Gopalan Nair

On tests of spatial pattern based on simulation envelopes

In the analysis of spatial point patterns, an important role is played by statistical tests based on simulation envelopes, such as the envelope of simulations of Ripleys K function. Recent ecological literature has correctly pointed out a common error in the interpretation of simulation envelopes. However, this has led to a widespread belief that the tests themselves are invalid. On the contrary, envelope-based statistical tests are correct statistical procedures, under appropriate conditions. In this paper, we explain the principles of Monte Carlo tests and their correct interpretation, canvas the benefits of graphical procedures, measure the statistical performance of several popular tests, and make practical recommendations. There are several caveats including the under-recognized problem that Monte Carlo tests of goodness of fit are probably conservative if the model parameters have to be estimated from data. Finally, we discuss whether graphs of simulation envelopes can be used to infer the scale of...

The stochastic h-index

A variant of the h-index, named the stochastic h-index, is proposed. This new index is obtained by adding to the h-index the probability, under a specific stochastic model, that the h-index will increase by one or more within a given time interval. The stochastic h-index thus extends the h-index to the real line and has a direct interpretation as the distance to the next higher index value. We show how the stochastic h-index can be evaluated and compare it with other variants of the h-index which purportedly indicate the distance to a higher h-index.

On two-stage Monte Carlo tests of composite hypotheses

A major weakness of the classical Monte Carlo test is that it is biased when the null hypothesis is composite. This problem persists even when the number of simulations tends to infinity. A standard remedy is to perform a double bootstrap test involving two stages of Monte Carlo simulation: under suitable conditions, this test is asymptotically exact for any fixed significance level. However, the two-stage test is shown to perform poorly in some common applications: for a given number of simulations, the test with the smallest achievable significance level can be strongly biased. A ‘balanced’ version of the two-stage test is proposed, which is exact, for all achievable significance levels, when the null hypothesis is simple, and which performs well for composite null hypotheses.

Poisson-saddlepoint approximation for Gibbs point processes with infinite-order interaction: in memory of Peter Hall

We develop a computational approximation to the intensity of a Gibbs spatial point process having interactions of any order. Limit theorems from stochastic geometry, and small-sample probabilities estimated once and for all by an extensive simulation study, are combined with scaling properties to form an approximation to the moment generating function of the sufficient statistic under a Poisson process. The approximate intensity is obtained as the solution of a self-consistency equation.

Efficient Estimation in Smooth Threshold Autoregressive (1) Models

Verifiable conditions are given for the existence of efficient estimation in Smooth Threshold Autoregressive models of order 1. The paper establishes local asymptotic normality in the semi-parametric setting which is then used to discuss adaptive and efficient estimates of the models. It is found that the adaptation is satisfied if the error densities are symmetric. Simulation results are presented to compare the conditional least squares estimate with the adaptive and efficient estimates for the models.