Husimi Q representation

In statistics, the multinomial test is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values. It is used for categorical data; see Read and Cressie.

We begin with a sample of $N$ items each of which has been observed to fall into one of $k$ categories. We can define $\mathbf{x}=(x_1,x_2,\dots ,x_k)$ as the observed numbers of items in each cell. Hence ${\sum }_{i=1}^k{x}_i=N$.

Next, we define a vector of parameters $H_0:\mathit{\pi }=({\pi }_1,{\pi }_2,\dots ,{\pi }_k)$, where :${\sum }_{i=1}^k{\pi }_i=1$. These are the parameter values under the null hypothesis.

The exact probability of the observed configuration $\mathbf{x}$ under the null hypothesis is given by

$\mathrm{Pr}(\mathbf{x}{\mathbf{)}}_{\mathbf{0}}=N!\prod _{i=1}^k\frac{{\pi }_i^{x_i}}{x_i!}.$

The significance probability for the test is the probability of occurrence of the data set observed, or of a data set less likely than that observed, if the null hypothesis is true. Using an exact test, this is calculated as

$\mathrm{Pr}(\mathbf{s}\mathbf{i}\mathbf{g})=\sum _{y:Pr(\mathbf{y})\le Pr(\mathbf{x}{\mathbf{)}}_{\mathbf{0}}}\mathrm{Pr}(\mathbf{y})$

where the sum ranges over all outcomes as likely as, or less likely than, that observed. In practice this becomes computationally onerous as $k$ and $N$ increase so it is probably only worth using exact tests for small samples. For larger samples, asymptotic approximations are accurate enough and easier to calculate.

One of these approximations is the likelihood ratio. We set up an alternative hypothesis under which each value ${\pi }_i$ is replaced by its maximum likelihood estimate $p_i=x_i/N$. The exact probability of the observed configuration $\mathbf{x}$ under the alternative hypothesis is given by

$\mathrm{Pr}(\mathbf{x}{\mathbf{)}}_{\mathbf{A}}=N!\prod _{i=1}^k\frac{p_i^{x_i}}{x_i!}.$

The natural logarithm of the ratio between these two probabilities multiplied by $-2$ is then the statistic for the likelihood ratio test

$-2\ln (LR)=-2{\sum }_{i=1}^k{x}_i\ln ({\pi }_i/p_i).$

If the null hypothesis is true, then as $N$ increases, the distribution of $-2\ln (LR)$ converges to that of chi-squared with $k-1$ degrees of freedom. However it has long been known (e.g. Lawley 1956) that for finite sample sizes, the moments of $-2\ln (LR)$ are greater than those of chi-squared, thus inflating the probability of type I errors (false positives). The difference between the moments of chi-squared and those of the test statistic are a function of $N^{-1}$. Williams (1976) showed that the first moment can be matched as far as $N^{-2}$ if the test statistic is divided by a factor given by

$q_1=1+\frac{\sum _{i=1}^k{\pi }_i^{-1}-1}{6N(k-1)}.$

In the special case where the null hypothesis is that all the values ${\pi }_i$ are equal to $1/k$ (i.e. it stipulates a uniform distribution), this simplifies to

$q_1=1+\frac{k+1}{6N}.$

Subsequently, Smith et al. (1981) derived a dividing factor which matches the first moment as far as $N^{-3}$. For the case of equal values of ${\pi }_i$, this factor is

$q_2=1+\frac{k+1}{6N}+\frac{k^2}{6N^2}.$

The null hypothesis can also be tested by using Pearson's chi-squared test

${\chi }^2=\sum _{i=1}^k\frac{(x_i-E_i{)}^2}{E_i}$

where $E_i=N{\pi }_i$ is the expected number of cases in category $i$ under the null hypothesis. This statistic also converges to a chi-squared distribution with $k-1$ degrees of freedom when the null hypothesis is true but does so from below, as it were, rather than from above as $-2\ln (LR)$ does, so may be preferable to the uncorrected version of $-2\ln (LR)$ for small samples.

References

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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang