Blue whale

The blue whale (Balaenoptera musculus) is a marine mammal belonging to the baleen whales (Mysticeti). At 30 metres (98 ft) in length and 180 tonnes (200 short tons) or more in weight, it is the largest extant animal and is the heaviest known to have existed.

Long and slender, the blue whale's body can be various shades of bluish-grey dorsally and somewhat lighter underneath. There are at least three distinct subspecies: B. m. musculus of the North Atlantic and North Pacific, B. m. intermedia of the Southern Ocean and B. m. brevicauda (also known as the pygmy blue whale) found in the Indian Ocean and South Pacific Ocean. B. m. indica, found in the Indian Ocean, may be another subspecies. As with other baleen whales, its diet consists almost exclusively of small crustaceans known as krill.

Blue whales were abundant in nearly all the oceans on Earth until the beginning of the twentieth century. For over a century, they were hunted almost to extinction by whalers until protected by the international community in 1966. A 2002 report estimated there were 5,000 to 12,000 blue whales worldwide, in at least five groups. More recent research into the Pygmy subspecies suggests this may be an overestimate. Before whaling, the largest population was in the Antarctic, numbering approximately 239,000 (range 202,000 to 311,000). There remain only much smaller (around 2,000) concentrations in each of the eastern North Pacific, Antarctic, and Indian Ocean groups. There are two more groups in the North Atlantic, and at least two in the Southern Hemisphere. As of 2014, the Californian blue whale population has rebounded to nearly its pre-hunting population.

Blue

Blue is the colour between violet and green on the optical spectrum of visible light. Human eyes perceive blue when observing light with a wavelength between 450 and 495 nanometres. Blues with a higher frequency and thus a shorter wavelength gradually look more violet, while those with a lower frequency and a longer wavelength gradually appear more green. Pure blue, in the middle, has a wavelength of 470 nanometres. In painting and traditional colour theory, blue is one of the three primary colours of pigments, along with red and yellow, which can be mixed to form a wide gamut of colours. Red and blue mixed together form violet, blue and yellow together form green. Blue is also a primary colour in the RGB colour model, used to create all the colours on the screen of a television or computer monitor.

The modern English word blue comes from Middle English bleu or blewe, from the Old French bleu, a word of Germanic origin, related to the Old High German word blao. The clear sky and the deep sea appear blue because of an optical effect known as Rayleigh scattering. When sunlight passes through the atmosphere, the blue wavelengths are scattered more widely by the oxygen and nitrogen molecules, and more blue comes to our eyes. Rayleigh scattering also explains blue eyes; there is no blue pigment in blue eyes. Distant objects appear more blue because of another optical effect called atmospheric perspective.

Whale

Whale is the common name for a widely distributed and diverse group of fully aquatic placental marine mammals. They are an informal grouping within the infraorder Cetacea, usually excluding dolphins and porpoises. Whales, dolphins and porpoises belong to the order Cetartiodactyla with even-toed ungulates and their closest living relatives are the hippopotamuses, having diverged about 40 million years ago. The two parvorders of whales, baleen whales (Mysticeti) and toothed whales (Odontoceti), are thought to have split apart around 34 million years ago. The whales comprise eight extant families: Balaenopteridae (the rorquals), Balaenidae (right whales), Cetotheriidae (the pygmy right whale), Eschrichtiidae (the gray whale), Monodontidae (belugas and narwhals), Physeteridae (the sperm whale), Kogiidae (the dwarf and pygmy sperm whale), and Ziphiidae (the beaked whales).

Gauss–Markov theorem

In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator. Here "best" means giving the lowest variance of the estimate, as compared to other unbiased, linear estimators. The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator (which also drops linearity) or ridge regression.

Statement

Suppose we have in matrix notation,

expanding to,

where are non-random but unobservable parameters, are non-random and observable (called the "explanatory variables"), are random, and so are random. The random variables are called the "disturbance", "noise" or simply "error" (will be contrasted with "residual" later in the article; see errors and residuals in statistics). Note that to include a constant in the model above, one can choose to introduce the constant as a variable with a newly introduced last column of X being unity i.e., for all .