Reliability studies of the skew normal distribution

It has been observed in various practical applications that data do not conform to the normal distribution, which is symmetric with no skewness. The skew normal distribution proposed by Azzalini(1985) is appropriate for the analysis of data which is unimodal but exhibits some skewness. The skew normal distribution includes the normal distribution as a special case where the skewness parameter is zero. In this thesis, we study the structural properties of the skew normal distribution, with an emphasis on the reliability properties of the model. More specifically, we obtain the failure rate, the mean residual life function, and the reliability function of a skew normal random variable. We also compare it with the normal distribution with respect to certain stochastic orderings. Appropriate machinery is developed to obtain the reliability of a component when the strength and stress follow the skew normal distribution. Finally, IQ score data from Roberts (1988) is analyzed to illustrate the procedure.

Optimal Test for Parameter Instability When the Unstable Process and the Error Distribution are Unknown

This paper proposes asymptotically optimal tests for unstable parameter process under the feasible circumstance that the researcher has little information about the unstable parameter process and the error distribution, and suggests conditions under which the knowledge of those processes does not provide asymptotic power gains. I first derive a test under known error distribution, which is asymptotically equivalent to LR tests for correctly identified unstable parameter processes under suitable conditions. The conditions are weak enough to cover a wide range of unstable processes such as various types of structural breaks and time varying parameter processes. The test is then extended to semiparametric models in which the underlying distribution in unknown but treated as unknown infinite dimensional nuisance parameter. The semiparametric test is adaptive in the sense that its asymptotic power function is equivalent to the power envelope under known error distribution.

Effects of informative censoring on the hazard function: Application to the proportional hazards regression model

The standard analyses of survival data involve the assumption that survival and censoring are independent. When censoring and survival are related, the phenomenon is known as informative censoring. This paper examines the effects of an informative censoring assumption on the hazard function and the estimated hazard ratio provided by the Cox model.^ The limiting factor in all analyses of informative censoring is the problem of non-identifiability. Non-identifiability implies that it is impossible to distinguish a situation in which censoring and death are independent from one in which there is dependence. However, it is possible that informative censoring occurs. Examination of the literature indicates how others have approached the problem and covers the relevant theoretical background.^ Three models are examined in detail. The first model uses conditionally independent marginal hazards to obtain the unconditional survival function and hazards. The second model is based on the Gumbel Type A method for combining independent marginal distributions into bivariate distributions using a dependency parameter. Finally, a formulation based on a compartmental model is presented and its results described. For the latter two approaches, the resulting hazard is used in the Cox model in a simulation study.^ The unconditional survival distribution formed from the first model involves dependency, but the crude hazard resulting from this unconditional distribution is identical to the marginal hazard, and inferences based on the hazard are valid. The hazard ratios formed from two distributions following the Gumbel Type A model are biased by a factor dependent on the amount of censoring in the two populations and the strength of the dependency of death and censoring in the two populations. The Cox model estimates this biased hazard ratio. In general, the hazard resulting from the compartmental model is not constant, even if the individual marginal hazards are constant, unless censoring is non-informative. The hazard ratio tends to a specific limit.^ Methods of evaluating situations in which informative censoring is present are described, and the relative utility of the three models examined is discussed. ^

Expression and function of $\beta$-1,4-galactosyltransferase during wild-type and T/{\it t\/} -mutant spermatogenesis

