, and formulate corresponding sufficiency conditions. These conditions are applied to exponential families of distributions. In Paper V, we investigate the Benjamini-Hochberg procedure as a means of confirming a selection of statistical decisions on the basis of a corresponding set of generalized p-values. Assuming independence, we show that control is imposed on the expected average loss among confirmed decisions. Connections to directional conclusions are given.

In radiation therapy treatment planning, recent works have used machine learning based on historically delivered plans to automate the process of producing clinically acceptable plans. Compared to traditional approaches such as repeated weighted-sum optimization or multicriteria optimization (MCO), automated planning methods have, in general, the benefits of low computational times and minimal user interaction, but on the other hand lack the flexibility associated with general-purpose frameworks such as MCO. Machine learning approaches can be especially sensitive to deviations in their dose prediction due to certain properties of the optimization functions usually used for dose mimicking and, moreover, suffer from the fact that there exists no general causality between prediction accuracy and optimized plan quality.In this thesis, we present a means of unifying ideas from machine learning planning methods with the well-established MCO framework. More precisely, given prior knowledge in the form of either a previously optimized plan or a set of historically delivered clinical plans, we are able to automatically generate Pareto optimal plans spanning a dose region corresponding to plans which are achievable as well as clinically acceptable. For the former case, this is achieved by introducing dose--volume constraints; for the latter case, this is achieved by fitting a weighted-data Gaussian mixture model on pre-defined dose statistics using the expectation--maximization algorithm, modifying it with exponential tilting and using specially developed optimization functions to take into account prediction uncertainties.Numerical results for conceptual demonstration are obtained for a prostate cancer case with treatment delivered by a volumetric-modulated arc therapy technique, where it is shown that the methods developed in the thesis are successful in automatically generating Pareto optimal plans of satisfactory quality and diversity, while excluding clinically irrelevant dose regions. For the case of using historical plans as prior knowledge, the computational times are significantly shorter than those typical of conventional MCO. / Inom strålterapiplanering har den senaste forskningen använt maskininlärning baserat på historiskt levererade planer för att automatisera den process i vilken kliniskt acceptabla planer produceras. Jämfört med traditionella angreppssätt, såsom upprepad optimering av en viktad målfunktion eller flermålsoptimering (MCO), har automatiska planeringsmetoder generellt sett fördelarna av lägre beräkningstider och minimal användarinteraktion, men saknar däremot flexibiliteten hos allmänna ramverk som exempelvis MCO. Maskininlärningsmetoder kan vara speciellt känsliga för avvikelser i dosprediktionssteget på grund av särskilda egenskaper hos de optimeringsfunktioner som vanligtvis används för att återskapa dosfördelningar, och lider dessutom av problemet att det inte finns något allmängiltigt orsakssamband mellan prediktionsnoggrannhet och kvalitet hos optimerad plan. I detta arbete presenterar vi ett sätt att förena idéer från maskininlärningsbaserade planeringsmetoder med det väletablerade MCO-ramverket. Mer precist kan vi, givet förkunskaper i form av antingen en tidigare optimerad plan eller en uppsättning av historiskt levererade kliniska planer, automatiskt generera Paretooptimala planer som täcker en dosregion motsvarande uppnåeliga såväl som kliniskt acceptabla planer. I det förra fallet görs detta genom att introducera dos--volym-bivillkor; i det senare fallet görs detta genom att anpassa en gaussisk blandningsmodell med viktade data med förväntning--maximering-algoritmen, modifiera den med exponentiell lutning och sedan använda speciellt utvecklade optimeringsfunktioner för att ta hänsyn till prediktionsosäkerheter.Numeriska resultat för konceptuell demonstration erhålls för ett fall av prostatacancer varvid behandlingen levererades med volymetriskt modulerad bågterapi, där det visas att metoderna utvecklade i detta arbete är framgångsrika i att automatiskt generera Paretooptimala planer med tillfredsställande kvalitet och variation medan kliniskt irrelevanta dosregioner utesluts. I fallet då historiska planer används som förkunskap är beräkningstiderna markant kortare än för konventionell MCO.