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UBC Theses and Dissertations

Statistical models for agroclimate risk analysis Hosseini, Mohamadreza


In order to model the binary process of precipitation and the dichotomized temperature process, we use the conditional probability of the present given the past. We find necessary and sufficient conditions for a collection of functions to correspond to the conditional probabilities of a discrete-time categorical stochastic process X₁,X₂,···. Moreover we find parametric representations for such processes and in particular rth-order Markov chains. To dichotomize the temperature process, quantiles are often used in the literature. We propose using a two-state definition of the quantiles by considering the "left quantile" and "right quantile" functions instead of the traditional definition. This has various advantages such as a symmetry relation between the quantiles of random variables X and -X. We show that the left (right) sample quantile tends to the left (right) distribution quantile at p ∈[0,1], if and only if the left and right distribution quantiles are identical at p and diverge almost surely otherwise. In order to measure the loss of estimating (or approximating) a quantile, we introduce a loss function that is invariant under strictly monotonic transformations and call it the "probability loss function". Using this loss function, we introduce measures of distance among random variables that are invariant under continuous strictly monotonic transformations. We use this distance measures to show optimal overall fits to a random variable are not necessarily optimal in the tails. This loss function is also used to find equivariant estimators of the parameters of distribution functions. We develop an algorithm to approximate quantiles of large datasets which works by partitioning the data or use existing partitions (possibly of non-equal size). We show the deterministic precision of this algorithm and how it can be adjusted to get customized precisions. Then we develop a framework to optimally summarize very large datasets using quantiles and combining such summaries in order to infer about the original dataset. Finally we show how these higher order Markov models can be used to construct confidence intervals for the probability of frost-free periods.

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