In the mouse, gamete recognition is mediated in part by the binding of sperm surface $\beta$1,4 galactosyltransferase (GalTase) to specific oligosaccharide residues on the zona pellucida ZP3. The expression of GalTase on the sperm surface is regulated by alleles within the distal segment of the T/t complex and results in a haploid-specific increase in GalTase expression on spermatids and sperm from t-bearing males, suggesting that differences in sperm GalTase activity may contribute to t-sperm transmission ratio distortion. In this study, the expression of GalTase RNA during wild-type and T/t-mutant spermatogenesis was characterized and the role of GalTase was analyzed in transmission ratio distortion. It was found that spermatogenic cells predominantly express the long form of the GalTase RNA, which encodes the GalTase protein that is preferentially targeted to the cell surface in somatic cells. In wild-type testes, GalTase RNA accumulates during the maturation of primary spermatocytes, reaches peak levels prior to meiosis, and decreases and meiosis. GalTase RNA accumulates to similar levels during the maturation of +/t and t/t primary spermatocytes, but unlike wild-type, the level of GalTase RNA in t-spermatocytes remains elevated during meiotic division. Consequently, spermatids in t-mutant testes inherit higher levels of GalTase RNA than do wild-type spermatids, which likely accounts for the haploid-specific increase in surface GalTase activity characteristic of spermatids from t-bearing mice.^ The functional significance of the increased GalTase activity during t-sperm transmission ratio distortion was determined by examining the distribution of GalTase RNA and surface GalTase protein in haploid spermatids from +/t males. Results show that +- and t-spermatids have similar levels of both GalTase RNA and protein, indicating that transmission ratio distortion in +/t mice is not likely due to haploid-specific differences in sperm surface GalTase activity.^ The presence of GalTase on the surface of an early spermatogenic cells before it is required on the mature sperm to perform its function during gamete binding suggests a separate function for GalTase in Sertoli-germ cell adhesion. Studies indicate that cell surface GalTase partly mediates the initial adhesion of pachytene spermatocytes, but not haploid spermatids, to Sertoli cells. ^

(Table 1) Two-way travel times and depths of picked internal reflection horizons (IRHs) of the NGRIP ice core

During the summer of 2003, a ground-penetrating radar survey around the North Greenland Icecore Project (NorthGRIP) deep ice-core drilling site (75°06' N, 42°20' W; 2957 m a.s.l.) was carried out using a shielded 250 MHz radar system. The drill site is located on an ice divide, roughly 300 km north-northwest of the summit of the Greenland ice sheet. More than 430 km of profiles were measured, covering a 10 km by 10 km area, with a grid centered on the drilling location, and eight profiles extending beyond this grid. Seven internal horizons within the upper 120 m of the ice sheet were continuously tracked, containing the last 400 years of accumulation history. Based on the age-depth and density-depth distribution of the deep core, the internal layers have been dated and the regional and temporal distribution of accumulation rate in the vicinity of NorthGRIP has been derived. The distribution of accumulation shows a relatively smoothly increasing trend from east to west from 145 kg/m**2/a to 200 kg/m**2/a over a distance of 50 km across the ice divide. The general trend is overlain by small-scale variations on the order of 2.5 kg/m**2/a/km, i.e. around 1.5% of the accumulation mean. The temporal variations of the seven periods defined by the seven tracked isochrones are on the order of +-4% of the mean of the last 400 years, i.e. at NorthGRIP ±7 kg/m**2/a. If the regional accumulation pattern has been stable for the last several thousand years during the Holocene, and ice flow has been comparable to today, advective effects along the particle trajectory upstream of NorthGRIP do not have a significant effect on the interpretation of climatically induced changes in accumulation rates derived from the deep ice core over the last 10 kyr.

(Table 2) Distribution of alkenone unsaturation index UK37 of the eastern North Atlantic

This paper reports the concentrations and within-class distributions of long-chain alkenones and alkyl alkenoates in the surface waters (0-50 m) of the eastern North Atlantic, and correlates their abundance and distribution with those of source organisms and with water temperature and other environmental variables. We collected these samples of >0.8 µm particulate material from the euphotic zone along the JGOFS 20°W longitude transect, from 61°N to 24°N, during seven cruises of the UK-JGOFS Biogeochemical Ocean Flux Study (BOFS) in 1989-1991; the biogeographical range of our 53 samples extends from the cold (<10°C), nutrient-rich and highly productive subarctic waters of the Iceland Basin to the warm (>25°C) oligotrophic subtropical waters off Africa. Surface water concentrations of total alkenone and alkenoates ranged from <50 ng/l in oligotrophic waters below 40°N to 2000-4500 ng/l in high latitude E. huxleyi blooms, and were well correlated with E. huxleyi cell densities, supporting the assumption that E. huxleyi is the predominant source of these compounds in the present day North Atlantic. The within-class distribution of the C37 and C38 alkenones and C36 alkenoates varied strongly as a function of temperature, and was largely unaffected by nutrient concentration, bloom status and other surface water properties. The biosynthetic response of the source organisms to growth temperature differed between the cold (<16°C) waters above 47°N and the warmer waters to the south. In cold (<16°C) waters above 47°N, the relative amounts of alkenoates and C38 alkenones synthesized was a strong function of growth temperature, while the unsaturation ratio of the alkenones (C37 and C38) was uncorrelated with temperature. Conversely, in warm (>16°C) waters below 47°N, the relative proportions of alkenoates and alkenones synthesized remained constant with increasing temperature while the unsaturation ratios of the C37 and C38 methyl alkenones (Uk37 and Uk38Me, respectively) increased linearly. The fitted regressions of Uk37 and Uk38Me versus temperature for waters >16°C were both highly significant (r**2 > 0.96) and had identical slopes (0.057) that were 50% higher than the slope (0.034) of the temperature calibration of Uk37 reported by Prahl and Wakeham (1987; doi:10.1038/330367a0) over the same temperature range. These observations suggest either a physiological adjustment in biochemical response to growth temperature above a 16-17°C threshold and/or variation between different E. huxleyi strains and/or related species inhabiting the cold and warm water regions of the eastern North Atlantic. Using our North Atlantic data set, we have produced multivariate temperature calibrations incorporating all major features of the alkenone and alkenoate data set. Predicted temperatures using multivariate calibrations are largely unbiased, with a standard error of approximately ±1°C over the entire data range. In contrast, simpler calibration models cannot adequately incorporate regional diversity and nonlinear trends with temperature. Our results indicate that calibrations based upon single variables, such as Uk37, can be strongly biased by unknown systematic errors arising from natural variability in the biosynthetic response of the source organisms to growth temperature. Multivariate temperature calibration can be expected to give more precise estimates of Integrated Production Temperatures (IPT) in the sedimentary record over a wider range of paleoenvironmental conditions, when derived using a calibration data set incorporating a similar range of natural variability in biosynthetic response.

Distribution of macrozooplankton at time series station MIGZOO-4

A large population of the colonial pelagic tunicate Pyrosoma atlanticum occurred in April 1991 in offshore waters of the Ligurian Sea (Northwestern Mediterranean). The high numbers of colonies caught allowed their vertical distribution and diel migration in the 0-965 m water column to be described as a function of their size. Daytime depths and amplitudes of the migration were correlated with colony size. The amplitude of the migration ranged from 90 m for 3-mm-length colonies to 760 m for 51-mm-length colonies, with a mean amplitude of 410 m for the whole population, all sizes pooled. The results of horizontal hauls at a given depth around sunrise and sunset showed a marked diurnal symmetry of the migratory cycle relative to noon, and that migration of the population was not cohesive. For example, the larger the colonies, the later after sunset they reached the upper layers during their upward migration.

Maceral distribution, stratigraphic position and vitrinite reflectance at DSDP Hole 81-555

Twenty-four sediment samples from late Paleocene to early Eocene were studied for maceral content, vitrinite reflectance, and spectral fluorescence in order to determine some parameters of the origin and diagenetic history of their organic fraction. The sediments had been obtained at Site 555 of DSDP Leg 81 in the northeastern North Atlantic. The bulk of the microscopically visible fraction is made up of humic materials; inertinites follow as a distant second; and liptinites are exceedingly rare. No unequivocal evidence of marine organic matter was found. Humic materials are highly decomposed, showing signs of aerobic (frequency of sclerotinites) as well as anaerobic (abundance of and intimate association with framboidal pyrite) microbial degradation. Vitrinite reflectance values vary between 0.26 and 0.35 Ro and show a slight increase with depth. These values, indicative of a low-rank lignite stage of coalification, contrast somewhat with the sporinite fluorescence spectra, which show the configuration typical for the peat stage. In either case, the evidence for such a low stage of coalification is surprising in view of the depth and age of the sediments.